1,1,189,95,2.3066323,"\int \sec ^2(c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\frac{3 \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\sin (c+d x) \sec ^{\frac{10}{3}}(c+d x) ((10 A+7 C) \cos (2 (c+d x))+5 (2 A+3 C))-2 i \sqrt[3]{2} (10 A+7 C) \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)\right)}{40 d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 C)}","\frac{3 (10 A+7 C) \sin (c+d x) (b \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{40 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{7/3}}{10 b^2 d}",1,"(3*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2)*((-2*I)*2^(1/3)*(10*A + 7*C)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] + (5*(2*A + 3*C) + (10*A + 7*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^(10/3)*Sin[c + d*x]))/(40*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/3))","C",1
2,1,185,92,1.7140217,"\int \sec (c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\frac{3 i e^{i (c+d x)} \cos ^3(c+d x) (b \sec (c+d x))^{4/3} \left((7 A+4 C) \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)-14 A \left(1+e^{2 i (c+d x)}\right)^2-4 C \left(5 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+1\right)\right) \left(A+C \sec ^2(c+d x)\right)}{7 b d \left(1+e^{2 i (c+d x)}\right)^2 (A \cos (2 (c+d x))+A+2 C)}","\frac{3 (7 A+4 C) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 b d}",1,"(((3*I)/7)*E^(I*(c + d*x))*Cos[c + d*x]^3*(-14*A*(1 + E^((2*I)*(c + d*x)))^2 - 4*C*(1 + 5*E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x))) + (7*A + 4*C)*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2))/(b*d*(1 + E^((2*I)*(c + d*x)))^2*(A + 2*C + A*Cos[2*(c + d*x)]))","C",1
3,1,162,88,1.2745651,"\int \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\frac{3 \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(C \sin (c+d x) \sec ^{\frac{4}{3}}(c+d x)-i \sqrt[3]{2} (4 A+C) \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)\right)}{2 d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 C)}","\frac{3 C \tan (c+d x) \sqrt[3]{b \sec (c+d x)}}{4 d}-\frac{3 b (4 A+C) \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}",1,"(3*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2)*((-I)*2^(1/3)*(4*A + C)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] + C*Sec[c + d*x]^(4/3)*Sin[c + d*x]))/(2*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/3))","C",1
4,1,93,89,0.1665944,"\int \cos (c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{4/3} \left(2 A \cos ^2(c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sec ^2(c+d x)\right)-C \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)\right)}{4 b d}","\frac{3 b C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}-\frac{3 b^2 (A-2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}",1,"(-3*Cot[c + d*x]*(2*A*Cos[c + d*x]^2*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[c + d*x]^2] - C*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(4/3)*Sqrt[-Tan[c + d*x]^2])/(4*b*d)","A",1
5,1,89,93,0.1465868,"\int \cos ^2(c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \sec (c+d x)} \left(A \cos ^2(c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\sec ^2(c+d x)\right)-5 C \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)\right)}{5 d}","\frac{3 A b^2 \tan (c+d x)}{5 d (b \sec (c+d x))^{5/3}}-\frac{3 b (2 A+5 C) \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}",1,"(-3*Cot[c + d*x]*(A*Cos[c + d*x]^2*Hypergeometric2F1[-5/6, 1/2, 1/6, Sec[c + d*x]^2] - 5*C*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2])/(5*d)","A",1
6,1,236,95,1.8879082,"\int \sec ^2(c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\frac{12 i e^{i (c+d x)} \cos ^3(c+d x) (b \sec (c+d x))^{4/3} \left((13 A+10 C) \left(1+e^{2 i (c+d x)}\right)^{13/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)-13 A \left(5 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+1\right) \left(1+e^{2 i (c+d x)}\right)^2-2 C \left(21 e^{2 i (c+d x)}+79 e^{4 i (c+d x)}+45 e^{6 i (c+d x)}+10 e^{8 i (c+d x)}+5\right)\right) \left(A+C \sec ^2(c+d x)\right)}{91 d \left(1+e^{2 i (c+d x)}\right)^4 (A \cos (2 (c+d x))+A+2 C)}","\frac{3 (13 A+10 C) \sin (c+d x) (b \sec (c+d x))^{7/3} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{91 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{10/3}}{13 b^2 d}",1,"(((12*I)/91)*E^(I*(c + d*x))*Cos[c + d*x]^3*(-13*A*(1 + E^((2*I)*(c + d*x)))^2*(1 + 5*E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x))) - 2*C*(5 + 21*E^((2*I)*(c + d*x)) + 79*E^((4*I)*(c + d*x)) + 45*E^((6*I)*(c + d*x)) + 10*E^((8*I)*(c + d*x))) + (13*A + 10*C)*(1 + E^((2*I)*(c + d*x)))^(13/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2))/(d*(1 + E^((2*I)*(c + d*x)))^4*(A + 2*C + A*Cos[2*(c + d*x)]))","C",1
7,1,192,92,2.9206448,"\int \sec (c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\frac{3 (b \sec (c+d x))^{7/3} \left(A+C \sec ^2(c+d x)\right) \left(\sin (c+d x) \sec ^{\frac{10}{3}}(c+d x) ((10 A+7 C) \cos (2 (c+d x))+5 (2 A+3 C))-2 i \sqrt[3]{2} (10 A+7 C) \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)\right)}{40 b d \sec ^{\frac{13}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 C)}","\frac{3 (10 A+7 C) \sin (c+d x) (b \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{40 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{7/3}}{10 b d}",1,"(3*(b*Sec[c + d*x])^(7/3)*(A + C*Sec[c + d*x]^2)*((-2*I)*2^(1/3)*(10*A + 7*C)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] + (5*(2*A + 3*C) + (10*A + 7*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^(10/3)*Sin[c + d*x]))/(40*b*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(13/3))","C",1
8,1,182,90,1.3240858,"\int (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\frac{3 i e^{i (c+d x)} \cos ^3(c+d x) (b \sec (c+d x))^{4/3} \left((7 A+4 C) \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)-14 A \left(1+e^{2 i (c+d x)}\right)^2-4 C \left(5 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+1\right)\right) \left(A+C \sec ^2(c+d x)\right)}{7 d \left(1+e^{2 i (c+d x)}\right)^2 (A \cos (2 (c+d x))+A+2 C)}","\frac{3 b (7 A+4 C) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 d}",1,"(((3*I)/7)*E^(I*(c + d*x))*Cos[c + d*x]^3*(-14*A*(1 + E^((2*I)*(c + d*x)))^2 - 4*C*(1 + 5*E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x))) + (7*A + 4*C)*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2))/(d*(1 + E^((2*I)*(c + d*x)))^2*(A + 2*C + A*Cos[2*(c + d*x)]))","C",1
9,1,163,91,1.4973115,"\int \cos (c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\frac{3 b \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(C \sin (c+d x) \sec ^{\frac{4}{3}}(c+d x)-i \sqrt[3]{2} (4 A+C) \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)\right)}{2 d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 C)}","\frac{3 b C \tan (c+d x) \sqrt[3]{b \sec (c+d x)}}{4 d}-\frac{3 b^2 (4 A+C) \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}",1,"(3*b*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2)*((-I)*2^(1/3)*(4*A + C)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] + C*Sec[c + d*x]^(4/3)*Sin[c + d*x]))/(2*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/3))","C",1
10,1,90,91,0.1720023,"\int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{4/3} \left(2 A \cos ^2(c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sec ^2(c+d x)\right)-C \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)\right)}{4 d}","\frac{3 b^2 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}-\frac{3 b^3 (A-2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}",1,"(-3*Cot[c + d*x]*(2*A*Cos[c + d*x]^2*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[c + d*x]^2] - C*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(4/3)*Sqrt[-Tan[c + d*x]^2])/(4*d)","A",1
11,1,207,95,3.1055263,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","\frac{3 i e^{-i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{8/3} \left((8 A+5 C) \left(1+e^{2 i (c+d x)}\right)^{8/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-5 \left(8 A \left(1+e^{2 i (c+d x)}\right)^2+C \left(14 e^{2 i (c+d x)}+5 e^{4 i (c+d x)}+1\right)\right)\right) \left(A+C \sec ^2(c+d x)\right)}{20 \sqrt[3]{2} d \sec ^{\frac{5}{3}}(c+d x) \sqrt[3]{b \sec (c+d x)} (A \cos (2 (c+d x))+A+2 C)}","\frac{3 (8 A+5 C) \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{16 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{5/3}}{8 b^2 d}",1,"(((3*I)/20)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(8/3)*(-5*(8*A*(1 + E^((2*I)*(c + d*x)))^2 + C*(1 + 14*E^((2*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)))) + (8*A + 5*C)*(1 + E^((2*I)*(c + d*x)))^(8/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))])*(A + C*Sec[c + d*x]^2))/(2^(1/3)*d*E^(I*(c + d*x))*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/3)*(b*Sec[c + d*x])^(1/3))","C",1
12,1,168,92,1.3324519,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","\frac{3 (b \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \left(2 C \sin (c+d x) \sec ^{\frac{5}{3}}(c+d x)-i 2^{2/3} (5 A+2 C) \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)\right)}{5 b d \sec ^{\frac{8}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 C)}","\frac{3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 b d}-\frac{3 (5 A+2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}",1,"(3*(b*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2)*((-I)*2^(2/3)*(5*A + 2*C)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))] + 2*C*Sec[c + d*x]^(5/3)*Sin[c + d*x]))/(5*b*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(8/3))","C",1
13,1,127,90,0.7782595,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(1/3),x]","-\frac{3 i \left((2 A-C) e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-5 \left(A e^{2 i (c+d x)}+A-C e^{2 i (c+d x)}\right)\right)}{5 d \left(1+e^{2 i (c+d x)}\right) \sqrt[3]{b \sec (c+d x)}}","\frac{3 C \tan (c+d x)}{2 d \sqrt[3]{b \sec (c+d x)}}-\frac{3 b (2 A-C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}",1,"(((-3*I)/5)*(-5*(A + A*E^((2*I)*(c + d*x)) - C*E^((2*I)*(c + d*x))) + (2*A - C)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))]))/(d*(1 + E^((2*I)*(c + d*x)))*(b*Sec[c + d*x])^(1/3))","C",1
14,1,121,88,0.6102101,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","-\frac{3 i e^{-i (c+d x)} \left(2 (A+4 C) e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+A \left(-1+e^{4 i (c+d x)}\right)\right)}{8 d \left(1+e^{2 i (c+d x)}\right) \sqrt[3]{b \sec (c+d x)}}","\frac{3 A b \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}",1,"(((-3*I)/8)*(A*(-1 + E^((4*I)*(c + d*x))) + 2*(A + 4*C)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*(b*Sec[c + d*x])^(1/3))","C",1
15,1,89,93,0.1153062,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(A \cos ^2(c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\sec ^2(c+d x)\right)+7 C \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)\right)}{7 d \sqrt[3]{b \sec (c+d x)}}","\frac{3 A b^2 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\frac{3 b (4 A+7 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{28 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}",1,"(-3*Cot[c + d*x]*(A*Cos[c + d*x]^2*Hypergeometric2F1[-7/6, 1/2, -1/6, Sec[c + d*x]^2] + 7*C*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(7*d*(b*Sec[c + d*x])^(1/3))","A",1
16,1,165,95,1.1946219,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 \left(A+C \sec ^2(c+d x)\right) \left(2 C \sin (c+d x) \sec ^{\frac{5}{3}}(c+d x)-i 2^{2/3} (5 A+2 C) \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)\right)}{5 d \sec ^{\frac{2}{3}}(c+d x) (b \sec (c+d x))^{4/3} (A \cos (2 (c+d x))+A+2 C)}","\frac{3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 b^2 d}-\frac{3 (5 A+2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}",1,"(3*(A + C*Sec[c + d*x]^2)*((-I)*2^(2/3)*(5*A + 2*C)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))] + 2*C*Sec[c + d*x]^(5/3)*Sin[c + d*x]))/(5*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(2/3)*(b*Sec[c + d*x])^(4/3))","C",1
17,1,130,92,0.6543214,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","-\frac{3 i \left((2 A-C) e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-5 \left(A e^{2 i (c+d x)}+A-C e^{2 i (c+d x)}\right)\right)}{5 b d \left(1+e^{2 i (c+d x)}\right) \sqrt[3]{b \sec (c+d x)}}","\frac{3 C \tan (c+d x)}{2 b d \sqrt[3]{b \sec (c+d x)}}-\frac{3 (2 A-C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}",1,"(((-3*I)/5)*(-5*(A + A*E^((2*I)*(c + d*x)) - C*E^((2*I)*(c + d*x))) + (2*A - C)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))]))/(b*d*(1 + E^((2*I)*(c + d*x)))*(b*Sec[c + d*x])^(1/3))","C",1
18,1,124,90,0.5339985,"\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(4/3),x]","-\frac{3 i e^{-i (c+d x)} \left(2 (A+4 C) e^{2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+A \left(-1+e^{4 i (c+d x)}\right)\right)}{8 b d \left(1+e^{2 i (c+d x)}\right) \sqrt[3]{b \sec (c+d x)}}","\frac{3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac{3 (A+4 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}",1,"(((-3*I)/8)*(A*(-1 + E^((4*I)*(c + d*x))) + 2*(A + 4*C)*E^((2*I)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))]))/(b*d*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*(b*Sec[c + d*x])^(1/3))","C",1
19,1,92,90,0.1292513,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(A \cos ^2(c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\sec ^2(c+d x)\right)+7 C \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)\right)}{7 b d \sqrt[3]{b \sec (c+d x)}}","\frac{3 A b \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\frac{3 (4 A+7 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{28 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}",1,"(-3*Cot[c + d*x]*(A*Cos[c + d*x]^2*Hypergeometric2F1[-7/6, 1/2, -1/6, Sec[c + d*x]^2] + 7*C*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(7*b*d*(b*Sec[c + d*x])^(1/3))","A",1
20,1,96,93,0.8451855,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\frac{\tan (c+d x) \left((7 A+10 C) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{3}{2};\sin ^2(c+d x)\right)+3 \sqrt[6]{\cos ^2(c+d x)} (2 A \cos (2 (c+d x))+9 A+10 C)\right)}{40 d \sqrt[6]{\cos ^2(c+d x)} (b \sec (c+d x))^{4/3}}","\frac{3 A b^2 \tan (c+d x)}{10 d (b \sec (c+d x))^{10/3}}-\frac{3 b (7 A+10 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{70 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{7/3}}",1,"((3*(Cos[c + d*x]^2)^(1/6)*(9*A + 10*C + 2*A*Cos[2*(c + d*x)]) + (7*A + 10*C)*Hypergeometric2F1[1/2, 5/6, 3/2, Sin[c + d*x]^2])*Tan[c + d*x])/(40*d*(Cos[c + d*x]^2)^(1/6)*(b*Sec[c + d*x])^(4/3))","A",1
21,1,303,146,3.770527,"\int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","-\frac{3 i 2^{m+\frac{7}{3}} e^{-\frac{1}{3} i (3 m+7) (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{7}{3}} (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 (A+2 C) e^{\frac{1}{3} i (3 m+10) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (-3 m-4);\frac{m}{2}+\frac{8}{3};-e^{2 i (c+d x)}\right)}{3 m+10}+\frac{A e^{\frac{1}{3} i (3 m+16) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (2-3 m);\frac{1}{6} (3 m+22);-e^{2 i (c+d x)}\right)}{3 m+16}+\frac{A e^{\frac{1}{3} i (3 m+4) (c+d x)} \, _2F_1\left(1,-\frac{m}{2}-\frac{5}{3};\frac{m}{2}+\frac{5}{3};-e^{2 i (c+d x)}\right)}{3 m+4}\right)}{d \sec ^{\frac{10}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 C)}","\frac{3 b (A (3 m+7)+C (3 m+4)) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (-3 m-1);\frac{1}{6} (5-3 m);\cos ^2(c+d x)\right)}{d (3 m+1) (3 m+7) \sqrt{\sin ^2(c+d x)}}+\frac{3 b C \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^{m+2}(c+d x)}{d (3 m+7)}",1,"((-3*I)*2^(7/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(7/3 + m)*((2*(A + 2*C)*E^((I/3)*(10 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (-4 - 3*m)/6, 8/3 + m/2, -E^((2*I)*(c + d*x))])/(10 + 3*m) + (A*E^((I/3)*(16 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (2 - 3*m)/6, (22 + 3*m)/6, -E^((2*I)*(c + d*x))])/(16 + 3*m) + (A*E^((I/3)*(4 + 3*m)*(c + d*x))*Hypergeometric2F1[1, -5/3 - m/2, 5/3 + m/2, -E^((2*I)*(c + d*x))])/(4 + 3*m))*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2))/(d*E^((I/3)*(7 + 3*m)*(c + d*x))*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(10/3))","C",0
22,1,303,146,3.2314612,"\int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2),x]","-\frac{3 i 2^{m+\frac{5}{3}} e^{-\frac{1}{3} i (3 m+5) (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{5}{3}} (b \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 (A+2 C) e^{\frac{1}{3} i (3 m+8) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (-3 m-2);\frac{m}{2}+\frac{7}{3};-e^{2 i (c+d x)}\right)}{3 m+8}+\frac{A e^{\frac{1}{3} i (3 m+2) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (-3 m-8);\frac{1}{6} (3 m+8);-e^{2 i (c+d x)}\right)}{3 m+2}+\frac{A e^{\frac{1}{3} i (3 m+14) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (4-3 m);\frac{1}{6} (3 m+20);-e^{2 i (c+d x)}\right)}{3 m+14}\right)}{d \sec ^{\frac{8}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 C)}","\frac{3 C \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m+1}(c+d x)}{d (3 m+5)}-\frac{3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (1-3 m);\frac{1}{6} (7-3 m);\cos ^2(c+d x)\right)}{d (1-3 m) (3 m+5) \sqrt{\sin ^2(c+d x)}}",1,"((-3*I)*2^(5/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(5/3 + m)*((A*E^((I/3)*(2 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (-8 - 3*m)/6, (8 + 3*m)/6, -E^((2*I)*(c + d*x))])/(2 + 3*m) + (2*(A + 2*C)*E^((I/3)*(8 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (-2 - 3*m)/6, 7/3 + m/2, -E^((2*I)*(c + d*x))])/(8 + 3*m) + (A*E^((I/3)*(14 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (4 - 3*m)/6, (20 + 3*m)/6, -E^((2*I)*(c + d*x))])/(14 + 3*m))*(b*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2))/(d*E^((I/3)*(5 + 3*m)*(c + d*x))*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(8/3))","C",0
23,1,303,144,2.8455908,"\int \sec ^m(c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","-\frac{3 i 2^{m+\frac{4}{3}} e^{-\frac{1}{3} i (3 m+4) (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{4}{3}} \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 (A+2 C) e^{\frac{1}{3} i (3 m+7) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (-3 m-1);\frac{1}{6} (3 m+13);-e^{2 i (c+d x)}\right)}{3 m+7}+\frac{A e^{\frac{1}{3} i (3 m+1) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (-3 m-7);\frac{1}{6} (3 m+7);-e^{2 i (c+d x)}\right)}{3 m+1}+\frac{A e^{\frac{1}{3} i (3 m+13) (c+d x)} \, _2F_1\left(1,\frac{1}{6} (5-3 m);\frac{1}{6} (3 m+19);-e^{2 i (c+d x)}\right)}{3 m+13}\right)}{d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 C)}","\frac{3 C \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^{m+1}(c+d x)}{d (3 m+4)}-\frac{3 (A (3 m+4)+3 C m+C) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (2-3 m);\frac{1}{6} (8-3 m);\cos ^2(c+d x)\right)}{d (2-3 m) (3 m+4) \sqrt{\sin ^2(c+d x)}}",1,"((-3*I)*2^(4/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(4/3 + m)*((A*E^((I/3)*(1 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (-7 - 3*m)/6, (7 + 3*m)/6, -E^((2*I)*(c + d*x))])/(1 + 3*m) + (2*(A + 2*C)*E^((I/3)*(7 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (-1 - 3*m)/6, (13 + 3*m)/6, -E^((2*I)*(c + d*x))])/(7 + 3*m) + (A*E^((I/3)*(13 + 3*m)*(c + d*x))*Hypergeometric2F1[1, (5 - 3*m)/6, (19 + 3*m)/6, -E^((2*I)*(c + d*x))])/(13 + 3*m))*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2))/(d*E^((I/3)*(4 + 3*m)*(c + d*x))*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/3))","C",0
24,1,311,147,8.5440569,"\int \frac{\sec ^m(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","-\frac{3 i 2^{m+\frac{2}{3}} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m-\frac{1}{3}} \left(1+e^{2 i (c+d x)}\right)^{m-\frac{1}{3}} \left(A+C \sec ^2(c+d x)\right) \left((3 m-1) e^{2 i (c+d x)} \left(2 (3 m+11) (A+2 C) \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);-e^{2 i (c+d x)}\right)+A (3 m+5) e^{2 i (c+d x)} \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m+11);\frac{1}{6} (3 m+17);-e^{2 i (c+d x)}\right)\right)+A \left(9 m^2+48 m+55\right) \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);-e^{2 i (c+d x)}\right)\right)}{d (3 m-1) (3 m+5) (3 m+11) \sec ^{\frac{5}{3}}(c+d x) \sqrt[3]{b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 C)}","\frac{3 (C (1-3 m)-A (3 m+2)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (4-3 m);\frac{1}{6} (10-3 m);\cos ^2(c+d x)\right)}{d (4-3 m) (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+2) \sqrt[3]{b \sec (c+d x)}}",1,"((-3*I)*2^(2/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-1/3 + m)*(1 + E^((2*I)*(c + d*x)))^(-1/3 + m)*(A*(55 + 48*m + 9*m^2)*Hypergeometric2F1[5/3 + m, (-1 + 3*m)/6, (5 + 3*m)/6, -E^((2*I)*(c + d*x))] + E^((2*I)*(c + d*x))*(-1 + 3*m)*(2*(A + 2*C)*(11 + 3*m)*Hypergeometric2F1[5/3 + m, (5 + 3*m)/6, (11 + 3*m)/6, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(5 + 3*m)*Hypergeometric2F1[5/3 + m, (11 + 3*m)/6, (17 + 3*m)/6, -E^((2*I)*(c + d*x))]))*(A + C*Sec[c + d*x]^2))/(d*(-1 + 3*m)*(5 + 3*m)*(11 + 3*m)*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/3)*(b*Sec[c + d*x])^(1/3))","C",1
25,1,311,145,8.5913427,"\int \frac{\sec ^m(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","-\frac{3 i 2^{m+\frac{1}{3}} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m-\frac{2}{3}} \left(1+e^{2 i (c+d x)}\right)^{m-\frac{2}{3}} \left(A+C \sec ^2(c+d x)\right) \left((3 m+10) \left(2 (3 m-2) (A+2 C) e^{2 i (c+d x)} \, _2F_1\left(m+\frac{4}{3},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};-e^{2 i (c+d x)}\right)+A (3 m+4) \, _2F_1\left(m+\frac{4}{3},\frac{1}{6} (3 m-2);\frac{1}{6} (3 m+4);-e^{2 i (c+d x)}\right)\right)+A \left(9 m^2+6 m-8\right) e^{4 i (c+d x)} \, _2F_1\left(\frac{m}{2}+\frac{5}{3},m+\frac{4}{3};\frac{m}{2}+\frac{8}{3};-e^{2 i (c+d x)}\right)\right)}{d (3 m-2) (3 m+4) (3 m+10) \sec ^{\frac{4}{3}}(c+d x) (b \sec (c+d x))^{2/3} (A \cos (2 c+2 d x)+A+2 C)}","\frac{3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+1) (b \sec (c+d x))^{2/3}}-\frac{3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (5-3 m);\frac{1}{6} (11-3 m);\cos ^2(c+d x)\right)}{d (5-3 m) (3 m+1) \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}",1,"((-3*I)*2^(1/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-2/3 + m)*(1 + E^((2*I)*(c + d*x)))^(-2/3 + m)*(A*E^((4*I)*(c + d*x))*(-8 + 6*m + 9*m^2)*Hypergeometric2F1[5/3 + m/2, 4/3 + m, 8/3 + m/2, -E^((2*I)*(c + d*x))] + (10 + 3*m)*(A*(4 + 3*m)*Hypergeometric2F1[4/3 + m, (-2 + 3*m)/6, (4 + 3*m)/6, -E^((2*I)*(c + d*x))] + 2*(A + 2*C)*E^((2*I)*(c + d*x))*(-2 + 3*m)*Hypergeometric2F1[4/3 + m, (4 + 3*m)/6, 5/3 + m/2, -E^((2*I)*(c + d*x))]))*(A + C*Sec[c + d*x]^2))/(d*(-2 + 3*m)*(4 + 3*m)*(10 + 3*m)*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(4/3)*(b*Sec[c + d*x])^(2/3))","C",1
26,1,340,148,3.9714605,"\int \frac{\sec ^m(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","-\frac{3 i 2^{m-\frac{1}{3}} e^{-\frac{1}{3} i (6 c+d (3 m+2) x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{2}{3}} \left(1+e^{2 i (c+d x)}\right)^{m+\frac{2}{3}} \left(A+C \sec ^2(c+d x)\right) \left(\frac{e^{\frac{1}{3} i (6 c+d (3 m+2) x)} \left(2 (3 m+8) (A+2 C) \, _2F_1\left(m+\frac{2}{3},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);-e^{2 i (c+d x)}\right)+A (3 m+2) e^{2 i (c+d x)} \, _2F_1\left(m+\frac{2}{3},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};-e^{2 i (c+d x)}\right)\right)}{(3 m+2) (3 m+8)}+\frac{A e^{\frac{1}{3} i d (3 m-4) x} \, _2F_1\left(m+\frac{2}{3},\frac{1}{6} (3 m-4);\frac{1}{6} (3 m+2);-e^{2 i (c+d x)}\right)}{3 m-4}\right)}{d \sec ^{\frac{2}{3}}(c+d x) (b \sec (c+d x))^{4/3} (A \cos (2 c+2 d x)+A+2 C)}","-\frac{3 (-3 A m+A+C (4-3 m)) \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (7-3 m);\frac{1}{6} (13-3 m);\cos ^2(c+d x)\right)}{b d (1-3 m) (7-3 m) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}-\frac{3 C \sin (c+d x) \sec ^m(c+d x)}{b d (1-3 m) \sqrt[3]{b \sec (c+d x)}}",1,"((-3*I)*2^(-1/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3 + m)*(1 + E^((2*I)*(c + d*x)))^(2/3 + m)*((A*E^((I/3)*d*(-4 + 3*m)*x)*Hypergeometric2F1[2/3 + m, (-4 + 3*m)/6, (2 + 3*m)/6, -E^((2*I)*(c + d*x))])/(-4 + 3*m) + (E^((I/3)*(6*c + d*(2 + 3*m)*x))*(2*(A + 2*C)*(8 + 3*m)*Hypergeometric2F1[2/3 + m, (2 + 3*m)/6, (8 + 3*m)/6, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(2 + 3*m)*Hypergeometric2F1[2/3 + m, (8 + 3*m)/6, 7/3 + m/2, -E^((2*I)*(c + d*x))]))/((2 + 3*m)*(8 + 3*m)))*(A + C*Sec[c + d*x]^2))/(d*E^((I/3)*(6*c + d*(2 + 3*m)*x))*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(2/3)*(b*Sec[c + d*x])^(4/3))","C",1
27,1,289,145,8.8993406,"\int \sec ^m(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\frac{i 2^{m+n+1} e^{-i (m+n+1) (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+n+1} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(\frac{2 (A+2 C) e^{i (m+n+2) (c+d x)} \, _2F_1\left(1,\frac{1}{2} (-m-n);\frac{1}{2} (m+n+4);-e^{2 i (c+d x)}\right)}{m+n+2}+\frac{A e^{i (m+n) (c+d x)} \, _2F_1\left(1,\frac{1}{2} (-m-n-2);\frac{1}{2} (m+n+2);-e^{2 i (c+d x)}\right)}{m+n}+\frac{A e^{i (m+n+4) (c+d x)} \, _2F_1\left(1,\frac{1}{2} (-m-n+2);\frac{1}{2} (m+n+6);-e^{2 i (c+d x)}\right)}{m+n+4}\right)}{d (A \cos (2 c+2 d x)+A+2 C)}","\frac{C \sin (c+d x) \sec ^{m+1}(c+d x) (b \sec (c+d x))^n}{d (m+n+1)}-\frac{(A (m+n+1)+C (m+n)) \sin (c+d x) \sec ^{m-1}(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-n+1);\frac{1}{2} (-m-n+3);\cos ^2(c+d x)\right)}{d (-m-n+1) (m+n+1) \sqrt{\sin ^2(c+d x)}}",1,"((-I)*2^(1 + m + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 + m + n)*((A*E^(I*(m + n)*(c + d*x))*Hypergeometric2F1[1, (-2 - m - n)/2, (2 + m + n)/2, -E^((2*I)*(c + d*x))])/(m + n) + (2*(A + 2*C)*E^(I*(2 + m + n)*(c + d*x))*Hypergeometric2F1[1, (-m - n)/2, (4 + m + n)/2, -E^((2*I)*(c + d*x))])/(2 + m + n) + (A*E^(I*(4 + m + n)*(c + d*x))*Hypergeometric2F1[1, (2 - m - n)/2, (6 + m + n)/2, -E^((2*I)*(c + d*x))])/(4 + m + n))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*E^(I*(1 + m + n)*(c + d*x))*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
28,1,222,120,1.8668996,"\int \sec ^2(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+3} e^{2 i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(A (n+4) \left(1+e^{2 i (c+d x)}\right)^2 \, _2F_1\left(1,-\frac{n}{2};\frac{n+4}{2};-e^{2 i (c+d x)}\right)+4 C (n+2) e^{2 i (c+d x)} \, _2F_1\left(1,-\frac{n}{2}-1;\frac{n+6}{2};-e^{2 i (c+d x)}\right)\right)}{d (n+2) (n+4) \left(1+e^{2 i (c+d x)}\right)^3 (A \cos (2 (c+d x))+A+2 C)}","\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \sec (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\cos ^2(c+d x)\right)}{b d (n+1) (n+3) \sqrt{\sin ^2(c+d x)}}+\frac{C \tan (c+d x) (b \sec (c+d x))^{n+2}}{b^2 d (n+3)}",1,"((-I)*2^(3 + n)*E^((2*I)*(c + d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*(4*C*E^((2*I)*(c + d*x))*(2 + n)*Hypergeometric2F1[1, -1 - n/2, (6 + n)/2, -E^((2*I)*(c + d*x))] + A*(1 + E^((2*I)*(c + d*x)))^2*(4 + n)*Hypergeometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*(1 + E^((2*I)*(c + d*x)))^3*(2 + n)*(4 + n)*(A + 2*C + A*Cos[2*(c + d*x)]))","C",0
29,1,215,109,1.8492842,"\int \sec (c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+2} e^{-i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+2} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(A (n+3) \left(1+e^{2 i (c+d x)}\right)^2 \, _2F_1\left(1,\frac{1-n}{2};\frac{n+3}{2};-e^{2 i (c+d x)}\right)+4 C (n+1) e^{2 i (c+d x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+5}{2};-e^{2 i (c+d x)}\right)\right)}{d (n+1) (n+3) (A \cos (2 (c+d x))+A+2 C)}","\frac{(A (n+2)+C (n+1)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(c+d x)\right)}{d n (n+2) \sqrt{\sin ^2(c+d x)}}+\frac{C \tan (c+d x) (b \sec (c+d x))^{n+1}}{b d (n+2)}",1,"((-I)*2^(2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2 + n)*(4*C*E^((2*I)*(c + d*x))*(1 + n)*Hypergeometric2F1[1, (-1 - n)/2, (5 + n)/2, -E^((2*I)*(c + d*x))] + A*(1 + E^((2*I)*(c + d*x)))^2*(3 + n)*Hypergeometric2F1[1, (1 - n)/2, (3 + n)/2, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*E^(I*(c + d*x))*(1 + n)*(3 + n)*(A + 2*C + A*Cos[2*(c + d*x)]))","C",0
30,1,273,113,6.2606936,"\int (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+1} e^{-i (n+1) (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+1} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(n e^{i (n+2) (c+d x)} \left(2 (n+4) (A+2 C) \, _2F_1\left(1,-\frac{n}{2};\frac{n+4}{2};-e^{2 i (c+d x)}\right)+A (n+2) e^{2 i (c+d x)} \, _2F_1\left(1,1-\frac{n}{2};\frac{n+6}{2};-e^{2 i (c+d x)}\right)\right)+A \left(n^2+6 n+8\right) e^{i n (c+d x)} \, _2F_1\left(1,-\frac{n}{2}-1;\frac{n+2}{2};-e^{2 i (c+d x)}\right)\right)}{d n (n+2) (n+4) (A \cos (2 c+2 d x)+A+2 C)}","\frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}-\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right)}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}}",1,"((-I)*2^(1 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 + n)*(A*E^(I*n*(c + d*x))*(8 + 6*n + n^2)*Hypergeometric2F1[1, -1 - n/2, (2 + n)/2, -E^((2*I)*(c + d*x))] + E^(I*(2 + n)*(c + d*x))*n*(A*E^((2*I)*(c + d*x))*(2 + n)*Hypergeometric2F1[1, 1 - n/2, (6 + n)/2, -E^((2*I)*(c + d*x))] + 2*(A + 2*C)*(4 + n)*Hypergeometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(c + d*x))]))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*E^(I*(1 + n)*(c + d*x))*n*(2 + n)*(4 + n)*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
31,1,119,117,0.2455041,"\int \cos (c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{-\tan ^2(c+d x)} (b \sec (c+d x))^n \left(A (n+1) \cos (c+d x) \cot (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\sec ^2(c+d x)\right)+C (n-1) \csc (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(c+d x)\right)\right)}{d (n-1) (n+1)}","\frac{b^2 (C (1-n)-A n) \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right)}{d (2-n) n \sqrt{\sin ^2(c+d x)}}+\frac{b C \tan (c+d x) (b \sec (c+d x))^{n-1}}{d n}",1,"((A*(1 + n)*Cos[c + d*x]*Cot[c + d*x]*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Sec[c + d*x]^2] + C*(-1 + n)*Csc[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[c + d*x]^2])*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(-1 + n)*(1 + n))","A",1
32,1,107,132,0.1898975,"\int \cos ^2(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^n \left(A n \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\sec ^2(c+d x)\right)+C (n-2) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(c+d x)\right)\right)}{d (n-2) n}","-\frac{b^3 (A (1-n)+C (2-n)) \sin (c+d x) (b \sec (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\cos ^2(c+d x)\right)}{d (1-n) (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C \tan (c+d x) (b \sec (c+d x))^{n-2}}{d (1-n)}",1,"(Cot[c + d*x]*(A*n*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Sec[c + d*x]^2] + C*(-2 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[c + d*x]^2])*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(-2 + n)*n)","A",1
33,1,118,132,0.2551647,"\int \cos ^3(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\frac{b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{n-1} \left(A (n-1) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\sec ^2(c+d x)\right)+C (n-3) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\sec ^2(c+d x)\right)\right)}{d (n-3) (n-1)}","-\frac{b^4 (A (2-n)+C (3-n)) \sin (c+d x) (b \sec (c+d x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\cos ^2(c+d x)\right)}{d (2-n) (4-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^3 C \tan (c+d x) (b \sec (c+d x))^{n-3}}{d (2-n)}",1,"(b*Cot[c + d*x]*(A*(-1 + n)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Sec[c + d*x]^2] + C*(-3 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(-1 + n)*Sqrt[-Tan[c + d*x]^2])/(d*(-3 + n)*(-1 + n))","A",1
34,1,341,142,3.2334411,"\int \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+\frac{9}{2}} e^{2 i c-\frac{1}{2} i d (2 n+1) x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{1}{2}} \left(1+e^{2 i (c+d x)}\right)^{n+\frac{1}{2}} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(\frac{e^{\frac{1}{2} i (4 c+d (2 n+9) x)} \left(2 (2 n+13) (A+2 C) \, _2F_1\left(n+\frac{9}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);-e^{2 i (c+d x)}\right)+A (2 n+9) e^{2 i (c+d x)} \, _2F_1\left(n+\frac{9}{2},\frac{1}{4} (2 n+13);\frac{1}{4} (2 n+17);-e^{2 i (c+d x)}\right)\right)}{(2 n+9) (2 n+13)}+\frac{A e^{\frac{1}{2} i d (2 n+5) x} \, _2F_1\left(n+\frac{9}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);-e^{2 i (c+d x)}\right)}{2 n+5}\right)}{d (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 (A (2 n+7)+C (2 n+5)) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (-2 n-3);\frac{1}{4} (1-2 n);\cos ^2(c+d x)\right)}{d (2 n+3) (2 n+7) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+7)}",1,"((-I)*2^(9/2 + n)*E^((2*I)*c - (I/2)*d*(1 + 2*n)*x)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + n)*(1 + E^((2*I)*(c + d*x)))^(1/2 + n)*((A*E^((I/2)*d*(5 + 2*n)*x)*Hypergeometric2F1[9/2 + n, (5 + 2*n)/4, (9 + 2*n)/4, -E^((2*I)*(c + d*x))])/(5 + 2*n) + (E^((I/2)*(4*c + d*(9 + 2*n)*x))*(2*(A + 2*C)*(13 + 2*n)*Hypergeometric2F1[9/2 + n, (9 + 2*n)/4, (13 + 2*n)/4, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(9 + 2*n)*Hypergeometric2F1[9/2 + n, (13 + 2*n)/4, (17 + 2*n)/4, -E^((2*I)*(c + d*x))]))/((9 + 2*n)*(13 + 2*n)))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
35,1,303,142,2.3716002,"\int \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+\frac{7}{2}} e^{-\frac{1}{2} i (2 n+5) (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{5}{2}} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(\frac{2 (A+2 C) e^{\frac{1}{2} i (2 n+7) (c+d x)} \, _2F_1\left(1,\frac{1}{4} (-2 n-3);\frac{1}{4} (2 n+11);-e^{2 i (c+d x)}\right)}{2 n+7}+\frac{A e^{\frac{1}{2} i (2 n+3) (c+d x)} \, _2F_1\left(1,\frac{1}{4} (-2 n-7);\frac{1}{4} (2 n+7);-e^{2 i (c+d x)}\right)}{2 n+3}+\frac{A e^{\frac{1}{2} i (2 n+11) (c+d x)} \, _2F_1\left(1,\frac{1}{4} (1-2 n);\frac{1}{4} (2 n+15);-e^{2 i (c+d x)}\right)}{2 n+11}\right)}{d (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (-2 n-1);\frac{1}{4} (3-2 n);\cos ^2(c+d x)\right)}{d (2 n+1) (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+5)}",1,"((-I)*2^(7/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(5/2 + n)*((A*E^((I/2)*(3 + 2*n)*(c + d*x))*Hypergeometric2F1[1, (-7 - 2*n)/4, (7 + 2*n)/4, -E^((2*I)*(c + d*x))])/(3 + 2*n) + (2*(A + 2*C)*E^((I/2)*(7 + 2*n)*(c + d*x))*Hypergeometric2F1[1, (-3 - 2*n)/4, (11 + 2*n)/4, -E^((2*I)*(c + d*x))])/(7 + 2*n) + (A*E^((I/2)*(11 + 2*n)*(c + d*x))*Hypergeometric2F1[1, (1 - 2*n)/4, (15 + 2*n)/4, -E^((2*I)*(c + d*x))])/(11 + 2*n))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*E^((I/2)*(5 + 2*n)*(c + d*x))*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
36,1,303,140,2.3807413,"\int \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+\frac{5}{2}} e^{-\frac{1}{2} i (2 n+3) (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{3}{2}} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(\frac{2 (A+2 C) e^{\frac{1}{2} i (2 n+5) (c+d x)} \, _2F_1\left(1,\frac{1}{4} (-2 n-1);\frac{1}{4} (2 n+9);-e^{2 i (c+d x)}\right)}{2 n+5}+\frac{A e^{\frac{1}{2} i (2 n+1) (c+d x)} \, _2F_1\left(1,\frac{1}{4} (-2 n-5);\frac{1}{4} (2 n+5);-e^{2 i (c+d x)}\right)}{2 n+1}+\frac{A e^{\frac{1}{2} i (2 n+9) (c+d x)} \, _2F_1\left(1,\frac{1}{4} (3-2 n);\frac{1}{4} (2 n+13);-e^{2 i (c+d x)}\right)}{2 n+9}\right)}{d (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+3)}-\frac{2 (A (2 n+3)+2 C n+C) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (1-2 n);\frac{1}{4} (5-2 n);\cos ^2(c+d x)\right)}{d (1-2 n) (2 n+3) \sqrt{\sin ^2(c+d x)} \sqrt{\sec (c+d x)}}",1,"((-I)*2^(5/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2 + n)*((A*E^((I/2)*(1 + 2*n)*(c + d*x))*Hypergeometric2F1[1, (-5 - 2*n)/4, (5 + 2*n)/4, -E^((2*I)*(c + d*x))])/(1 + 2*n) + (2*(A + 2*C)*E^((I/2)*(5 + 2*n)*(c + d*x))*Hypergeometric2F1[1, (-1 - 2*n)/4, (9 + 2*n)/4, -E^((2*I)*(c + d*x))])/(5 + 2*n) + (A*E^((I/2)*(9 + 2*n)*(c + d*x))*Hypergeometric2F1[1, (3 - 2*n)/4, (13 + 2*n)/4, -E^((2*I)*(c + d*x))])/(9 + 2*n))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*E^((I/2)*(3 + 2*n)*(c + d*x))*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
37,1,311,141,4.8745428,"\int \frac{(b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","-\frac{i 2^{n+\frac{3}{2}} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n-\frac{1}{2}} \left(1+e^{2 i (c+d x)}\right)^{n-\frac{1}{2}} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left((2 n-1) e^{2 i (c+d x)} \left(2 (2 n+7) (A+2 C) \, _2F_1\left(n+\frac{3}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);-e^{2 i (c+d x)}\right)+A (2 n+3) e^{2 i (c+d x)} \, _2F_1\left(n+\frac{3}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);-e^{2 i (c+d x)}\right)\right)+A \left(4 n^2+20 n+21\right) \, _2F_1\left(n+\frac{3}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);-e^{2 i (c+d x)}\right)\right)}{d (2 n-1) (2 n+3) (2 n+7) (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n}{d (2 n+1)}-\frac{2 (2 A n+A-C (1-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (3-2 n);\frac{1}{4} (7-2 n);\cos ^2(c+d x)\right)}{d (3-2 n) (2 n+1) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{3}{2}}(c+d x)}",1,"((-I)*2^(3/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-1/2 + n)*(1 + E^((2*I)*(c + d*x)))^(-1/2 + n)*(A*(21 + 20*n + 4*n^2)*Hypergeometric2F1[3/2 + n, (-1 + 2*n)/4, (3 + 2*n)/4, -E^((2*I)*(c + d*x))] + E^((2*I)*(c + d*x))*(-1 + 2*n)*(2*(A + 2*C)*(7 + 2*n)*Hypergeometric2F1[3/2 + n, (3 + 2*n)/4, (7 + 2*n)/4, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(3 + 2*n)*Hypergeometric2F1[3/2 + n, (7 + 2*n)/4, (11 + 2*n)/4, -E^((2*I)*(c + d*x))]))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*(-1 + 2*n)*(3 + 2*n)*(7 + 2*n)*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",1
38,1,343,140,3.0724011,"\int \frac{(b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","-\frac{i 2^{n+\frac{1}{2}} e^{-\frac{1}{2} i (4 c+d (2 n+1) x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{1}{2}} \left(1+e^{2 i (c+d x)}\right)^{n+\frac{1}{2}} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(\frac{e^{\frac{1}{2} i (4 c+d (2 n+1) x)} \left(2 (2 n+5) (A+2 C) \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);-e^{2 i (c+d x)}\right)+A (2 n+1) e^{2 i (c+d x)} \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);-e^{2 i (c+d x)}\right)\right)}{d (2 n+1) (2 n+5)}+\frac{A e^{\frac{1}{2} i d (2 n-3) x} \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);-e^{2 i (c+d x)}\right)}{d (2 n-3)}\right)}{A \cos (2 c+2 d x)+A+2 C}","-\frac{2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right)}{d (1-2 n) (5-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt{\sec (c+d x)}}",1,"((-I)*2^(1/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + n)*(1 + E^((2*I)*(c + d*x)))^(1/2 + n)*((A*E^((I/2)*d*(-3 + 2*n)*x)*Hypergeometric2F1[1/2 + n, (-3 + 2*n)/4, (1 + 2*n)/4, -E^((2*I)*(c + d*x))])/(d*(-3 + 2*n)) + (E^((I/2)*(4*c + d*(1 + 2*n)*x))*(2*(A + 2*C)*(5 + 2*n)*Hypergeometric2F1[1/2 + n, (1 + 2*n)/4, (5 + 2*n)/4, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(1 + 2*n)*Hypergeometric2F1[1/2 + n, (5 + 2*n)/4, (9 + 2*n)/4, -E^((2*I)*(c + d*x))]))/(d*(1 + 2*n)*(5 + 2*n)))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(E^((I/2)*(4*c + d*(1 + 2*n)*x))*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",1
39,1,338,142,4.0696468,"\int \frac{(b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","-\frac{i 2^{n-\frac{1}{2}} e^{-\frac{1}{2} i (4 c+d (2 n-1) x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n-\frac{1}{2}} \left(1+e^{2 i (c+d x)}\right)^{n-\frac{1}{2}} \sec ^{-n-2}(c+d x) \left(A+C \sec ^2(c+d x)\right) (b \sec (c+d x))^n \left(\frac{e^{\frac{1}{2} i (4 c+d (2 n-1) x)} \left(2 (2 n+3) (A+2 C) \, _2F_1\left(n-\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);-e^{2 i (c+d x)}\right)+A (2 n-1) e^{2 i (c+d x)} \, _2F_1\left(n-\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);-e^{2 i (c+d x)}\right)\right)}{4 n^2+4 n-3}+\frac{A e^{\frac{1}{2} i d (2 n-5) x} \, _2F_1\left(n-\frac{1}{2},\frac{1}{4} (2 n-5);\frac{1}{4} (2 n-1);-e^{2 i (c+d x)}\right)}{2 n-5}\right)}{d (A \cos (2 c+2 d x)+A+2 C)}","-\frac{2 (A (3-2 n)+C (5-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (7-2 n);\frac{1}{4} (11-2 n);\cos ^2(c+d x)\right)}{d (3-2 n) (7-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{7}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (3-2 n) \sec ^{\frac{3}{2}}(c+d x)}",1,"((-I)*2^(-1/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(-1/2 + n)*(1 + E^((2*I)*(c + d*x)))^(-1/2 + n)*((A*E^((I/2)*d*(-5 + 2*n)*x)*Hypergeometric2F1[-1/2 + n, (-5 + 2*n)/4, (-1 + 2*n)/4, -E^((2*I)*(c + d*x))])/(-5 + 2*n) + (E^((I/2)*(4*c + d*(-1 + 2*n)*x))*(2*(A + 2*C)*(3 + 2*n)*Hypergeometric2F1[-1/2 + n, (-1 + 2*n)/4, (3 + 2*n)/4, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(-1 + 2*n)*Hypergeometric2F1[-1/2 + n, (3 + 2*n)/4, (7 + 2*n)/4, -E^((2*I)*(c + d*x))]))/(-3 + 4*n + 4*n^2))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/(d*E^((I/2)*(4*c + d*(-1 + 2*n)*x))*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
40,1,129,167,0.2869291,"\int \sec ^m(c+d x) (b \sec (c+d x))^n \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \left(B (m+n+2) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sec ^2(c+d x)\right)+C (m+n+1) \sec (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);\sec ^2(c+d x)\right)\right)}{d (m+n+1) (m+n+2)}","\frac{B \sin (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-n);\frac{1}{2} (-m-n+2);\cos ^2(c+d x)\right)}{d (m+n) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) \sec ^{m+1}(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-n-1);\frac{1}{2} (-m-n+1);\cos ^2(c+d x)\right)}{d (m+n+1) \sqrt{\sin ^2(c+d x)}}",1,"(Csc[c + d*x]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(B*(2 + m + n)*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Sec[c + d*x]^2] + C*(1 + m + n)*Hypergeometric2F1[1/2, (2 + m + n)/2, (4 + m + n)/2, Sec[c + d*x]^2]*Sec[c + d*x])*Sqrt[-Tan[c + d*x]^2])/(d*(1 + m + n)*(2 + m + n))","A",1
41,1,346,154,5.0082988,"\int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sec ^{\frac{2}{3}}(c+d x) \left(2 \tan (c+d x) \sec ^2(c+d x) (4 (11 A+8 C) \cos (2 (c+d x))+44 A+55 B \cos (c+d x)+72 C)+275 B \csc (c) \cos (d x)\right)-\frac{i 2^{2/3} e^{-i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(16 \left(-1+e^{2 i c}\right) (11 A+8 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+275 B \left(-1+e^{2 i c}\right) \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i (c+d x)}\right)+275 B \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{440 d \sec ^{\frac{8}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{3 (11 A+8 C) \sin (c+d x) (b \sec (c+d x))^{5/3} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{55 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{8/3} \, _2F_1\left(-\frac{4}{3},\frac{1}{2};-\frac{1}{3};\cos ^2(c+d x)\right)}{8 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{8/3}}{11 b^2 d}",1,"(3*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-I)*2^(2/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(275*B*(1 + E^((2*I)*(c + d*x))) + 275*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[-1/6, 2/3, 5/6, -E^((2*I)*(c + d*x))] + 16*(11*A + 8*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sec[c + d*x]^(2/3)*(275*B*Cos[d*x]*Csc[c] + 2*(44*A + 72*C + 55*B*Cos[c + d*x] + 4*(11*A + 8*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])))/(440*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(8/3))","C",1
42,1,265,151,6.3199236,"\int \sec (c+d x) (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 i \left(\frac{b e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left((8 A+5 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{8/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-40 A e^{i (c+d x)}-80 A e^{3 i (c+d x)}-40 A e^{5 i (c+d x)}-16 B \left(1+e^{2 i (c+d x)}\right)^{8/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)-16 B e^{4 i (c+d x)}+16 B-5 C e^{i (c+d x)}-70 C e^{3 i (c+d x)}-25 C e^{5 i (c+d x)}\right)}{40 \sqrt[3]{2} d \left(1+e^{2 i (c+d x)}\right)^2}","\frac{3 (8 A+5 C) \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{16 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{5/3} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{5/3}}{8 b d}",1,"(((3*I)/40)*((b*E^(I*(c + d*x)))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(16*B - 40*A*E^(I*(c + d*x)) - 5*C*E^(I*(c + d*x)) - 80*A*E^((3*I)*(c + d*x)) - 70*C*E^((3*I)*(c + d*x)) - 16*B*E^((4*I)*(c + d*x)) - 40*A*E^((5*I)*(c + d*x)) - 25*C*E^((5*I)*(c + d*x)) - 16*B*(1 + E^((2*I)*(c + d*x)))^(8/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))] + (8*A + 5*C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(8/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))]))/(2^(1/3)*d*(1 + E^((2*I)*(c + d*x)))^2)","C",1
43,1,311,146,2.1701125,"\int (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{3 \cos (c+d x) (5 B \csc (c) \cos (d x) \cos (c+d x)+2 C \sin (c+d x))}{d}-\frac{3 i 2^{2/3} e^{-i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(\left(-1+e^{2 i c}\right) (5 A+2 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+5 B \left(-1+e^{2 i c}\right) \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i (c+d x)}\right)+5 B \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sec ^{\frac{8}{3}}(c+d x)}\right)}{5 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{3 b (5 A+2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 d}",1,"((b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-3*I)*2^(2/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(5*B*(1 + E^((2*I)*(c + d*x))) + 5*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[-1/6, 2/3, 5/6, -E^((2*I)*(c + d*x))] + (5*A + 2*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sec[c + d*x]^(8/3)) + (3*Cos[c + d*x]*(5*B*Cos[d*x]*Cos[c + d*x]*Csc[c] + 2*C*Sin[c + d*x]))/d))/(5*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","C",1
44,1,120,148,0.212923,"\int \cos (c+d x) (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{5/3} \left(10 A \cos ^2(c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)-5 B \cos (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)-2 C \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sec ^2(c+d x)\right)\right)}{10 b d}","-\frac{3 b^2 (2 A-C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}-\frac{3 b B \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 b C \tan (c+d x)}{2 d \sqrt[3]{b \sec (c+d x)}}",1,"(-3*Cot[c + d*x]*(10*A*Cos[c + d*x]^2*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2] - 5*B*Cos[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2] - 2*C*Hypergeometric2F1[1/2, 5/6, 11/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(5/3)*Sqrt[-Tan[c + d*x]^2])/(10*b*d)","A",1
45,1,116,150,0.1780062,"\int \cos ^2(c+d x) (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{2/3} \left(A \cos ^2(c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\sec ^2(c+d x)\right)+4 B \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)-2 C \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)\right)}{4 d}","\frac{3 A b^2 \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac{3 b (A+4 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}-\frac{3 b^2 B \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}",1,"(-3*Cot[c + d*x]*(A*Cos[c + d*x]^2*Hypergeometric2F1[-2/3, 1/2, 1/3, Sec[c + d*x]^2] + 4*B*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2] - 2*C*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(2/3)*Sqrt[-Tan[c + d*x]^2])/(4*d)","A",1
46,1,118,154,0.2319978,"\int \cos ^3(c+d x) (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(4 A \cos ^2(c+d x) \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\sec ^2(c+d x)\right)+7 B \cos (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\sec ^2(c+d x)\right)+28 C \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)\right)}{28 d \sqrt[3]{b \sec (c+d x)}}","\frac{3 A b^3 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\frac{3 b^2 (4 A+7 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{28 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}-\frac{3 b^3 B \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{7/3}}",1,"(-3*b*Cot[c + d*x]*(4*A*Cos[c + d*x]^2*Hypergeometric2F1[-7/6, 1/2, -1/6, Sec[c + d*x]^2] + 7*B*Cos[c + d*x]*Hypergeometric2F1[-2/3, 1/2, 1/3, Sec[c + d*x]^2] + 28*C*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(28*d*(b*Sec[c + d*x])^(1/3))","A",1
47,1,444,154,6.6337072,"\int \sec ^2(c+d x) (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 b \csc (c) e^{-i d x} \sqrt[3]{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(40 \sqrt[3]{2} \left(-1+e^{2 i c}\right) (13 A+10 C) e^{2 i d x} \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt[3]{\sec (c+d x)} \left(80 e^{i (c+d x)} \left(13 A \left(5 e^{2 i (c+d x)}+2 e^{4 i (c+d x)}+1\right) \left(1+e^{2 i (c+d x)}\right)^2+2 C \left(21 e^{2 i (c+d x)}+79 e^{4 i (c+d x)}+45 e^{6 i (c+d x)}+10 e^{8 i (c+d x)}+5\right)\right)+637 B \left(1+e^{2 i (c+d x)}\right)^{13/3} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+91 B \left(-30 e^{2 i (c+d x)}+30 e^{6 i (c+d x)}+7 e^{8 i (c+d x)}-7\right)\right)}{2 \left(1+e^{2 i (c+d x)}\right)^4}\right)}{1820 d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{3 (13 A+10 C) \sin (c+d x) (b \sec (c+d x))^{7/3} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{91 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{10/3} \, _2F_1\left(-\frac{5}{3},\frac{1}{2};-\frac{2}{3};\cos ^2(c+d x)\right)}{10 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{10/3}}{13 b^2 d}",1,"(3*b*Csc[c]*(40*2^(1/3)*(13*A + 10*C)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(91*B*(-7 - 30*E^((2*I)*(c + d*x)) + 30*E^((6*I)*(c + d*x)) + 7*E^((8*I)*(c + d*x))) + 80*E^(I*(c + d*x))*(13*A*(1 + E^((2*I)*(c + d*x)))^2*(1 + 5*E^((2*I)*(c + d*x)) + 2*E^((4*I)*(c + d*x))) + 2*C*(5 + 21*E^((2*I)*(c + d*x)) + 79*E^((4*I)*(c + d*x)) + 45*E^((6*I)*(c + d*x)) + 10*E^((8*I)*(c + d*x)))) + 637*B*(1 + E^((2*I)*(c + d*x)))^(13/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))])*Sec[c + d*x]^(1/3))/(2*E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^4))*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(1820*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/3))","C",1
48,1,465,151,7.0211169,"\int \sec (c+d x) (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\frac{\cos ^4(c+d x) (b \sec (c+d x))^{7/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{3 \sec (c) \sec (c+d x) (70 A \sin (d x)+40 B \sin (c)+49 C \sin (d x))}{140 d}+\frac{3 (10 A+7 C) \tan (c)}{20 d}+\frac{3 \sec (c) \sec ^2(c+d x) (10 B \sin (d x)+7 C \sin (c))}{35 d}+\frac{24 B \csc (c) \cos (d x)}{7 d}+\frac{3 C \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}\right)}{A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C}-\frac{3 i e^{-i (c+d x)} \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} (b \sec (c+d x))^{7/3} \left(7 \left(-1+e^{2 i c}\right) (10 A+7 C) e^{i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+160 B \left(-1+e^{2 i c}\right) \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i (c+d x)}\right)+160 B \left(1+e^{2 i (c+d x)}\right)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{70\ 2^{2/3} \left(-1+e^{2 i c}\right) d \sec ^{\frac{13}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}}{b}","\frac{3 (10 A+7 C) \sin (c+d x) (b \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{40 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{7/3} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{7/3}}{10 b d}",1,"((((-3*I)/70)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(160*B*(1 + E^((2*I)*(c + d*x))) + 160*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[-1/3, 1/3, 2/3, -E^((2*I)*(c + d*x))] + 7*(10*A + 7*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(7/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(2^(2/3)*d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(13/3)) + (Cos[c + d*x]^4*(b*Sec[c + d*x])^(7/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((24*B*Cos[d*x]*Csc[c])/(7*d) + (3*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (3*Sec[c]*Sec[c + d*x]^2*(7*C*Sin[c] + 10*B*Sin[d*x]))/(35*d) + (3*Sec[c]*Sec[c + d*x]*(40*B*Sin[c] + 70*A*Sin[d*x] + 49*C*Sin[d*x]))/(140*d) + (3*(10*A + 7*C)*Tan[c])/(20*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))/b","C",1
49,1,290,146,2.5107491,"\int (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 i b \sqrt[3]{b \sec (c+d x)} \left(-14 A e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)+28 A e^{i (c+d x)}+56 A e^{3 i (c+d x)}+28 A e^{5 i (c+d x)}+7 B \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+7 B e^{4 i (c+d x)}-7 B-8 C e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)+8 C e^{i (c+d x)}+40 C e^{3 i (c+d x)}+16 C e^{5 i (c+d x)}\right)}{28 d \left(1+e^{2 i (c+d x)}\right)^2}","\frac{3 b (7 A+4 C) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 d}",1,"(((-3*I)/28)*b*(-7*B + 28*A*E^(I*(c + d*x)) + 8*C*E^(I*(c + d*x)) + 56*A*E^((3*I)*(c + d*x)) + 40*C*E^((3*I)*(c + d*x)) + 7*B*E^((4*I)*(c + d*x)) + 28*A*E^((5*I)*(c + d*x)) + 16*C*E^((5*I)*(c + d*x)) + 7*B*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] - 14*A*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))] - 8*C*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))])*(b*Sec[c + d*x])^(1/3))/(d*(1 + E^((2*I)*(c + d*x)))^2)","C",1
50,1,303,146,2.6120161,"\int \cos (c+d x) (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 b \sqrt[3]{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sqrt[3]{\sec (c+d x)} (4 B \csc (c) \cos (d x)+C \tan (c+d x))-\frac{i \sqrt[3]{2} e^{-i (c+d x)} \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\left(-1+e^{2 i c}\right) (4 A+C) e^{i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+4 B \left(-1+e^{2 i c}\right) \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i (c+d x)}\right)+4 B \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{2 d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{3 b^2 (4 A+C) \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 b B \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{d \sqrt{\sin ^2(c+d x)}}+\frac{3 b C \tan (c+d x) \sqrt[3]{b \sec (c+d x)}}{4 d}",1,"(3*b*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-I)*2^(1/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(4*B*(1 + E^((2*I)*(c + d*x))) + 4*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[-1/3, 1/3, 2/3, -E^((2*I)*(c + d*x))] + (4*A + C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sec[c + d*x]^(1/3)*(4*B*Cos[d*x]*Csc[c] + C*Tan[c + d*x])))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/3))","C",1
51,1,117,150,0.2346456,"\int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{4/3} \left(2 A \cos ^2(c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sec ^2(c+d x)\right)-4 B \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)-C \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)\right)}{4 d}","-\frac{3 b^3 (A-2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}-\frac{3 b^2 B \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 b^2 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}",1,"(-3*Cot[c + d*x]*(2*A*Cos[c + d*x]^2*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[c + d*x]^2] - 4*B*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2] - C*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(4/3)*Sqrt[-Tan[c + d*x]^2])/(4*d)","A",1
52,1,118,154,0.1897483,"\int \cos ^3(c+d x) (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \sec (c+d x)} \left(2 A \cos ^2(c+d x) \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\sec ^2(c+d x)\right)+5 B \cos (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sec ^2(c+d x)\right)-10 C \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)\right)}{10 d}","\frac{3 A b^3 \tan (c+d x)}{5 d (b \sec (c+d x))^{5/3}}-\frac{3 b^2 (2 A+5 C) \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{10 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}-\frac{3 b^3 B \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}",1,"(-3*b*Cot[c + d*x]*(2*A*Cos[c + d*x]^2*Hypergeometric2F1[-5/6, 1/2, 1/6, Sec[c + d*x]^2] + 5*B*Cos[c + d*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[c + d*x]^2] - 10*C*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2])/(10*d)","A",1
53,1,304,154,3.2369587,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","\frac{3 b e^{-i c} \left(-1+e^{2 i c}\right) \csc (c) \left(2 (7 A+4 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)-28 A e^{i (c+d x)}-56 A e^{3 i (c+d x)}-28 A e^{5 i (c+d x)}-7 B \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)-7 B e^{4 i (c+d x)}+7 B-8 C e^{i (c+d x)}-40 C e^{3 i (c+d x)}-16 C e^{5 i (c+d x)}\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{28 d \left(1+e^{2 i (c+d x)}\right)^2 (b \sec (c+d x))^{5/3} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{3 (7 A+4 C) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 b^2 d}",1,"(3*b*(-1 + E^((2*I)*c))*Csc[c]*(7*B - 28*A*E^(I*(c + d*x)) - 8*C*E^(I*(c + d*x)) - 56*A*E^((3*I)*(c + d*x)) - 40*C*E^((3*I)*(c + d*x)) - 7*B*E^((4*I)*(c + d*x)) - 28*A*E^((5*I)*(c + d*x)) - 16*C*E^((5*I)*(c + d*x)) - 7*B*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] + 2*(7*A + 4*C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(28*d*E^(I*c)*(1 + E^((2*I)*(c + d*x)))^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(5/3))","C",1
54,1,305,147,1.9365465,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","\frac{3 \sqrt[3]{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sqrt[3]{\sec (c+d x)} (4 B \csc (c) \cos (d x)+C \tan (c+d x))-\frac{i \sqrt[3]{2} e^{-i (c+d x)} \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\left(-1+e^{2 i c}\right) (4 A+C) e^{i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+4 B \left(-1+e^{2 i c}\right) \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i (c+d x)}\right)+4 B \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{2 b d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (4 A+C) \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 B \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) \sqrt[3]{b \sec (c+d x)}}{4 b d}",1,"(3*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-I)*2^(1/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(4*B*(1 + E^((2*I)*(c + d*x))) + 4*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[-1/3, 1/3, 2/3, -E^((2*I)*(c + d*x))] + (4*A + C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sec[c + d*x]^(1/3)*(4*B*Cos[d*x]*Csc[c] + C*Tan[c + d*x])))/(2*b*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/3))","C",1
55,1,173,142,1.9392571,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(2/3),x]","\frac{3 e^{-i d x} (\sin (d x)-i \cos (d x)) \sqrt[3]{b \sec (c+d x)} \left((A-2 C) e^{i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)-2 A \cos (c+d x)+4 B \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+4 i C \sin (c+d x)+4 C \cos (c+d x)\right)}{4 b d}","-\frac{3 b (A-2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\cos ^2(c+d x)\right)}{5 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{5/3}}-\frac{3 B \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{2 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 C \tan (c+d x)}{d (b \sec (c+d x))^{2/3}}",1,"(3*(b*Sec[c + d*x])^(1/3)*((-I)*Cos[d*x] + Sin[d*x])*(-2*A*Cos[c + d*x] + 4*C*Cos[c + d*x] + 4*B*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] + (A - 2*C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))] + (4*I)*C*Sin[c + d*x]))/(4*b*d*E^(I*d*x))","C",1
56,1,305,147,1.7056698,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","\frac{3 \sqrt[3]{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sqrt[3]{\sec (c+d x)} (4 B \csc (c) \cos (d x)+C \tan (c+d x))-\frac{i \sqrt[3]{2} e^{-i (c+d x)} \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(\left(-1+e^{2 i c}\right) (4 A+C) e^{i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+4 B \left(-1+e^{2 i c}\right) \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i (c+d x)}\right)+4 B \left(1+e^{2 i (c+d x)}\right)\right)}{-1+e^{2 i c}}\right)}{2 b d \sec ^{\frac{7}{3}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (4 A+C) \sin (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 B \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) \sqrt[3]{b \sec (c+d x)}}{4 b d}",1,"(3*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-I)*2^(1/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(4*B*(1 + E^((2*I)*(c + d*x))) + 4*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[-1/3, 1/3, 2/3, -E^((2*I)*(c + d*x))] + (4*A + C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))]))/(E^(I*(c + d*x))*(-1 + E^((2*I)*c))) + Sec[c + d*x]^(1/3)*(4*B*Cos[d*x]*Csc[c] + C*Tan[c + d*x])))/(2*b*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/3))","C",1
57,1,304,154,3.4957953,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","\frac{3 b e^{-i c} \left(-1+e^{2 i c}\right) \csc (c) \left(2 (7 A+4 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{5}{3};-e^{2 i (c+d x)}\right)-28 A e^{i (c+d x)}-56 A e^{3 i (c+d x)}-28 A e^{5 i (c+d x)}-7 B \left(1+e^{2 i (c+d x)}\right)^{7/3} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)-7 B e^{4 i (c+d x)}+7 B-8 C e^{i (c+d x)}-40 C e^{3 i (c+d x)}-16 C e^{5 i (c+d x)}\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{28 d \left(1+e^{2 i (c+d x)}\right)^2 (b \sec (c+d x))^{5/3} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{3 (7 A+4 C) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\cos ^2(c+d x)\right)}{7 b d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{4 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{4/3}}{7 b^2 d}",1,"(3*b*(-1 + E^((2*I)*c))*Csc[c]*(7*B - 28*A*E^(I*(c + d*x)) - 8*C*E^(I*(c + d*x)) - 56*A*E^((3*I)*(c + d*x)) - 40*C*E^((3*I)*(c + d*x)) - 7*B*E^((4*I)*(c + d*x)) - 28*A*E^((5*I)*(c + d*x)) - 16*C*E^((5*I)*(c + d*x)) - 7*B*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))] + 2*(7*A + 4*C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(7/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -E^((2*I)*(c + d*x))])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(28*d*E^(I*c)*(1 + E^((2*I)*(c + d*x)))^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(5/3))","C",1
58,1,333,154,2.662966,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","\frac{\left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{3 (7 (10 A+7 C) \sin (c+d x)+4 \tan (c+d x) (10 B+7 C \sec (c+d x))+160 B \csc (c) \cos (d x) \cos (c+d x))}{d}-\frac{3 i \sqrt[3]{2} e^{-i (c+d x)} \sqrt[3]{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \left(7 \left(-1+e^{2 i c}\right) (10 A+7 C) e^{i (c+d x)} \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{6},\frac{1}{3};\frac{7}{6};-e^{2 i (c+d x)}\right)+160 B \left(-1+e^{2 i c}\right) \sqrt[3]{1+e^{2 i (c+d x)}} \, _2F_1\left(-\frac{1}{3},\frac{1}{3};\frac{2}{3};-e^{2 i (c+d x)}\right)+160 B \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sec ^{\frac{4}{3}}(c+d x)}\right)}{140 (b \sec (c+d x))^{2/3} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{3 (10 A+7 C) \sin (c+d x) (b \sec (c+d x))^{4/3} \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\cos ^2(c+d x)\right)}{40 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{7/3} \, _2F_1\left(-\frac{7}{6},\frac{1}{2};-\frac{1}{6};\cos ^2(c+d x)\right)}{7 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{7/3}}{10 b^3 d}",1,"((A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-3*I)*2^(1/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3)*(160*B*(1 + E^((2*I)*(c + d*x))) + 160*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[-1/3, 1/3, 2/3, -E^((2*I)*(c + d*x))] + 7*(10*A + 7*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sec[c + d*x]^(4/3)) + (3*(160*B*Cos[d*x]*Cos[c + d*x]*Csc[c] + 7*(10*A + 7*C)*Sin[c + d*x] + 4*(10*B + 7*C*Sec[c + d*x])*Tan[c + d*x]))/d))/(140*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(2/3))","C",1
59,1,299,154,1.9623913,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\frac{\left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{3 (5 B \csc (c) \cos (d x)+2 C \tan (c+d x))}{d}-\frac{3 i 2^{2/3} e^{-i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(\left(-1+e^{2 i c}\right) (5 A+2 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+5 B \left(-1+e^{2 i c}\right) \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i (c+d x)}\right)+5 B \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sec ^{\frac{2}{3}}(c+d x)}\right)}{5 (b \sec (c+d x))^{4/3} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (5 A+2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 b^2 d}",1,"((A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-3*I)*2^(2/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(5*B*(1 + E^((2*I)*(c + d*x))) + 5*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[-1/6, 2/3, 5/6, -E^((2*I)*(c + d*x))] + (5*A + 2*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sec[c + d*x]^(2/3)) + (3*(5*B*Cos[d*x]*Csc[c] + 2*C*Tan[c + d*x]))/d))/(5*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(4/3))","C",1
60,1,175,149,2.189727,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 e^{-i d x} (\sin (d x)-i \cos (d x)) (b \sec (c+d x))^{2/3} \left((2 A-C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-10 A \cos (c+d x)+5 B \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+5 i C \sin (c+d x)+5 C \cos (c+d x)\right)}{10 b^2 d}","-\frac{3 (2 A-C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}-\frac{3 B \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 C \tan (c+d x)}{2 b d \sqrt[3]{b \sec (c+d x)}}",1,"(3*(b*Sec[c + d*x])^(2/3)*((-I)*Cos[d*x] + Sin[d*x])*(-10*A*Cos[c + d*x] + 5*C*Cos[c + d*x] + 5*B*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))] + (2*A - C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))] + (5*I)*C*Sin[c + d*x]))/(10*b^2*d*E^(I*d*x))","C",1
61,1,298,146,2.0028181,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(4/3),x]","\frac{\left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{30 \cos (c+d x) (4 B \cot (c)-A \sin (c+d x))}{d}+\frac{3 i 2^{2/3} e^{-i d x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right)^{2/3} \left(e^{i d x} \left(8 B e^{i (c+d x)} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-5 \left(-1+e^{2 i c}\right) (A+4 C) \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)\right)+40 B e^{i c} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sec ^{\frac{2}{3}}(c+d x)}\right)}{20 (b \sec (c+d x))^{4/3} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (A+4 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{4 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 A \tan (c+d x)}{4 d (b \sec (c+d x))^{4/3}}-\frac{3 B \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{4 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}",1,"((A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((3*I)*2^(2/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(2/3)*(40*B*E^(I*c)*Hypergeometric2F1[-1/6, 2/3, 5/6, -E^((2*I)*(c + d*x))] + E^(I*d*x)*(-5*(A + 4*C)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))] + 8*B*E^(I*(c + d*x))*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))])))/(d*E^(I*d*x)*(-1 + E^((2*I)*c))*Sec[c + d*x]^(2/3)) - (30*Cos[c + d*x]*(4*B*Cot[c] - A*Sin[c + d*x]))/d))/(20*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(4/3))","C",1
62,1,175,149,1.2133754,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 e^{-i d x} (\sin (d x)-i \cos (d x)) (b \sec (c+d x))^{2/3} \left((2 A-C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-10 A \cos (c+d x)+5 B \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+5 i C \sin (c+d x)+5 C \cos (c+d x)\right)}{10 b^2 d}","-\frac{3 (2 A-C) \sin (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\cos ^2(c+d x)\right)}{8 d \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{4/3}}-\frac{3 B \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 C \tan (c+d x)}{2 b d \sqrt[3]{b \sec (c+d x)}}",1,"(3*(b*Sec[c + d*x])^(2/3)*((-I)*Cos[d*x] + Sin[d*x])*(-10*A*Cos[c + d*x] + 5*C*Cos[c + d*x] + 5*B*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))] + (2*A - C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))] + (5*I)*C*Sin[c + d*x]))/(10*b^2*d*E^(I*d*x))","C",1
63,1,299,154,1.9441411,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\frac{\left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{3 (5 B \csc (c) \cos (d x)+2 C \tan (c+d x))}{d}-\frac{3 i 2^{2/3} e^{-i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(\left(-1+e^{2 i c}\right) (5 A+2 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right)+5 B \left(-1+e^{2 i c}\right) \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(-\frac{1}{6},\frac{2}{3};\frac{5}{6};-e^{2 i (c+d x)}\right)+5 B \left(1+e^{2 i (c+d x)}\right)\right)}{\left(-1+e^{2 i c}\right) d \sec ^{\frac{2}{3}}(c+d x)}\right)}{5 (b \sec (c+d x))^{4/3} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (5 A+2 C) \sin (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\cos ^2(c+d x)\right)}{5 b d \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{2 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{2/3}}{5 b^2 d}",1,"((A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-3*I)*2^(2/3)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(5*B*(1 + E^((2*I)*(c + d*x))) + 5*B*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[-1/6, 2/3, 5/6, -E^((2*I)*(c + d*x))] + (5*A + 2*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))]))/(d*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Sec[c + d*x]^(2/3)) + (3*(5*B*Cos[d*x]*Csc[c] + 2*C*Tan[c + d*x]))/d))/(5*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(b*Sec[c + d*x])^(4/3))","C",1
64,1,699,154,6.3922962,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","-\frac{6 i 2^{2/3} B \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right)^{2/3} \, _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};-e^{2 i (c+d x)}\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{5 d \sec ^{\frac{2}{3}}(c+d x) (b \sec (c+d x))^{4/3} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{3 A \csc (c) e^{-i d x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right)^{2/3} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-5 \sqrt[3]{1+e^{2 i (c+d x)}}\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{5 \sqrt[3]{2} d \sec ^{\frac{2}{3}}(c+d x) (b \sec (c+d x))^{4/3} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{3 C \csc (c) e^{-i d x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{2/3} \left(1+e^{2 i (c+d x)}\right)^{2/3} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{2}{3},\frac{5}{6};\frac{11}{6};-e^{2 i (c+d x)}\right)-5 \sqrt[3]{1+e^{2 i (c+d x)}}\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{8 \sqrt[3]{2} d \sec ^{\frac{2}{3}}(c+d x) (b \sec (c+d x))^{4/3} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{\left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{3 (8 A+5 C) \csc (c) \cos (d x)}{8 d}+\frac{3 \sec (c) \sec (c+d x) (8 B \sin (d x)+5 C \sin (c))}{20 d}+\frac{6 B \tan (c)}{5 d}+\frac{3 C \sec (c) \sin (d x) \sec ^2(c+d x)}{4 d}\right)}{(b \sec (c+d x))^{4/3} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{3 (8 A+5 C) \sin (c+d x) (b \sec (c+d x))^{2/3} \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\cos ^2(c+d x)\right)}{16 b^2 d \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{5/3} \, _2F_1\left(-\frac{5}{6},\frac{1}{2};\frac{1}{6};\cos ^2(c+d x)\right)}{5 b^3 d \sqrt{\sin ^2(c+d x)}}+\frac{3 C \tan (c+d x) (b \sec (c+d x))^{5/3}}{8 b^3 d}",1,"(((-6*I)/5)*2^(2/3)*B*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -E^((2*I)*(c + d*x))]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(2/3)*(b*Sec[c + d*x])^(4/3)) + (3*A*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(2/3)*Csc[c]*(-5*(1 + E^((2*I)*(c + d*x)))^(1/3) + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*2^(1/3)*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(2/3)*(b*Sec[c + d*x])^(4/3)) + (3*C*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3)*(1 + E^((2*I)*(c + d*x)))^(2/3)*Csc[c]*(-5*(1 + E^((2*I)*(c + d*x)))^(1/3) + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[2/3, 5/6, 11/6, -E^((2*I)*(c + d*x))])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(8*2^(1/3)*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(2/3)*(b*Sec[c + d*x])^(4/3)) + ((A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((3*(8*A + 5*C)*Cos[d*x]*Csc[c])/(8*d) + (3*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(4*d) + (3*Sec[c]*Sec[c + d*x]*(5*C*Sin[c] + 8*B*Sin[d*x]))/(20*d) + (6*B*Tan[c])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(b*Sec[c + d*x])^(4/3))","C",1
65,1,484,230,8.5500042,"\int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 i 2^{m+\frac{7}{3}} e^{-\frac{1}{3} i d (3 m+4) x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{4}{3}} \left(1+e^{2 i (c+d x)}\right)^{m+\frac{4}{3}} (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 (A+2 C) e^{\frac{1}{3} i (6 c+d (3 m+10) x)} \, _2F_1\left(\frac{m}{2}+\frac{5}{3},m+\frac{10}{3};\frac{m}{2}+\frac{8}{3};-e^{2 i (c+d x)}\right)}{3 m+10}+\frac{A e^{4 i c+\frac{1}{3} i d (3 m+16) x} \, _2F_1\left(\frac{m}{2}+\frac{8}{3},m+\frac{10}{3};\frac{1}{6} (3 m+22);-e^{2 i (c+d x)}\right)}{3 m+16}+\frac{A e^{\frac{1}{3} i d (3 m+4) x} \, _2F_1\left(m+\frac{10}{3},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};-e^{2 i (c+d x)}\right)}{3 m+4}+\frac{2 B e^{\frac{1}{3} i (3 c+d (3 m+7) x)} \, _2F_1\left(m+\frac{10}{3},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);-e^{2 i (c+d x)}\right)}{3 m+7}+\frac{2 B e^{\frac{1}{3} i (9 c+d (3 m+13) x)} \, _2F_1\left(m+\frac{10}{3},\frac{1}{6} (3 m+13);\frac{1}{6} (3 m+19);-e^{2 i (c+d x)}\right)}{3 m+13}\right)}{d \sec ^{\frac{10}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{3 b (A (3 m+7)+C (3 m+4)) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (-3 m-1);\frac{1}{6} (5-3 m);\cos ^2(c+d x)\right)}{d (3 m+1) (3 m+7) \sqrt{\sin ^2(c+d x)}}+\frac{3 b B \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^{m+1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (-3 m-4);\frac{1}{6} (2-3 m);\cos ^2(c+d x)\right)}{d (3 m+4) \sqrt{\sin ^2(c+d x)}}+\frac{3 b C \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^{m+2}(c+d x)}{d (3 m+7)}",1,"((-3*I)*2^(7/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(4/3 + m)*(1 + E^((2*I)*(c + d*x)))^(4/3 + m)*((2*(A + 2*C)*E^((I/3)*(6*c + d*(10 + 3*m)*x))*Hypergeometric2F1[5/3 + m/2, 10/3 + m, 8/3 + m/2, -E^((2*I)*(c + d*x))])/(10 + 3*m) + (A*E^((4*I)*c + (I/3)*d*(16 + 3*m)*x)*Hypergeometric2F1[8/3 + m/2, 10/3 + m, (22 + 3*m)/6, -E^((2*I)*(c + d*x))])/(16 + 3*m) + (A*E^((I/3)*d*(4 + 3*m)*x)*Hypergeometric2F1[10/3 + m, (4 + 3*m)/6, 5/3 + m/2, -E^((2*I)*(c + d*x))])/(4 + 3*m) + (2*B*E^((I/3)*(3*c + d*(7 + 3*m)*x))*Hypergeometric2F1[10/3 + m, (7 + 3*m)/6, (13 + 3*m)/6, -E^((2*I)*(c + d*x))])/(7 + 3*m) + (2*B*E^((I/3)*(9*c + d*(13 + 3*m)*x))*Hypergeometric2F1[10/3 + m, (13 + 3*m)/6, (19 + 3*m)/6, -E^((2*I)*(c + d*x))])/(13 + 3*m))*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^((I/3)*d*(4 + 3*m)*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(10/3))","C",0
66,1,547,227,7.0789124,"\int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 i 2^{m+\frac{5}{3}} e^{-\frac{1}{3} i d (3 m+2) x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{2}{3}} \left(1+e^{2 i (c+d x)}\right)^{m+\frac{2}{3}} (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{A e^{4 i c+\frac{1}{3} i d (3 m+14) x} \, _2F_1\left(\frac{m}{2}+\frac{7}{3},m+\frac{8}{3};\frac{1}{6} (3 m+20);-e^{2 i (c+d x)}\right)}{3 m+14}+\frac{A e^{\frac{1}{3} i d (3 m+2) x} \, _2F_1\left(m+\frac{8}{3},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);-e^{2 i (c+d x)}\right)}{3 m+2}+\frac{2 A e^{\frac{1}{3} i (6 c+d (3 m+8) x)} \, _2F_1\left(m+\frac{8}{3},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};-e^{2 i (c+d x)}\right)}{3 m+8}+\frac{2 B e^{\frac{1}{3} i (3 c+d (3 m+5) x)} \, _2F_1\left(m+\frac{8}{3},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);-e^{2 i (c+d x)}\right)}{3 m+5}+\frac{2 B e^{\frac{1}{3} i (9 c+d (3 m+11) x)} \, _2F_1\left(m+\frac{8}{3},\frac{1}{6} (3 m+11);\frac{1}{6} (3 m+17);-e^{2 i (c+d x)}\right)}{3 m+11}+\frac{4 C e^{\frac{1}{3} i (6 c+d (3 m+8) x)} \, _2F_1\left(m+\frac{8}{3},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};-e^{2 i (c+d x)}\right)}{3 m+8}\right)}{d \sec ^{\frac{8}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (A (3 m+5)+C (3 m+2)) \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (1-3 m);\frac{1}{6} (7-3 m);\cos ^2(c+d x)\right)}{d (1-3 m) (3 m+5) \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (-3 m-2);\frac{1}{6} (4-3 m);\cos ^2(c+d x)\right)}{d (3 m+2) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) (b \sec (c+d x))^{2/3} \sec ^{m+1}(c+d x)}{d (3 m+5)}",1,"((-3*I)*2^(5/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3 + m)*(1 + E^((2*I)*(c + d*x)))^(2/3 + m)*((A*E^((4*I)*c + (I/3)*d*(14 + 3*m)*x)*Hypergeometric2F1[7/3 + m/2, 8/3 + m, (20 + 3*m)/6, -E^((2*I)*(c + d*x))])/(14 + 3*m) + (A*E^((I/3)*d*(2 + 3*m)*x)*Hypergeometric2F1[8/3 + m, (2 + 3*m)/6, (8 + 3*m)/6, -E^((2*I)*(c + d*x))])/(2 + 3*m) + (2*B*E^((I/3)*(3*c + d*(5 + 3*m)*x))*Hypergeometric2F1[8/3 + m, (5 + 3*m)/6, (11 + 3*m)/6, -E^((2*I)*(c + d*x))])/(5 + 3*m) + (2*A*E^((I/3)*(6*c + d*(8 + 3*m)*x))*Hypergeometric2F1[8/3 + m, (8 + 3*m)/6, 7/3 + m/2, -E^((2*I)*(c + d*x))])/(8 + 3*m) + (4*C*E^((I/3)*(6*c + d*(8 + 3*m)*x))*Hypergeometric2F1[8/3 + m, (8 + 3*m)/6, 7/3 + m/2, -E^((2*I)*(c + d*x))])/(8 + 3*m) + (2*B*E^((I/3)*(9*c + d*(11 + 3*m)*x))*Hypergeometric2F1[8/3 + m, (11 + 3*m)/6, (17 + 3*m)/6, -E^((2*I)*(c + d*x))])/(11 + 3*m))*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^((I/3)*d*(2 + 3*m)*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(8/3))","C",0
67,1,494,225,7.3248319,"\int \sec ^m(c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{3 i 2^{m+\frac{4}{3}} e^{-\frac{1}{3} i d (3 m+1) x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{1}{3}} \left(1+e^{2 i (c+d x)}\right)^{m+\frac{1}{3}} \sqrt[3]{b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{e^{i c} \left(\frac{e^{\frac{1}{3} i (3 c+d (3 m+7) x)} \left(2 (3 m+13) (A+2 C) \, _2F_1\left(m+\frac{7}{3},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);-e^{2 i (c+d x)}\right)+A (3 m+7) e^{2 i (c+d x)} \, _2F_1\left(m+\frac{7}{3},\frac{1}{6} (3 m+13);\frac{1}{6} (3 m+19);-e^{2 i (c+d x)}\right)\right)}{(3 m+7) (3 m+13)}+\frac{2 B e^{\frac{1}{3} i d (3 m+4) x} \, _2F_1\left(m+\frac{7}{3},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};-e^{2 i (c+d x)}\right)}{3 m+4}\right)}{d}+\frac{A e^{\frac{1}{3} i x (3 d m+d)} \, _2F_1\left(m+\frac{7}{3},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);-e^{2 i (c+d x)}\right)}{3 d m+d}+\frac{2 B e^{\frac{1}{3} i (9 c+d (3 m+10) x)} \, _2F_1\left(\frac{m}{2}+\frac{5}{3},m+\frac{7}{3};\frac{m}{2}+\frac{8}{3};-e^{2 i (c+d x)}\right)}{d (3 m+10)}\right)}{\sec ^{\frac{7}{3}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (A (3 m+4)+3 C m+C) \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (2-3 m);\frac{1}{6} (8-3 m);\cos ^2(c+d x)\right)}{d (2-3 m) (3 m+4) \sqrt{\sin ^2(c+d x)}}+\frac{3 B \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (-3 m-1);\frac{1}{6} (5-3 m);\cos ^2(c+d x)\right)}{d (3 m+1) \sqrt{\sin ^2(c+d x)}}+\frac{3 C \sin (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^{m+1}(c+d x)}{d (3 m+4)}",1,"((-3*I)*2^(4/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3 + m)*(1 + E^((2*I)*(c + d*x)))^(1/3 + m)*((2*B*E^((I/3)*(9*c + d*(10 + 3*m)*x))*Hypergeometric2F1[5/3 + m/2, 7/3 + m, 8/3 + m/2, -E^((2*I)*(c + d*x))])/(d*(10 + 3*m)) + (A*E^((I/3)*(d + 3*d*m)*x)*Hypergeometric2F1[7/3 + m, (1 + 3*m)/6, (7 + 3*m)/6, -E^((2*I)*(c + d*x))])/(d + 3*d*m) + (E^(I*c)*((2*B*E^((I/3)*d*(4 + 3*m)*x)*Hypergeometric2F1[7/3 + m, (4 + 3*m)/6, 5/3 + m/2, -E^((2*I)*(c + d*x))])/(4 + 3*m) + (E^((I/3)*(3*c + d*(7 + 3*m)*x))*(2*(A + 2*C)*(13 + 3*m)*Hypergeometric2F1[7/3 + m, (7 + 3*m)/6, (13 + 3*m)/6, -E^((2*I)*(c + d*x))] + A*E^((2*I)*(c + d*x))*(7 + 3*m)*Hypergeometric2F1[7/3 + m, (13 + 3*m)/6, (19 + 3*m)/6, -E^((2*I)*(c + d*x))]))/((7 + 3*m)*(13 + 3*m))))/d)*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(E^((I/3)*d*(1 + 3*m)*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/3))","C",0
68,1,548,228,11.9978887,"\int \frac{\sec ^m(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^m*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","-\frac{3 i 2^{m+\frac{2}{3}} e^{-\frac{1}{3} i (3 c+d (3 m+2) x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{2}{3}} \left(1+e^{2 i (c+d x)}\right)^{m+\frac{2}{3}} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(e^{i c} (3 m-1) \left((3 m+2) e^{\frac{1}{3} i (3 c+d (3 m+5) x)} \left((3 m+5) e^{i (c+d x)} \left(A (3 m+8) e^{i (c+d x)} \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m+11);\frac{1}{6} (3 m+17);-e^{2 i (c+d x)}\right)+2 B (3 m+11) \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m+8);\frac{m}{2}+\frac{7}{3};-e^{2 i (c+d x)}\right)\right)+2 \left(9 m^2+57 m+88\right) (A+2 C) \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);-e^{2 i (c+d x)}\right)\right)+2 B \left(27 m^3+216 m^2+549 m+440\right) e^{\frac{1}{3} i d (3 m+2) x} \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);-e^{2 i (c+d x)}\right)\right)+A \left(81 m^4+702 m^3+2079 m^2+2418 m+880\right) e^{\frac{1}{3} i d (3 m-1) x} \, _2F_1\left(m+\frac{5}{3},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);-e^{2 i (c+d x)}\right)\right)}{d (3 m-1) (3 m+2) (3 m+5) (3 m+8) (3 m+11) \sec ^{\frac{5}{3}}(c+d x) \sqrt[3]{b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{3 (C (1-3 m)-A (3 m+2)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (4-3 m);\frac{1}{6} (10-3 m);\cos ^2(c+d x)\right)}{d (4-3 m) (3 m+2) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}-\frac{3 B \sin (c+d x) \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (1-3 m);\frac{1}{6} (7-3 m);\cos ^2(c+d x)\right)}{d (1-3 m) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}+\frac{3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+2) \sqrt[3]{b \sec (c+d x)}}",1,"((-3*I)*2^(2/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(2/3 + m)*(1 + E^((2*I)*(c + d*x)))^(2/3 + m)*(A*E^((I/3)*d*(-1 + 3*m)*x)*(880 + 2418*m + 2079*m^2 + 702*m^3 + 81*m^4)*Hypergeometric2F1[5/3 + m, (-1 + 3*m)/6, (5 + 3*m)/6, -E^((2*I)*(c + d*x))] + E^(I*c)*(-1 + 3*m)*(2*B*E^((I/3)*d*(2 + 3*m)*x)*(440 + 549*m + 216*m^2 + 27*m^3)*Hypergeometric2F1[5/3 + m, (2 + 3*m)/6, (8 + 3*m)/6, -E^((2*I)*(c + d*x))] + E^((I/3)*(3*c + d*(5 + 3*m)*x))*(2 + 3*m)*(2*(A + 2*C)*(88 + 57*m + 9*m^2)*Hypergeometric2F1[5/3 + m, (5 + 3*m)/6, (11 + 3*m)/6, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(5 + 3*m)*(2*B*(11 + 3*m)*Hypergeometric2F1[5/3 + m, (8 + 3*m)/6, 7/3 + m/2, -E^((2*I)*(c + d*x))] + A*E^(I*(c + d*x))*(8 + 3*m)*Hypergeometric2F1[5/3 + m, (11 + 3*m)/6, (17 + 3*m)/6, -E^((2*I)*(c + d*x))]))))*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^((I/3)*(3*c + d*(2 + 3*m)*x))*(-1 + 3*m)*(2 + 3*m)*(5 + 3*m)*(8 + 3*m)*(11 + 3*m)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/3)*(b*Sec[c + d*x])^(1/3))","C",0
69,1,545,226,10.8062614,"\int \frac{\sec ^m(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^m*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","-\frac{3 i 2^{m+\frac{1}{3}} e^{-\frac{1}{3} i (3 c+d (3 m+1) x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+\frac{1}{3}} \left(1+e^{2 i (c+d x)}\right)^{m+\frac{1}{3}} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left((3 m+10) \left(2 (3 m-2) e^{\frac{1}{3} i (3 c+d (3 m+1) x)} \left((3 m+1) e^{i (c+d x)} \left((3 m+7) (A+2 C) \, _2F_1\left(m+\frac{4}{3},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};-e^{2 i (c+d x)}\right)+B (3 m+4) e^{i (c+d x)} \, _2F_1\left(m+\frac{4}{3},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);-e^{2 i (c+d x)}\right)\right)+B \left(9 m^2+33 m+28\right) \, _2F_1\left(m+\frac{4}{3},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);-e^{2 i (c+d x)}\right)\right)+A \left(27 m^3+108 m^2+117 m+28\right) e^{\frac{1}{3} i d (3 m-2) x} \, _2F_1\left(m+\frac{4}{3},\frac{1}{6} (3 m-2);\frac{1}{6} (3 m+4);-e^{2 i (c+d x)}\right)\right)+A \left(81 m^4+270 m^3+135 m^2-150 m-56\right) e^{4 i c+\frac{1}{3} i d (3 m+10) x} \, _2F_1\left(\frac{m}{2}+\frac{5}{3},m+\frac{4}{3};\frac{m}{2}+\frac{8}{3};-e^{2 i (c+d x)}\right)\right)}{d (3 m-2) (3 m+1) (3 m+4) (3 m+7) (3 m+10) \sec ^{\frac{4}{3}}(c+d x) (b \sec (c+d x))^{2/3} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{3 (3 A m+A-C (2-3 m)) \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (5-3 m);\frac{1}{6} (11-3 m);\cos ^2(c+d x)\right)}{d (5-3 m) (3 m+1) \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}-\frac{3 B \sin (c+d x) \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (2-3 m);\frac{1}{6} (8-3 m);\cos ^2(c+d x)\right)}{d (2-3 m) \sqrt{\sin ^2(c+d x)} (b \sec (c+d x))^{2/3}}+\frac{3 C \sin (c+d x) \sec ^{m+1}(c+d x)}{d (3 m+1) (b \sec (c+d x))^{2/3}}",1,"((-3*I)*2^(1/3 + m)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/3 + m)*(1 + E^((2*I)*(c + d*x)))^(1/3 + m)*(A*E^((4*I)*c + (I/3)*d*(10 + 3*m)*x)*(-56 - 150*m + 135*m^2 + 270*m^3 + 81*m^4)*Hypergeometric2F1[5/3 + m/2, 4/3 + m, 8/3 + m/2, -E^((2*I)*(c + d*x))] + (10 + 3*m)*(A*E^((I/3)*d*(-2 + 3*m)*x)*(28 + 117*m + 108*m^2 + 27*m^3)*Hypergeometric2F1[4/3 + m, (-2 + 3*m)/6, (4 + 3*m)/6, -E^((2*I)*(c + d*x))] + 2*E^((I/3)*(3*c + d*(1 + 3*m)*x))*(-2 + 3*m)*(B*(28 + 33*m + 9*m^2)*Hypergeometric2F1[4/3 + m, (1 + 3*m)/6, (7 + 3*m)/6, -E^((2*I)*(c + d*x))] + E^(I*(c + d*x))*(1 + 3*m)*((A + 2*C)*(7 + 3*m)*Hypergeometric2F1[4/3 + m, (4 + 3*m)/6, 5/3 + m/2, -E^((2*I)*(c + d*x))] + B*E^(I*(c + d*x))*(4 + 3*m)*Hypergeometric2F1[4/3 + m, (7 + 3*m)/6, (13 + 3*m)/6, -E^((2*I)*(c + d*x))]))))*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^((I/3)*(3*c + d*(1 + 3*m)*x))*(-2 + 3*m)*(1 + 3*m)*(4 + 3*m)*(7 + 3*m)*(10 + 3*m)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(4/3)*(b*Sec[c + d*x])^(2/3))","C",0
70,-1,0,234,180.0011256,"\int \frac{\sec ^m(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^m*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\text{\$Aborted}","-\frac{3 (-3 A m+A+C (4-3 m)) \sin (c+d x) \sec ^{m-2}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (7-3 m);\frac{1}{6} (13-3 m);\cos ^2(c+d x)\right)}{b d (1-3 m) (7-3 m) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}-\frac{3 B \sin (c+d x) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (4-3 m);\frac{1}{6} (10-3 m);\cos ^2(c+d x)\right)}{b d (4-3 m) \sqrt{\sin ^2(c+d x)} \sqrt[3]{b \sec (c+d x)}}-\frac{3 C \sin (c+d x) \sec ^m(c+d x)}{b d (1-3 m) \sqrt[3]{b \sec (c+d x)}}",1,"$Aborted","F",-1
71,1,292,226,3.1724307,"\int \sec ^m(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{i 2^{m+n+1} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{m+n} \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{A \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,\frac{1}{2} (-m-n+2);\frac{1}{2} (m+n+2);-e^{2 i (c+d x)}\right)}{m+n}+\frac{2 B e^{i (c+d x)} \, _2F_1\left(1,\frac{1}{2} (-m-n+1);\frac{1}{2} (m+n+3);-e^{2 i (c+d x)}\right)}{m+n+1}+\frac{4 C e^{2 i (c+d x)} \, _2F_1\left(1,\frac{1}{2} (-m-n);\frac{1}{2} (m+n+4);-e^{2 i (c+d x)}\right)}{(m+n+2) \left(1+e^{2 i (c+d x)}\right)}\right)}{d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(A (m+n+1)+C (m+n)) \sin (c+d x) \sec ^{m-1}(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-n+1);\frac{1}{2} (-m-n+3);\cos ^2(c+d x)\right)}{d (-m-n+1) (m+n+1) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-m-n);\frac{1}{2} (-m-n+2);\cos ^2(c+d x)\right)}{d (m+n) \sqrt{\sin ^2(c+d x)}}+\frac{C \sin (c+d x) \sec ^{m+1}(c+d x) (b \sec (c+d x))^n}{d (m+n+1)}",1,"((-I)*2^(1 + m + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(m + n)*((4*C*E^((2*I)*(c + d*x))*Hypergeometric2F1[1, (-m - n)/2, (4 + m + n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*(c + d*x)))*(2 + m + n)) + (2*B*E^(I*(c + d*x))*Hypergeometric2F1[1, (1 - m - n)/2, (3 + m + n)/2, -E^((2*I)*(c + d*x))])/(1 + m + n) + (A*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1, (2 - m - n)/2, (2 + m + n)/2, -E^((2*I)*(c + d*x))])/(m + n))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","C",0
72,1,296,189,2.765857,"\int \sec ^2(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+3} e^{2 i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{A \left(1+e^{2 i (c+d x)}\right)^2 \, _2F_1\left(1,-\frac{n}{2};\frac{n+4}{2};-e^{2 i (c+d x)}\right)}{n+2}+\frac{2 B e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+5}{2};-e^{2 i (c+d x)}\right)}{n+3}+\frac{4 C e^{2 i (c+d x)} \, _2F_1\left(1,-\frac{n}{2}-1;\frac{n+6}{2};-e^{2 i (c+d x)}\right)}{n+4}\right)}{d \left(1+e^{2 i (c+d x)}\right)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A (n+3)+C (n+2)) \sin (c+d x) (b \sec (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\cos ^2(c+d x)\right)}{b d (n+1) (n+3) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) (b \sec (c+d x))^{n+2} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-2);-\frac{n}{2};\cos ^2(c+d x)\right)}{b^2 d (n+2) \sqrt{\sin ^2(c+d x)}}+\frac{C \tan (c+d x) (b \sec (c+d x))^{n+2}}{b^2 d (n+3)}",1,"((-I)*2^(3 + n)*E^((2*I)*(c + d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*((2*B*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*Hypergeometric2F1[1, (-1 - n)/2, (5 + n)/2, -E^((2*I)*(c + d*x))])/(3 + n) + (4*C*E^((2*I)*(c + d*x))*Hypergeometric2F1[1, -1 - n/2, (6 + n)/2, -E^((2*I)*(c + d*x))])/(4 + n) + (A*(1 + E^((2*I)*(c + d*x)))^2*Hypergeometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(c + d*x))])/(2 + n))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(1 + E^((2*I)*(c + d*x)))^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","C",0
73,1,285,182,2.2708162,"\int \sec (c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+2} e^{i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^n \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{A \, _2F_1\left(1,\frac{1-n}{2};\frac{n+3}{2};-e^{2 i (c+d x)}\right)}{n+1}+\frac{2 B e^{i (c+d x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+4}{2};-e^{2 i (c+d x)}\right)}{(n+2) \left(1+e^{2 i (c+d x)}\right)}+\frac{4 C e^{2 i (c+d x)} \, _2F_1\left(1,\frac{1}{2} (-n-1);\frac{n+5}{2};-e^{2 i (c+d x)}\right)}{(n+3) \left(1+e^{2 i (c+d x)}\right)^2}\right)}{d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A (n+2)+C (n+1)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(c+d x)\right)}{d n (n+2) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) (b \sec (c+d x))^{n+1} \, _2F_1\left(\frac{1}{2},\frac{1}{2} (-n-1);\frac{1-n}{2};\cos ^2(c+d x)\right)}{b d (n+1) \sqrt{\sin ^2(c+d x)}}+\frac{C \tan (c+d x) (b \sec (c+d x))^{n+1}}{b d (n+2)}",1,"((-I)*2^(2 + n)*E^(I*(c + d*x))*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^n*((4*C*E^((2*I)*(c + d*x))*Hypergeometric2F1[1, (-1 - n)/2, (5 + n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*(c + d*x)))^2*(3 + n)) + (A*Hypergeometric2F1[1, (1 - n)/2, (3 + n)/2, -E^((2*I)*(c + d*x))])/(1 + n) + (2*B*E^(I*(c + d*x))*Hypergeometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(c + d*x))])/((1 + E^((2*I)*(c + d*x)))*(2 + n)))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","C",0
74,1,292,175,1.7687199,"\int (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+1} e^{-i (c+d x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+1} \sec ^{-n}(c+d x) (b \sec (c+d x))^n \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left((n+1) \left(A (n+2) \left(1+e^{2 i (c+d x)}\right)^2 \, _2F_1\left(1,1-\frac{n}{2};\frac{n+2}{2};-e^{2 i (c+d x)}\right)+4 C n e^{2 i (c+d x)} \, _2F_1\left(1,-\frac{n}{2};\frac{n+4}{2};-e^{2 i (c+d x)}\right)\right)+2 B n (n+2) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \, _2F_1\left(1,\frac{1-n}{2};\frac{n+3}{2};-e^{2 i (c+d x)}\right)\right)}{d n (n+1) (n+2) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{b (A n+A+C n) \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right)}{d (1-n) (n+1) \sqrt{\sin ^2(c+d x)}}+\frac{B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},-\frac{n}{2};\frac{2-n}{2};\cos ^2(c+d x)\right)}{d n \sqrt{\sin ^2(c+d x)}}+\frac{C \tan (c+d x) (b \sec (c+d x))^n}{d (n+1)}",1,"((-I)*2^(1 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1 + n)*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(2*B*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*n*(2 + n)*Hypergeometric2F1[1, (1 - n)/2, (3 + n)/2, -E^((2*I)*(c + d*x))] + (1 + n)*(A*(1 + E^((2*I)*(c + d*x)))^2*(2 + n)*Hypergeometric2F1[1, 1 - n/2, (2 + n)/2, -E^((2*I)*(c + d*x))] + 4*C*E^((2*I)*(c + d*x))*n*Hypergeometric2F1[1, -1/2*n, (4 + n)/2, -E^((2*I)*(c + d*x))]))*(b*Sec[c + d*x])^n)/(d*E^(I*(c + d*x))*n*(1 + n)*(2 + n)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^n)","C",0
75,1,161,191,0.3836834,"\int \cos (c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{-\tan ^2(c+d x)} (b \sec (c+d x))^n \left(A n (n+1) \cos (c+d x) \cot (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\sec ^2(c+d x)\right)+(n-1) \csc (c+d x) \left(B (n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(c+d x)\right)+C n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(c+d x)\right)\right)\right)}{d (n-1) n (n+1)}","\frac{b^2 (C (1-n)-A n) \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right)}{d (2-n) n \sqrt{\sin ^2(c+d x)}}-\frac{b B \sin (c+d x) (b \sec (c+d x))^{n-1} \, _2F_1\left(\frac{1}{2},\frac{1-n}{2};\frac{3-n}{2};\cos ^2(c+d x)\right)}{d (1-n) \sqrt{\sin ^2(c+d x)}}+\frac{b C \tan (c+d x) (b \sec (c+d x))^{n-1}}{d n}",1,"((A*n*(1 + n)*Cos[c + d*x]*Cot[c + d*x]*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Sec[c + d*x]^2] + (-1 + n)*Csc[c + d*x]*(B*(1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[c + d*x]^2] + C*n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[c + d*x]^2]))*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(-1 + n)*n*(1 + n))","A",1
76,1,155,208,0.2915062,"\int \cos ^2(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^n \left(A (n-1) n \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\sec ^2(c+d x)\right)+(n-2) \left(B n \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\sec ^2(c+d x)\right)+C (n-1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(c+d x)\right)\right)\right)}{d (n-2) (n-1) n}","-\frac{b^3 (A (1-n)+C (2-n)) \sin (c+d x) (b \sec (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\cos ^2(c+d x)\right)}{d (1-n) (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 B \sin (c+d x) (b \sec (c+d x))^{n-2} \, _2F_1\left(\frac{1}{2},\frac{2-n}{2};\frac{4-n}{2};\cos ^2(c+d x)\right)}{d (2-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^2 C \tan (c+d x) (b \sec (c+d x))^{n-2}}{d (1-n)}",1,"(Cot[c + d*x]*(A*(-1 + n)*n*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Sec[c + d*x]^2] + (-2 + n)*(B*n*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Sec[c + d*x]^2] + C*(-1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[c + d*x]^2]))*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(-2 + n)*(-1 + n)*n)","A",1
77,1,168,208,0.657293,"\int \cos ^3(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{n-1} \left(A \left(n^2-3 n+2\right) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-3}{2};\frac{n-1}{2};\sec ^2(c+d x)\right)+(n-3) \left(B (n-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\sec ^2(c+d x)\right)+C (n-2) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\sec ^2(c+d x)\right)\right)\right)}{d (n-3) (n-2) (n-1)}","-\frac{b^4 (A (2-n)+C (3-n)) \sin (c+d x) (b \sec (c+d x))^{n-4} \, _2F_1\left(\frac{1}{2},\frac{4-n}{2};\frac{6-n}{2};\cos ^2(c+d x)\right)}{d (2-n) (4-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^3 B \sin (c+d x) (b \sec (c+d x))^{n-3} \, _2F_1\left(\frac{1}{2},\frac{3-n}{2};\frac{5-n}{2};\cos ^2(c+d x)\right)}{d (3-n) \sqrt{\sin ^2(c+d x)}}-\frac{b^3 C \tan (c+d x) (b \sec (c+d x))^{n-3}}{d (2-n)}",1,"(b*Cot[c + d*x]*(A*(2 - 3*n + n^2)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (-3 + n)/2, (-1 + n)/2, Sec[c + d*x]^2] + (-3 + n)*(B*(-1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Sec[c + d*x]^2] + C*(-2 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Sec[c + d*x]^2]))*(b*Sec[c + d*x])^(-1 + n)*Sqrt[-Tan[c + d*x]^2])/(d*(-3 + n)*(-2 + n)*(-1 + n))","A",1
78,-1,0,223,180.001151,"\int \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{\$Aborted}","\frac{2 (A (2 n+7)+C (2 n+5)) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (-2 n-3);\frac{1}{4} (1-2 n);\cos ^2(c+d x)\right)}{d (2 n+3) (2 n+7) \sqrt{\sin ^2(c+d x)}}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (-2 n-5);\frac{1}{4} (-2 n-1);\cos ^2(c+d x)\right)}{d (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+7)}",1,"$Aborted","F",-1
79,1,487,223,7.5744985,"\int \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+\frac{7}{2}} e^{-\frac{1}{2} i d (2 n+3) x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{3}{2}} \left(1+e^{2 i (c+d x)}\right)^{n+\frac{3}{2}} \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(e^{2 i c} \left(\frac{2 (A+2 C) e^{\frac{1}{2} i d (2 n+7) x} \, _2F_1\left(n+\frac{7}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);-e^{2 i (c+d x)}\right)}{2 n+7}+\frac{A e^{\frac{1}{2} i (4 c+d (2 n+11) x)} \, _2F_1\left(n+\frac{7}{2},\frac{1}{4} (2 n+11);\frac{1}{4} (2 n+15);-e^{2 i (c+d x)}\right)}{2 n+11}+\frac{2 B e^{\frac{1}{2} i (2 c+d (2 n+9) x)} \, _2F_1\left(n+\frac{7}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);-e^{2 i (c+d x)}\right)}{2 n+9}\right)+\frac{A e^{\frac{1}{2} i d (2 n+3) x} \, _2F_1\left(n+\frac{7}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);-e^{2 i (c+d x)}\right)}{2 n+3}+\frac{2 B e^{\frac{1}{2} i (2 c+d (2 n+5) x)} \, _2F_1\left(n+\frac{7}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);-e^{2 i (c+d x)}\right)}{2 n+5}\right)}{d (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 (A (2 n+5)+C (2 n+3)) \sin (c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (-2 n-1);\frac{1}{4} (3-2 n);\cos ^2(c+d x)\right)}{d (2 n+1) (2 n+5) \sqrt{\sin ^2(c+d x)}}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (-2 n-3);\frac{1}{4} (1-2 n);\cos ^2(c+d x)\right)}{d (2 n+3) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+5)}",1,"((-I)*2^(7/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(3/2 + n)*(1 + E^((2*I)*(c + d*x)))^(3/2 + n)*((A*E^((I/2)*d*(3 + 2*n)*x)*Hypergeometric2F1[7/2 + n, (3 + 2*n)/4, (7 + 2*n)/4, -E^((2*I)*(c + d*x))])/(3 + 2*n) + (2*B*E^((I/2)*(2*c + d*(5 + 2*n)*x))*Hypergeometric2F1[7/2 + n, (5 + 2*n)/4, (9 + 2*n)/4, -E^((2*I)*(c + d*x))])/(5 + 2*n) + E^((2*I)*c)*((2*(A + 2*C)*E^((I/2)*d*(7 + 2*n)*x)*Hypergeometric2F1[7/2 + n, (7 + 2*n)/4, (11 + 2*n)/4, -E^((2*I)*(c + d*x))])/(7 + 2*n) + (2*B*E^((I/2)*(2*c + d*(9 + 2*n)*x))*Hypergeometric2F1[7/2 + n, (9 + 2*n)/4, (13 + 2*n)/4, -E^((2*I)*(c + d*x))])/(9 + 2*n) + (A*E^((I/2)*(4*c + d*(11 + 2*n)*x))*Hypergeometric2F1[7/2 + n, (11 + 2*n)/4, (15 + 2*n)/4, -E^((2*I)*(c + d*x))])/(11 + 2*n)))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^((I/2)*d*(3 + 2*n)*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
80,1,492,221,8.0118028,"\int \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{i 2^{n+\frac{5}{2}} e^{-\frac{1}{2} i d (2 n+1) x} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{1}{2}} \left(1+e^{2 i (c+d x)}\right)^{n+\frac{1}{2}} \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{e^{i c} \left(e^{i c} \left(\frac{2 (A+2 C) e^{\frac{1}{2} i d (2 n+5) x} \, _2F_1\left(n+\frac{5}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);-e^{2 i (c+d x)}\right)}{2 n+5}+\frac{A e^{\frac{1}{2} i (4 c+d (2 n+9) x)} \, _2F_1\left(n+\frac{5}{2},\frac{1}{4} (2 n+9);\frac{1}{4} (2 n+13);-e^{2 i (c+d x)}\right)}{2 n+9}+\frac{2 B e^{\frac{1}{2} i (2 c+d (2 n+7) x)} \, _2F_1\left(n+\frac{5}{2},\frac{1}{4} (2 n+7);\frac{1}{4} (2 n+11);-e^{2 i (c+d x)}\right)}{2 n+7}\right)+\frac{2 B e^{\frac{1}{2} i d (2 n+3) x} \, _2F_1\left(n+\frac{5}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);-e^{2 i (c+d x)}\right)}{2 n+3}\right)}{d}+\frac{A e^{\frac{1}{2} i x (2 d n+d)} \, _2F_1\left(n+\frac{5}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);-e^{2 i (c+d x)}\right)}{2 d n+d}\right)}{A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C}","-\frac{2 (A (2 n+3)+2 C n+C) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (1-2 n);\frac{1}{4} (5-2 n);\cos ^2(c+d x)\right)}{d (1-2 n) (2 n+3) \sqrt{\sin ^2(c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (-2 n-1);\frac{1}{4} (3-2 n);\cos ^2(c+d x)\right)}{d (2 n+1) \sqrt{\sin ^2(c+d x)}}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n}{d (2 n+3)}",1,"((-I)*2^(5/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + n)*(1 + E^((2*I)*(c + d*x)))^(1/2 + n)*((A*E^((I/2)*(d + 2*d*n)*x)*Hypergeometric2F1[5/2 + n, (1 + 2*n)/4, (5 + 2*n)/4, -E^((2*I)*(c + d*x))])/(d + 2*d*n) + (E^(I*c)*((2*B*E^((I/2)*d*(3 + 2*n)*x)*Hypergeometric2F1[5/2 + n, (3 + 2*n)/4, (7 + 2*n)/4, -E^((2*I)*(c + d*x))])/(3 + 2*n) + E^(I*c)*((2*(A + 2*C)*E^((I/2)*d*(5 + 2*n)*x)*Hypergeometric2F1[5/2 + n, (5 + 2*n)/4, (9 + 2*n)/4, -E^((2*I)*(c + d*x))])/(5 + 2*n) + (2*B*E^((I/2)*(2*c + d*(7 + 2*n)*x))*Hypergeometric2F1[5/2 + n, (7 + 2*n)/4, (11 + 2*n)/4, -E^((2*I)*(c + d*x))])/(7 + 2*n) + (A*E^((I/2)*(4*c + d*(9 + 2*n)*x))*Hypergeometric2F1[5/2 + n, (9 + 2*n)/4, (13 + 2*n)/4, -E^((2*I)*(c + d*x))])/(9 + 2*n))))/d)*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(E^((I/2)*d*(1 + 2*n)*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
81,-1,0,222,180.0012714,"\int \frac{(b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\text{\$Aborted}","-\frac{2 (2 A n+A-C (1-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (3-2 n);\frac{1}{4} (7-2 n);\cos ^2(c+d x)\right)}{d (3-2 n) (2 n+1) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (1-2 n);\frac{1}{4} (5-2 n);\cos ^2(c+d x)\right)}{d (1-2 n) \sqrt{\sin ^2(c+d x)} \sqrt{\sec (c+d x)}}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (b \sec (c+d x))^n}{d (2 n+1)}",1,"$Aborted","F",-1
82,1,502,221,9.0442195,"\int \frac{(b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","-\frac{i 2^{n+\frac{1}{2}} e^{-\frac{1}{2} i (4 c+d (2 n+1) x)} \left(\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right)^{n+\frac{1}{2}} \left(1+e^{2 i (c+d x)}\right)^{n+\frac{1}{2}} \sec ^{-n-2}(c+d x) (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(e^{2 i c} \left(\frac{e^{\frac{1}{2} i (2 c+d (2 n+3) x)} \left(A (2 n+3) e^{i (c+d x)} \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);-e^{2 i (c+d x)}\right)+2 B (2 n+5) \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);-e^{2 i (c+d x)}\right)\right)}{d (2 n+3) (2 n+5)}+\frac{2 (A+2 C) e^{\frac{1}{2} i d (2 n+1) x} \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);-e^{2 i (c+d x)}\right)}{2 d n+d}\right)+\frac{A e^{\frac{1}{2} i d (2 n-3) x} \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);-e^{2 i (c+d x)}\right)}{d (2 n-3)}+\frac{2 B e^{\frac{1}{2} i (2 c+d (2 n-1) x)} \, _2F_1\left(n+\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);-e^{2 i (c+d x)}\right)}{d (2 n-1)}\right)}{A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C}","-\frac{2 (-2 A n+A+C (3-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right)}{d (1-2 n) (5-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (3-2 n);\frac{1}{4} (7-2 n);\cos ^2(c+d x)\right)}{d (3-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (1-2 n) \sqrt{\sec (c+d x)}}",1,"((-I)*2^(1/2 + n)*(E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x))))^(1/2 + n)*(1 + E^((2*I)*(c + d*x)))^(1/2 + n)*((A*E^((I/2)*d*(-3 + 2*n)*x)*Hypergeometric2F1[1/2 + n, (-3 + 2*n)/4, (1 + 2*n)/4, -E^((2*I)*(c + d*x))])/(d*(-3 + 2*n)) + (2*B*E^((I/2)*(2*c + d*(-1 + 2*n)*x))*Hypergeometric2F1[1/2 + n, (-1 + 2*n)/4, (3 + 2*n)/4, -E^((2*I)*(c + d*x))])/(d*(-1 + 2*n)) + E^((2*I)*c)*((2*(A + 2*C)*E^((I/2)*d*(1 + 2*n)*x)*Hypergeometric2F1[1/2 + n, (1 + 2*n)/4, (5 + 2*n)/4, -E^((2*I)*(c + d*x))])/(d + 2*d*n) + (E^((I/2)*(2*c + d*(3 + 2*n)*x))*(2*B*(5 + 2*n)*Hypergeometric2F1[1/2 + n, (3 + 2*n)/4, (7 + 2*n)/4, -E^((2*I)*(c + d*x))] + A*E^(I*(c + d*x))*(3 + 2*n)*Hypergeometric2F1[1/2 + n, (5 + 2*n)/4, (9 + 2*n)/4, -E^((2*I)*(c + d*x))]))/(d*(3 + 2*n)*(5 + 2*n))))*Sec[c + d*x]^(-2 - n)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(E^((I/2)*(4*c + d*(1 + 2*n)*x))*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
83,-1,0,223,180.0005493,"\int \frac{(b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\text{\$Aborted}","-\frac{2 (A (3-2 n)+C (5-2 n)) \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (7-2 n);\frac{1}{4} (11-2 n);\cos ^2(c+d x)\right)}{d (3-2 n) (7-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{7}{2}}(c+d x)}-\frac{2 B \sin (c+d x) (b \sec (c+d x))^n \, _2F_1\left(\frac{1}{2},\frac{1}{4} (5-2 n);\frac{1}{4} (9-2 n);\cos ^2(c+d x)\right)}{d (5-2 n) \sqrt{\sin ^2(c+d x)} \sec ^{\frac{5}{2}}(c+d x)}-\frac{2 C \sin (c+d x) (b \sec (c+d x))^n}{d (3-2 n) \sec ^{\frac{3}{2}}(c+d x)}",1,"$Aborted","F",-1
84,1,93,140,0.7542641,"\int \sec ^3(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a \left(15 (4 A+3 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 (A+2 C) \tan ^2(c+d x)+15 (A+C)+3 C \tan ^4(c+d x)\right)+15 (4 A+3 C) \sec (c+d x)+30 C \sec ^3(c+d x)\right)\right)}{120 d}","\frac{a (5 A+4 C) \tan ^3(c+d x)}{15 d}+\frac{a (5 A+4 C) \tan (c+d x)}{5 d}+\frac{a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^4(c+d x)}{5 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(15*(4*A + 3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(4*A + 3*C)*Sec[c + d*x] + 30*C*Sec[c + d*x]^3 + 8*(15*(A + C) + 5*(A + 2*C)*Tan[c + d*x]^2 + 3*C*Tan[c + d*x]^4))))/(120*d)","A",1
85,1,75,117,0.4427547,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a \left(3 (4 A+3 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (4 A+3 C) \sec (c+d x)+24 (A+C)+8 C \tan ^2(c+d x)+6 C \sec ^3(c+d x)\right)\right)}{24 d}","\frac{a (3 A+2 C) \tan (c+d x)}{3 d}+\frac{a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(3*(4*A + 3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(A + C) + 3*(4*A + 3*C)*Sec[c + d*x] + 6*C*Sec[c + d*x]^3 + 8*C*Tan[c + d*x]^2)))/(24*d)","A",1
86,1,56,86,0.2814556,"\int \sec (c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a \left(3 (2 A+C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(6 (A+C)+2 C \tan ^2(c+d x)+3 C \sec (c+d x)\right)\right)}{6 d}","\frac{a (3 A+2 C) \tan (c+d x)}{3 d}+\frac{a (2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a C \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*(3*(2*A + C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(A + C) + 3*C*Sec[c + d*x] + 2*C*Tan[c + d*x]^2)))/(6*d)","A",1
87,1,67,58,0.0278929,"\int (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a (2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a A x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tan (c+d x) \sec (c+d x)}{2 d}",1,"a*A*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/(2*d) + (a*C*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
88,1,54,42,0.0297803,"\int \cos (c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a A x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a A \sin (c+d x)}{d}+a A x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*A*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Cos[d*x]*Sin[c])/d + (a*A*Cos[c]*Sin[d*x])/d + (a*C*Tan[c + d*x])/d","A",1
89,1,52,58,0.0850263,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a \left(4 A \sin (c+d x)+A \sin (2 (c+d x))+2 A c+2 A d x+4 C \tanh ^{-1}(\sin (c+d x))+4 C d x\right)}{4 d}","\frac{a A \sin (c+d x)}{d}+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+2 C)+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*(2*A*c + 2*A*d*x + 4*C*d*x + 4*C*ArcTanh[Sin[c + d*x]] + 4*A*Sin[c + d*x] + A*Sin[2*(c + d*x)]))/(4*d)","A",1
90,1,59,77,0.13442,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a (3 (3 A+4 C) \sin (c+d x)+3 A \sin (2 (c+d x))+A \sin (3 (c+d x))+6 A c+6 A d x+12 C d x)}{12 d}","\frac{a (2 A+3 C) \sin (c+d x)}{3 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+2 C)",1,"(a*(6*A*c + 6*A*d*x + 12*C*d*x + 3*(3*A + 4*C)*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)","A",1
91,1,77,95,0.2311695,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a (24 (3 A+4 C) \sin (c+d x)+24 (A+C) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+36 A c+36 A d x+48 c C+48 C d x)}{96 d}","\frac{a (A+C) \sin (c+d x)}{d}+\frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a A \sin ^3(c+d x)}{3 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C)",1,"(a*(36*A*c + 48*c*C + 36*A*d*x + 48*C*d*x + 24*(3*A + 4*C)*Sin[c + d*x] + 24*(A + C)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
92,1,86,131,0.2858375,"\int \cos ^5(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a \left(-160 (2 A+C) \sin ^3(c+d x)+480 (A+C) \sin (c+d x)+15 (4 (3 A+4 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+A \sin (4 (c+d x)))+96 A \sin ^5(c+d x)\right)}{480 d}","-\frac{a (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \sin (c+d x)}{5 d}+\frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C)",1,"(a*(480*(A + C)*Sin[c + d*x] - 160*(2*A + C)*Sin[c + d*x]^3 + 96*A*Sin[c + d*x]^5 + 15*(4*(3*A + 4*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + A*Sin[4*(c + d*x)])))/(480*d)","A",1
93,1,321,172,1.9422913,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(240 (4 A+3 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-120 (3 A+C) \sin (2 c+d x)+120 A \sin (c+2 d x)+120 A \sin (3 c+2 d x)+440 A \sin (2 c+3 d x)-60 A \sin (4 c+3 d x)+60 A \sin (3 c+4 d x)+60 A \sin (5 c+4 d x)+100 A \sin (4 c+5 d x)+40 (16 A+15 C) \sin (d x)+210 C \sin (c+2 d x)+210 C \sin (3 c+2 d x)+360 C \sin (2 c+3 d x)+45 C \sin (3 c+4 d x)+45 C \sin (5 c+4 d x)+72 C \sin (4 c+5 d x))\right)}{1920 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a^2 (4 A+3 C) \tan (c+d x)}{3 d}+\frac{a^2 (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^2 (4 A+3 C) \tan (c+d x) \sec (c+d x)}{12 d}+\frac{(10 A+3 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{30 d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^2}{5 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{10 a d}",1,"-1/1920*(a^2*(1 + Cos[c + d*x])^2*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^4*Sec[c + d*x]^5*(240*(4*A + 3*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(40*(16*A + 15*C)*Sin[d*x] - 120*(3*A + C)*Sin[2*c + d*x] + 120*A*Sin[c + 2*d*x] + 210*C*Sin[c + 2*d*x] + 120*A*Sin[3*c + 2*d*x] + 210*C*Sin[3*c + 2*d*x] + 440*A*Sin[2*c + 3*d*x] + 360*C*Sin[2*c + 3*d*x] - 60*A*Sin[4*c + 3*d*x] + 60*A*Sin[3*c + 4*d*x] + 45*C*Sin[3*c + 4*d*x] + 60*A*Sin[5*c + 4*d*x] + 45*C*Sin[5*c + 4*d*x] + 100*A*Sin[4*c + 5*d*x] + 72*C*Sin[4*c + 5*d*x])))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
94,1,291,132,1.5183197,"\int \sec (c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(24 (12 A+7 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-48 (3 A+2 C) \sin (c)+12 A \sin (2 c+d x)+144 A \sin (c+2 d x)-48 A \sin (3 c+2 d x)+12 A \sin (2 c+3 d x)+12 A \sin (4 c+3 d x)+48 A \sin (3 c+4 d x)+3 (4 A+15 C) \sin (d x)+45 C \sin (2 c+d x)+128 C \sin (c+2 d x)+21 C \sin (2 c+3 d x)+21 C \sin (4 c+3 d x)+32 C \sin (3 c+4 d x))\right)}{384 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a^2 (12 A+7 C) \tan (c+d x)}{6 d}+\frac{a^2 (12 A+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (12 A+7 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d}-\frac{C \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d}",1,"-1/384*(a^2*(1 + Cos[c + d*x])^2*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^4*Sec[c + d*x]^4*(24*(12*A + 7*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-48*(3*A + 2*C)*Sin[c] + 3*(4*A + 15*C)*Sin[d*x] + 12*A*Sin[2*c + d*x] + 45*C*Sin[2*c + d*x] + 144*A*Sin[c + 2*d*x] + 128*C*Sin[c + 2*d*x] - 48*A*Sin[3*c + 2*d*x] + 12*A*Sin[2*c + 3*d*x] + 21*C*Sin[2*c + 3*d*x] + 12*A*Sin[4*c + 3*d*x] + 21*C*Sin[4*c + 3*d*x] + 48*A*Sin[3*c + 4*d*x] + 32*C*Sin[3*c + 4*d*x])))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
95,1,1090,96,6.574793,"\int (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{(-2 A-C) \cos ^4(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(2 A+C) \cos ^4(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A x \cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C)}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{6 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(7 C \cos \left(\frac{c}{2}\right)-5 C \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(-7 C \cos \left(\frac{c}{2}\right)-5 C \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{C \cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{C \cos ^4(c+d x) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{a^2 (A+C) \tan (c+d x)}{d}+\frac{a^2 (2 A+C) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 A x+\frac{C \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{3 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(A*x*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(2*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((-2*A - C)*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((2*A + C)*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(12*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(7*C*Cos[c/2] - 5*C*Sin[c/2]))/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2]))/(6*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (C*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(12*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(-7*C*Cos[c/2] - 5*C*Sin[c/2]))/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2]))/(6*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
96,1,330,112,2.545925,"\int \cos (c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \cos ^4(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(A+C \sec ^2(c+d x)\right) \left(-\frac{2 (2 A+3 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (2 A+3 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 A \sin (c) \cos (d x)}{d}+\frac{4 A \cos (c) \sin (d x)}{d}+8 A x+\frac{8 C \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{8 C \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{8 (A \cos (2 (c+d x))+A+2 C)}","-\frac{a^2 (2 A-3 C) \tan (c+d x)}{2 d}+\frac{a^2 (2 A+3 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(2 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}+2 a^2 A x+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^2}{d}",1,"(a^2*Cos[c + d*x]^4*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(8*A*x - (2*(2*A + 3*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(2*A + 3*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*A*Cos[d*x]*Sin[c])/d + (4*A*Cos[c]*Sin[d*x])/d + C/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (8*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - C/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (8*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(8*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
97,1,292,119,1.2501045,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","-\frac{a^2 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(4 \cos (d x) \left(3 A d x-4 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 C d x\right)+4 \cos (2 c+d x) \left(3 A d x-4 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 C d x\right)+A \sin (2 c+d x)+8 A \sin (c+2 d x)+8 A \sin (3 c+2 d x)+A \sin (2 c+3 d x)+A \sin (4 c+3 d x)+A \sin (d x)+16 C \sin (d x)\right)}{16 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan \left(\frac{1}{2} (c+d x)\right)+1\right)}","\frac{a^2 (3 A-2 C) \sin (c+d x)}{2 d}-\frac{(A-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}+\frac{1}{2} a^2 x (3 A+2 C)+\frac{2 a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}",1,"-1/16*(a^2*Sec[(c + d*x)/2]^2*(4*Cos[d*x]*(3*A*d*x + 2*C*d*x - 4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4*Cos[2*c + d*x]*(3*A*d*x + 2*C*d*x - 4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + A*Sin[d*x] + 16*C*Sin[d*x] + A*Sin[2*c + d*x] + 8*A*Sin[c + 2*d*x] + 8*A*Sin[3*c + 2*d*x] + A*Sin[2*c + 3*d*x] + A*Sin[4*c + 3*d*x]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(-1 + Tan[(c + d*x)/2])*(1 + Tan[(c + d*x)/2]))","B",1
98,1,109,110,0.2197504,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \left(3 (7 A+4 C) \sin (c+d x)+6 A \sin (2 (c+d x))+A \sin (3 (c+d x))+12 A d x-12 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 C d x\right)}{12 d}","\frac{a^2 (A+C) \sin (c+d x)}{d}+\frac{A \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{3 d}+a^2 x (A+2 C)+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^2*(12*A*d*x + 24*C*d*x - 12*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(7*A + 4*C)*Sin[c + d*x] + 6*A*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)","A",1
99,1,73,136,0.2419572,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 (48 (3 A+4 C) \sin (c+d x)+24 (2 A+C) \sin (2 (c+d x))+16 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+84 A d x+144 C d x)}{96 d}","\frac{a^2 (7 A+12 C) \sin (c+d x)}{6 d}+\frac{a^2 (7 A+12 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (7 A+12 C)+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^2}{4 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{6 d}",1,"(a^2*(84*A*d*x + 144*C*d*x + 48*(3*A + 4*C)*Sin[c + d*x] + 24*(2*A + C)*Sin[2*(c + d*x)] + 16*A*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
100,1,97,169,0.4124563,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 (30 (11 A+14 C) \sin (c+d x)+120 (A+C) \sin (2 (c+d x))+45 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+3 A \sin (5 (c+d x))+120 A c+180 A d x+20 C \sin (3 (c+d x))+240 C d x)}{240 d}","\frac{a^2 (18 A+25 C) \sin (c+d x)}{15 d}+\frac{a^2 (9 A+10 C) \sin (c+d x) \cos ^2(c+d x)}{30 d}+\frac{a^2 (3 A+4 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{10 d}+\frac{1}{4} a^2 x (3 A+4 C)+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}",1,"(a^2*(120*A*c + 180*A*d*x + 240*C*d*x + 30*(11*A + 14*C)*Sin[c + d*x] + 120*(A + C)*Sin[2*(c + d*x)] + 45*A*Sin[3*(c + d*x)] + 20*C*Sin[3*(c + d*x)] + 15*A*Sin[4*(c + d*x)] + 3*A*Sin[5*(c + d*x)]))/(240*d)","A",1
101,1,123,194,0.6943078,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 (240 (5 A+6 C) \sin (c+d x)+15 (31 A+32 C) \sin (2 (c+d x))+200 A \sin (3 (c+d x))+75 A \sin (4 (c+d x))+24 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+240 A c+660 A d x+160 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+840 C d x)}{960 d}","-\frac{2 a^2 (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{5 d}+\frac{a^2 (9 A+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (11 A+14 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{15 d}+\frac{1}{16} a^2 x (11 A+14 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}",1,"(a^2*(240*A*c + 660*A*d*x + 840*C*d*x + 240*(5*A + 6*C)*Sin[c + d*x] + 15*(31*A + 32*C)*Sin[2*(c + d*x)] + 200*A*Sin[3*(c + d*x)] + 160*C*Sin[3*(c + d*x)] + 75*A*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 24*A*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
102,1,387,197,3.2414166,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(480 (30 A+23 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-160 (45 A+34 C) \sin (c)+1140 A \sin (2 c+d x)+8160 A \sin (c+2 d x)-2640 A \sin (3 c+2 d x)+1590 A \sin (2 c+3 d x)+1590 A \sin (4 c+3 d x)+4080 A \sin (3 c+4 d x)-240 A \sin (5 c+4 d x)+450 A \sin (4 c+5 d x)+450 A \sin (6 c+5 d x)+720 A \sin (5 c+6 d x)+30 (38 A+75 C) \sin (d x)+2250 C \sin (2 c+d x)+7680 C \sin (c+2 d x)-480 C \sin (3 c+2 d x)+1955 C \sin (2 c+3 d x)+1955 C \sin (4 c+3 d x)+3264 C \sin (3 c+4 d x)+345 C \sin (4 c+5 d x)+345 C \sin (6 c+5 d x)+544 C \sin (5 c+6 d x))\right)}{30720 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a^3 (30 A+23 C) \tan ^3(c+d x)}{120 d}+\frac{a^3 (30 A+23 C) \tan (c+d x)}{10 d}+\frac{a^3 (30 A+23 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 a^3 (30 A+23 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{(30 A+7 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{120 d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{10 a d}",1,"-1/30720*(a^3*(1 + Cos[c + d*x])^3*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^6*Sec[c + d*x]^6*(480*(30*A + 23*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-160*(45*A + 34*C)*Sin[c] + 30*(38*A + 75*C)*Sin[d*x] + 1140*A*Sin[2*c + d*x] + 2250*C*Sin[2*c + d*x] + 8160*A*Sin[c + 2*d*x] + 7680*C*Sin[c + 2*d*x] - 2640*A*Sin[3*c + 2*d*x] - 480*C*Sin[3*c + 2*d*x] + 1590*A*Sin[2*c + 3*d*x] + 1955*C*Sin[2*c + 3*d*x] + 1590*A*Sin[4*c + 3*d*x] + 1955*C*Sin[4*c + 3*d*x] + 4080*A*Sin[3*c + 4*d*x] + 3264*C*Sin[3*c + 4*d*x] - 240*A*Sin[5*c + 4*d*x] + 450*A*Sin[4*c + 5*d*x] + 345*C*Sin[4*c + 5*d*x] + 450*A*Sin[6*c + 5*d*x] + 345*C*Sin[6*c + 5*d*x] + 720*A*Sin[5*c + 6*d*x] + 544*C*Sin[5*c + 6*d*x])))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
103,1,323,157,2.2021359,"\int \sec (c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(240 (20 A+13 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (7 A+3 C) \sin (2 c+d x)+360 A \sin (c+2 d x)+360 A \sin (3 c+2 d x)+1840 A \sin (2 c+3 d x)-360 A \sin (4 c+3 d x)+180 A \sin (3 c+4 d x)+180 A \sin (5 c+4 d x)+440 A \sin (4 c+5 d x)+80 (34 A+29 C) \sin (d x)+750 C \sin (c+2 d x)+750 C \sin (3 c+2 d x)+1520 C \sin (2 c+3 d x)+195 C \sin (3 c+4 d x)+195 C \sin (5 c+4 d x)+304 C \sin (4 c+5 d x))\right)}{7680 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a^3 (20 A+13 C) \tan ^3(c+d x)}{60 d}+\frac{a^3 (20 A+13 C) \tan (c+d x)}{5 d}+\frac{a^3 (20 A+13 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (20 A+13 C) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d}-\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}",1,"-1/7680*(a^3*(1 + Cos[c + d*x])^3*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^6*Sec[c + d*x]^5*(240*(20*A + 13*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(34*A + 29*C)*Sin[d*x] - 240*(7*A + 3*C)*Sin[2*c + d*x] + 360*A*Sin[c + 2*d*x] + 750*C*Sin[c + 2*d*x] + 360*A*Sin[3*c + 2*d*x] + 750*C*Sin[3*c + 2*d*x] + 1840*A*Sin[2*c + 3*d*x] + 1520*C*Sin[2*c + 3*d*x] - 360*A*Sin[4*c + 3*d*x] + 180*A*Sin[3*c + 4*d*x] + 195*C*Sin[3*c + 4*d*x] + 180*A*Sin[5*c + 4*d*x] + 195*C*Sin[5*c + 4*d*x] + 440*A*Sin[4*c + 5*d*x] + 304*C*Sin[4*c + 5*d*x])))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
104,1,363,147,2.0612881,"\int (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(\sec (c) (4 A \sin (2 c+d x)+72 A \sin (c+2 d x)-24 A \sin (3 c+2 d x)+4 A \sin (2 c+3 d x)+4 A \sin (4 c+3 d x)+24 A \sin (3 c+4 d x)+24 A d x \cos (c)+16 A d x \cos (c+2 d x)+16 A d x \cos (3 c+2 d x)+4 A d x \cos (3 c+4 d x)+4 A d x \cos (5 c+4 d x)-72 A \sin (c)+4 A \sin (d x)+23 C \sin (2 c+d x)+88 C \sin (c+2 d x)-8 C \sin (3 c+2 d x)+15 C \sin (2 c+3 d x)+15 C \sin (4 c+3 d x)+24 C \sin (3 c+4 d x)-72 C \sin (c)+23 C \sin (d x))-8 (28 A+15 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{256 d (A \cos (2 (c+d x))+A+2 C)}","\frac{5 a^3 (4 A+3 C) \tan (c+d x)}{8 d}+\frac{a^3 (28 A+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 A+5 C) \tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{8 d}+a^3 A x+\frac{C \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{4 a d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(1 + Cos[c + d*x])^3*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^6*Sec[c + d*x]^4*(-8*(28*A + 15*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(24*A*d*x*Cos[c] + 16*A*d*x*Cos[c + 2*d*x] + 16*A*d*x*Cos[3*c + 2*d*x] + 4*A*d*x*Cos[3*c + 4*d*x] + 4*A*d*x*Cos[5*c + 4*d*x] - 72*A*Sin[c] - 72*C*Sin[c] + 4*A*Sin[d*x] + 23*C*Sin[d*x] + 4*A*Sin[2*c + d*x] + 23*C*Sin[2*c + d*x] + 72*A*Sin[c + 2*d*x] + 88*C*Sin[c + 2*d*x] - 24*A*Sin[3*c + 2*d*x] - 8*C*Sin[3*c + 2*d*x] + 4*A*Sin[2*c + 3*d*x] + 15*C*Sin[2*c + 3*d*x] + 4*A*Sin[4*c + 3*d*x] + 15*C*Sin[4*c + 3*d*x] + 24*A*Sin[3*c + 4*d*x] + 24*C*Sin[3*c + 4*d*x])))/(256*d*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
105,1,1250,145,6.482069,"\int \cos (c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{(-6 A-5 C) \cos ^5(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(6 A+5 C) \cos ^5(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 A x \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (d x) \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sin (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (c) \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sin (d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+11 C \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+11 C \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(5 C \cos \left(\frac{c}{2}\right)-4 C \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(-5 C \cos \left(\frac{c}{2}\right)-4 C \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{C \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{C \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{a^3 (6 A+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(6 A-5 C) \tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{6 d}+3 a^3 A x+\frac{5 a^3 C \tan (c+d x)}{2 d}-\frac{(3 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 a d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}",1,"(3*A*x*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(4*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((-6*A - 5*C)*Cos[c + d*x]^5*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(8*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((6*A + 5*C)*Cos[c + d*x]^5*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(8*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[d*x]*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sin[c])/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[c]*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sin[d*x])/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(5*C*Cos[c/2] - 4*C*Sin[c/2]))/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(12*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (C*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(-5*C*Cos[c/2] - 4*C*Sin[c/2]))/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(12*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
106,1,364,162,4.5130601,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 \cos ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(A+C \sec ^2(c+d x)\right) \left(-\frac{2 (2 A+7 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (2 A+7 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{12 A \sin (c) \cos (d x)}{d}+\frac{A \sin (2 c) \cos (2 d x)}{d}+\frac{12 A \cos (c) \sin (d x)}{d}+\frac{A \cos (2 c) \sin (2 d x)}{d}+2 x (7 A+2 C)+\frac{12 C \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{12 C \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{16 (A \cos (2 (c+d x))+A+2 C)}","\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}+\frac{a^3 (2 A+7 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(A-4 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (7 A+2 C)-\frac{(A-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{2 d}",1,"(a^3*Cos[c + d*x]^5*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(2*(7*A + 2*C)*x - (2*(2*A + 7*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(2*A + 7*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (12*A*Cos[d*x]*Sin[c])/d + (A*Cos[2*d*x]*Sin[2*c])/d + (12*A*Cos[c]*Sin[d*x])/d + (A*Cos[2*c]*Sin[2*d*x])/d + C/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (12*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - C/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (12*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(16*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
107,1,1014,156,6.2373718,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","a^3 \left(-\frac{3 C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(5 A+6 C) x \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 (\cos (2 c+2 d x) A+A+2 C)}+\frac{(15 A+4 C) \cos (d x) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 A \cos (2 d x) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin (2 c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (3 d x) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin (3 c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(15 A+4 C) \cos (c) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin (d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 A \cos (2 c) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin (2 d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (3 c) \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin (3 d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{C \cos ^2(c+d x) (\cos (c+d x)+1)^3 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}\right)","-\frac{(5 A-6 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{6 d}+\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{1}{2} a^3 x (5 A+6 C)+\frac{3 a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"a^3*(((5*A + 6*C)*x*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2))/(8*(A + 2*C + A*Cos[2*c + 2*d*x])) - (3*C*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2))/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (3*C*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2))/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((15*A + 4*C)*Cos[d*x]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[c])/(16*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (3*A*Cos[2*d*x]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[2*c])/(16*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[3*d*x]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[3*c])/(48*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((15*A + 4*C)*Cos[c]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[d*x])/(16*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (3*A*Cos[2*c]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[2*d*x])/(16*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[3*c]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[3*d*x])/(48*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (C*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])))","B",1
108,1,124,169,0.3060801,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 \left(8 (13 A+12 C) \sin (c+d x)+8 (4 A+C) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+A \sin (4 (c+d x))+60 A d x-32 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+112 C d x\right)}{32 d}","\frac{5 a^3 (3 A+4 C) \sin (c+d x)}{8 d}+\frac{(5 A+4 C) \sin (c+d x) \cos (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{8 d}+\frac{1}{8} a^3 x (15 A+28 C)+\frac{a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{4 a d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(60*A*d*x + 112*C*d*x - 32*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*(13*A + 12*C)*Sin[c + d*x] + 8*(4*A + C)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + A*Sin[4*(c + d*x)]))/(32*d)","A",1
109,1,97,161,0.3298771,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 (60 (23 A+30 C) \sin (c+d x)+120 (4 A+3 C) \sin (2 (c+d x))+170 A \sin (3 (c+d x))+45 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+780 A d x+40 C \sin (3 (c+d x))+1200 C d x)}{480 d}","-\frac{a^3 (13 A+20 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (13 A+20 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (13 A+20 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (13 A+20 C)+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^3}{5 d}+\frac{3 A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}",1,"(a^3*(780*A*d*x + 1200*C*d*x + 60*(23*A + 30*C)*Sin[c + d*x] + 120*(4*A + 3*C)*Sin[2*(c + d*x)] + 170*A*Sin[3*(c + d*x)] + 40*C*Sin[3*(c + d*x)] + 45*A*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)","A",1
110,1,123,216,0.4692487,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 (120 (21 A+26 C) \sin (c+d x)+15 (63 A+64 C) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+135 A \sin (4 (c+d x))+36 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+900 A c+1380 A d x+240 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+1800 C d x)}{960 d}","\frac{a^3 (34 A+45 C) \sin (c+d x)}{15 d}+\frac{a^3 (73 A+90 C) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac{a^3 (23 A+30 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(31 A+30 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{120 d}+\frac{1}{16} a^3 x (23 A+30 C)+\frac{A \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{10 a d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}",1,"(a^3*(900*A*c + 1380*A*d*x + 1800*C*d*x + 120*(21*A + 26*C)*Sin[c + d*x] + 15*(63*A + 64*C)*Sin[2*(c + d*x)] + 380*A*Sin[3*(c + d*x)] + 240*C*Sin[3*(c + d*x)] + 135*A*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 36*A*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
111,1,419,228,5.2491442,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^7(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(6720 (14 A+11 C) \cos ^7(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-140 (217 A+122 C) \sin (2 c+d x)+10710 A \sin (c+2 d x)+10710 A \sin (3 c+2 d x)+41244 A \sin (2 c+3 d x)-7560 A \sin (4 c+3 d x)+7560 A \sin (3 c+4 d x)+7560 A \sin (5 c+4 d x)+15848 A \sin (4 c+5 d x)-420 A \sin (6 c+5 d x)+1470 A \sin (5 c+6 d x)+1470 A \sin (7 c+6 d x)+2324 A \sin (6 c+7 d x)+560 (91 A+83 C) \sin (d x)+16415 C \sin (c+2 d x)+16415 C \sin (3 c+2 d x)+37296 C \sin (2 c+3 d x)-840 C \sin (4 c+3 d x)+7700 C \sin (3 c+4 d x)+7700 C \sin (5 c+4 d x)+12712 C \sin (4 c+5 d x)+1155 C \sin (5 c+6 d x)+1155 C \sin (7 c+6 d x)+1816 C \sin (6 c+7 d x))\right)}{215040 d (A \cos (2 (c+d x))+A+2 C)}","\frac{8 a^4 (14 A+11 C) \tan ^3(c+d x)}{105 d}+\frac{16 a^4 (14 A+11 C) \tan (c+d x)}{35 d}+\frac{a^4 (14 A+11 C) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{a^4 (14 A+11 C) \tan (c+d x) \sec ^3(c+d x)}{70 d}+\frac{27 a^4 (14 A+11 C) \tan (c+d x) \sec (c+d x)}{140 d}+\frac{(21 A+4 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{105 d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^5}{21 a d}",1,"-1/215040*(a^4*(1 + Cos[c + d*x])^4*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^7*(6720*(14*A + 11*C)*Cos[c + d*x]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(560*(91*A + 83*C)*Sin[d*x] - 140*(217*A + 122*C)*Sin[2*c + d*x] + 10710*A*Sin[c + 2*d*x] + 16415*C*Sin[c + 2*d*x] + 10710*A*Sin[3*c + 2*d*x] + 16415*C*Sin[3*c + 2*d*x] + 41244*A*Sin[2*c + 3*d*x] + 37296*C*Sin[2*c + 3*d*x] - 7560*A*Sin[4*c + 3*d*x] - 840*C*Sin[4*c + 3*d*x] + 7560*A*Sin[3*c + 4*d*x] + 7700*C*Sin[3*c + 4*d*x] + 7560*A*Sin[5*c + 4*d*x] + 7700*C*Sin[5*c + 4*d*x] + 15848*A*Sin[4*c + 5*d*x] + 12712*C*Sin[4*c + 5*d*x] - 420*A*Sin[6*c + 5*d*x] + 1470*A*Sin[5*c + 6*d*x] + 1155*C*Sin[5*c + 6*d*x] + 1470*A*Sin[7*c + 6*d*x] + 1155*C*Sin[7*c + 6*d*x] + 2324*A*Sin[6*c + 7*d*x] + 1816*C*Sin[6*c + 7*d*x])))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
112,1,387,188,3.7077674,"\int \sec (c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(3360 (10 A+7 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-640 (25 A+18 C) \sin (c)+1860 A \sin (2 c+d x)+17280 A \sin (c+2 d x)-6720 A \sin (3 c+2 d x)+2670 A \sin (2 c+3 d x)+2670 A \sin (4 c+3 d x)+8640 A \sin (3 c+4 d x)-960 A \sin (5 c+4 d x)+810 A \sin (4 c+5 d x)+810 A \sin (6 c+5 d x)+1600 A \sin (5 c+6 d x)+30 (62 A+125 C) \sin (d x)+3750 C \sin (2 c+d x)+15360 C \sin (c+2 d x)-1920 C \sin (3 c+2 d x)+3845 C \sin (2 c+3 d x)+3845 C \sin (4 c+3 d x)+6912 C \sin (3 c+4 d x)+735 C \sin (4 c+5 d x)+735 C \sin (6 c+5 d x)+1152 C \sin (5 c+6 d x))\right)}{61440 d (A \cos (2 (c+d x))+A+2 C)}","\frac{2 a^4 (10 A+7 C) \tan ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+7 C) \tan (c+d x)}{5 d}+\frac{7 a^4 (10 A+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (10 A+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}-\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d}",1,"-1/61440*(a^4*(1 + Cos[c + d*x])^4*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^6*(3360*(10*A + 7*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-640*(25*A + 18*C)*Sin[c] + 30*(62*A + 125*C)*Sin[d*x] + 1860*A*Sin[2*c + d*x] + 3750*C*Sin[2*c + d*x] + 17280*A*Sin[c + 2*d*x] + 15360*C*Sin[c + 2*d*x] - 6720*A*Sin[3*c + 2*d*x] - 1920*C*Sin[3*c + 2*d*x] + 2670*A*Sin[2*c + 3*d*x] + 3845*C*Sin[2*c + 3*d*x] + 2670*A*Sin[4*c + 3*d*x] + 3845*C*Sin[4*c + 3*d*x] + 8640*A*Sin[3*c + 4*d*x] + 6912*C*Sin[3*c + 4*d*x] - 960*A*Sin[5*c + 4*d*x] + 810*A*Sin[4*c + 5*d*x] + 735*C*Sin[4*c + 5*d*x] + 810*A*Sin[6*c + 5*d*x] + 735*C*Sin[6*c + 5*d*x] + 1600*A*Sin[5*c + 6*d*x] + 1152*C*Sin[5*c + 6*d*x])))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
113,1,418,177,3.1560396,"\int (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(\sec (c) (-780 A \sin (2 c+d x)+120 A \sin (c+2 d x)+120 A \sin (3 c+2 d x)+820 A \sin (2 c+3 d x)-180 A \sin (4 c+3 d x)+60 A \sin (3 c+4 d x)+60 A \sin (5 c+4 d x)+200 A \sin (4 c+5 d x)+150 A d x \cos (2 c+d x)+75 A d x \cos (2 c+3 d x)+75 A d x \cos (4 c+3 d x)+15 A d x \cos (4 c+5 d x)+15 A d x \cos (6 c+5 d x)+1220 A \sin (d x)+150 A d x \cos (d x)-480 C \sin (2 c+d x)+330 C \sin (c+2 d x)+330 C \sin (3 c+2 d x)+800 C \sin (2 c+3 d x)-30 C \sin (4 c+3 d x)+105 C \sin (3 c+4 d x)+105 C \sin (5 c+4 d x)+166 C \sin (4 c+5 d x)+1180 C \sin (d x))-240 (12 A+7 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3840 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a^4 (10 A+7 C) \tan (c+d x)}{2 d}+\frac{a^4 (12 A+7 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(8 A+7 C) \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{6 d}+a^4 A x+\frac{(5 A+7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{15 d}+\frac{a C \tan (c+d x) (a \sec (c+d x)+a)^3}{5 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}",1,"(a^4*(1 + Cos[c + d*x])^4*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^5*(-240*(12*A + 7*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(150*A*d*x*Cos[d*x] + 150*A*d*x*Cos[2*c + d*x] + 75*A*d*x*Cos[2*c + 3*d*x] + 75*A*d*x*Cos[4*c + 3*d*x] + 15*A*d*x*Cos[4*c + 5*d*x] + 15*A*d*x*Cos[6*c + 5*d*x] + 1220*A*Sin[d*x] + 1180*C*Sin[d*x] - 780*A*Sin[2*c + d*x] - 480*C*Sin[2*c + d*x] + 120*A*Sin[c + 2*d*x] + 330*C*Sin[c + 2*d*x] + 120*A*Sin[3*c + 2*d*x] + 330*C*Sin[3*c + 2*d*x] + 820*A*Sin[2*c + 3*d*x] + 800*C*Sin[2*c + 3*d*x] - 180*A*Sin[4*c + 3*d*x] - 30*C*Sin[4*c + 3*d*x] + 60*A*Sin[3*c + 4*d*x] + 105*C*Sin[3*c + 4*d*x] + 60*A*Sin[5*c + 4*d*x] + 105*C*Sin[5*c + 4*d*x] + 200*A*Sin[4*c + 5*d*x] + 166*C*Sin[4*c + 5*d*x])))/(3840*d*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
114,1,379,181,2.5777733,"\int \cos (c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(A \cos ^2(c+d x)+C\right) \left(\sec (c) (24 A \sin (2 c+d x)+288 A \sin (c+2 d x)-96 A \sin (3 c+2 d x)+30 A \sin (2 c+3 d x)+30 A \sin (4 c+3 d x)+96 A \sin (3 c+4 d x)+6 A \sin (4 c+5 d x)+6 A \sin (6 c+5 d x)+288 A d x \cos (c)+192 A d x \cos (c+2 d x)+192 A d x \cos (3 c+2 d x)+48 A d x \cos (3 c+4 d x)+48 A d x \cos (5 c+4 d x)-288 A \sin (c)+24 A \sin (d x)+105 C \sin (2 c+d x)+544 C \sin (c+2 d x)-96 C \sin (3 c+2 d x)+81 C \sin (2 c+3 d x)+81 C \sin (4 c+3 d x)+160 C \sin (3 c+4 d x)-480 C \sin (c)+105 C \sin (d x))-24 (52 A+35 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1536 d (A \cos (2 (c+d x))+A+2 C)}","\frac{5 a^4 (4 A+7 C) \tan (c+d x)}{8 d}+\frac{a^4 (52 A+35 C) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{(12 A-35 C) \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{24 d}+4 a^4 A x-\frac{(12 A-7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 d}-\frac{a (4 A-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^4}{d}",1,"(a^4*(C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(-24*(52*A + 35*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(288*A*d*x*Cos[c] + 192*A*d*x*Cos[c + 2*d*x] + 192*A*d*x*Cos[3*c + 2*d*x] + 48*A*d*x*Cos[3*c + 4*d*x] + 48*A*d*x*Cos[5*c + 4*d*x] - 288*A*Sin[c] - 480*C*Sin[c] + 24*A*Sin[d*x] + 105*C*Sin[d*x] + 24*A*Sin[2*c + d*x] + 105*C*Sin[2*c + d*x] + 288*A*Sin[c + 2*d*x] + 544*C*Sin[c + 2*d*x] - 96*A*Sin[3*c + 2*d*x] - 96*C*Sin[3*c + 2*d*x] + 30*A*Sin[2*c + 3*d*x] + 81*C*Sin[2*c + 3*d*x] + 30*A*Sin[4*c + 3*d*x] + 81*C*Sin[4*c + 3*d*x] + 96*A*Sin[3*c + 4*d*x] + 160*C*Sin[3*c + 4*d*x] + 6*A*Sin[4*c + 5*d*x] + 6*A*Sin[6*c + 5*d*x])))/(1536*d*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
115,1,1420,192,6.4648534,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{(-2 A-3 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(2 A+3 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(13 A+2 C) x \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (2 d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (2 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (2 c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (2 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{\cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+20 C \sin \left(\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+20 C \sin \left(\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \left(13 C \cos \left(\frac{c}{2}\right)-11 C \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \left(-13 C \cos \left(\frac{c}{2}\right)-11 C \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{5 a^4 (A-2 C) \sin (c+d x)}{2 d}+\frac{2 a^4 (2 A+3 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 A+22 C) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (13 A+2 C)-\frac{(A-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 d}-\frac{a (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}",1,"((13*A + 2*C)*x*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(16*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((-2*A - 3*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((2*A + 3*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(4*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[c])/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[2*d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[2*c])/(32*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[d*x])/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[2*c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[2*d*x])/(32*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(48*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(13*C*Cos[c/2] - 11*C*Sin[c/2]))/(96*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 20*C*Sin[(d*x)/2]))/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(48*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(-13*C*Cos[c/2] - 11*C*Sin[c/2]))/(96*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 20*C*Sin[(d*x)/2]))/(24*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
116,1,1250,198,6.2514371,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{(-2 A-13 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(2 A+13 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(3 A+2 C) x \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (2 c+2 d x) A+A+2 C)}+\frac{(27 A+4 C) \cos (d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (2 d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (2 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (3 d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (3 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{(27 A+4 C) \cos (c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (2 c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (2 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \cos (3 c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin (3 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}-\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}","\frac{5 a^4 (2 A-C) \sin (c+d x)}{2 d}+\frac{a^4 (2 A+13 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(4 A-9 C) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{3 d}+2 a^4 x (3 A+2 C)-\frac{(2 A-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}+\frac{2 a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"((3*A + 2*C)*x*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(4*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((-2*A - 13*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(16*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((2*A + 13*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(16*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((27*A + 4*C)*Cos[d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[c])/(32*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[2*d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[2*c])/(8*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[3*d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[3*c])/(96*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((27*A + 4*C)*Cos[c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[d*x])/(32*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[2*c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[2*d*x])/(8*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Cos[3*c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[3*d*x])/(96*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(32*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(32*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
117,1,375,200,2.4008782,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 \cos ^2(c+d x) (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \left(A+C \sec ^2(c+d x)\right) \left(\frac{96 (7 A+4 C) \sin (c) \cos (d x)}{d}+\frac{24 (7 A+C) \sin (2 c) \cos (2 d x)}{d}+\frac{96 (7 A+4 C) \cos (c) \sin (d x)}{d}+\frac{24 (7 A+C) \cos (2 c) \sin (2 d x)}{d}+\frac{32 A \sin (3 c) \cos (3 d x)}{d}+\frac{3 A \sin (4 c) \cos (4 d x)}{d}+\frac{32 A \cos (3 c) \sin (3 d x)}{d}+\frac{3 A \cos (4 c) \sin (4 d x)}{d}+12 x (35 A+52 C)+\frac{96 C \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{96 C \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{384 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{384 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}\right)}{768 (A \cos (2 (c+d x))+A+2 C)}","\frac{5 a^4 (7 A+4 C) \sin (c+d x)}{8 d}-\frac{(35 A-12 C) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{24 d}+\frac{1}{8} a^4 x (35 A+52 C)+\frac{4 a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(7 A+4 C) \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{8 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"(a^4*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*(A + C*Sec[c + d*x]^2)*(12*(35*A + 52*C)*x - (384*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (384*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (96*(7*A + 4*C)*Cos[d*x]*Sin[c])/d + (24*(7*A + C)*Cos[2*d*x]*Sin[2*c])/d + (32*A*Cos[3*d*x]*Sin[3*c])/d + (3*A*Cos[4*d*x]*Sin[4*c])/d + (96*(7*A + 4*C)*Cos[c]*Sin[d*x])/d + (24*(7*A + C)*Cos[2*c]*Sin[2*d*x])/d + (32*A*Cos[3*c]*Sin[3*d*x])/d + (3*A*Cos[4*c]*Sin[4*d*x])/d + (96*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (96*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(768*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
118,1,147,207,0.5573723,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 \left(30 (49 A+54 C) \sin (c+d x)+240 (2 A+C) \sin (2 (c+d x))+145 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+3 A \sin (5 (c+d x))+840 A d x+20 C \sin (3 (c+d x))-240 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+240 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+1440 C d x\right)}{240 d}","\frac{a^4 (7 A+10 C) \sin (c+d x)}{2 d}+\frac{(7 A+8 C) \sin (c+d x) \cos (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (7 A+12 C)+\frac{a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(7 A+5 C) \sin (c+d x) \cos ^2(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{15 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{5 d}",1,"(a^4*(840*A*d*x + 1440*C*d*x - 240*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 240*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 30*(49*A + 54*C)*Sin[c + d*x] + 240*(2*A + C)*Sin[2*(c + d*x)] + 145*A*Sin[3*(c + d*x)] + 20*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 3*A*Sin[5*(c + d*x)]))/(240*d)","A",1
119,1,119,192,0.400576,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 (480 (11 A+14 C) \sin (c+d x)+15 (127 A+112 C) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+225 A \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+2940 A d x+320 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+4200 C d x)}{960 d}","-\frac{2 a^4 (7 A+10 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (7 A+10 C) \sin (c+d x)}{5 d}+\frac{a^4 (7 A+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (7 A+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (7 A+10 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}+\frac{2 A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d}",1,"(a^4*(2940*A*d*x + 4200*C*d*x + 480*(11*A + 14*C)*Sin[c + d*x] + 15*(127*A + 112*C)*Sin[2*(c + d*x)] + 720*A*Sin[3*(c + d*x)] + 320*C*Sin[3*(c + d*x)] + 225*A*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 48*A*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
120,1,145,254,0.7278816,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 (105 (323 A+392 C) \sin (c+d x)+420 (31 A+32 C) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+651 A \sin (5 (c+d x))+140 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+11760 A c+18480 A d x+4060 C \sin (3 (c+d x))+840 C \sin (4 (c+d x))+84 C \sin (5 (c+d x))+23520 C d x)}{6720 d}","\frac{a^4 (454 A+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (247 A+308 C) \sin (c+d x) \cos ^2(c+d x)}{210 d}+\frac{a^4 (11 A+14 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{(109 A+126 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{210 d}+\frac{1}{4} a^4 x (11 A+14 C)+\frac{(8 A+7 C) \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{35 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}+\frac{2 a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{21 d}",1,"(a^4*(11760*A*c + 18480*A*d*x + 23520*C*d*x + 105*(323*A + 392*C)*Sin[c + d*x] + 420*(31*A + 32*C)*Sin[2*(c + d*x)] + 5495*A*Sin[3*(c + d*x)] + 4060*C*Sin[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 840*C*Sin[4*(c + d*x)] + 651*A*Sin[5*(c + d*x)] + 84*C*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 15*A*Sin[7*(c + d*x)]))/(6720*d)","A",1
121,1,792,165,6.3883848,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(204 A \sin \left(c-\frac{d x}{2}\right)-60 A \sin \left(c+\frac{d x}{2}\right)+84 A \sin \left(2 c+\frac{d x}{2}\right)+36 A \sin \left(c+\frac{3 d x}{2}\right)+36 A \sin \left(2 c+\frac{3 d x}{2}\right)+132 A \sin \left(3 c+\frac{3 d x}{2}\right)-156 A \sin \left(c+\frac{5 d x}{2}\right)-60 A \sin \left(2 c+\frac{5 d x}{2}\right)-60 A \sin \left(3 c+\frac{5 d x}{2}\right)+36 A \sin \left(4 c+\frac{5 d x}{2}\right)-12 A \sin \left(2 c+\frac{7 d x}{2}\right)+12 A \sin \left(3 c+\frac{7 d x}{2}\right)+12 A \sin \left(4 c+\frac{7 d x}{2}\right)+36 A \sin \left(5 c+\frac{7 d x}{2}\right)-48 A \sin \left(3 c+\frac{9 d x}{2}\right)-24 A \sin \left(4 c+\frac{9 d x}{2}\right)-24 A \sin \left(5 c+\frac{9 d x}{2}\right)-60 A \sin \left(\frac{d x}{2}\right)-60 A \sin \left(\frac{3 d x}{2}\right)+219 C \sin \left(c-\frac{d x}{2}\right)+21 C \sin \left(c+\frac{d x}{2}\right)+165 C \sin \left(2 c+\frac{d x}{2}\right)+5 C \sin \left(c+\frac{3 d x}{2}\right)+69 C \sin \left(2 c+\frac{3 d x}{2}\right)+165 C \sin \left(3 c+\frac{3 d x}{2}\right)-211 C \sin \left(c+\frac{5 d x}{2}\right)-115 C \sin \left(2 c+\frac{5 d x}{2}\right)-51 C \sin \left(3 c+\frac{5 d x}{2}\right)+45 C \sin \left(4 c+\frac{5 d x}{2}\right)-19 C \sin \left(2 c+\frac{7 d x}{2}\right)+5 C \sin \left(3 c+\frac{7 d x}{2}\right)+21 C \sin \left(4 c+\frac{7 d x}{2}\right)+45 C \sin \left(5 c+\frac{7 d x}{2}\right)-64 C \sin \left(3 c+\frac{9 d x}{2}\right)-40 C \sin \left(4 c+\frac{9 d x}{2}\right)-24 C \sin \left(5 c+\frac{9 d x}{2}\right)-75 C \sin \left(\frac{d x}{2}\right)-91 C \sin \left(\frac{3 d x}{2}\right)\right) \left(A+C \sec ^2(c+d x)\right)}{192 d (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}-\frac{3 (4 A+5 C) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos (c+d x) \left(A+C \sec ^2(c+d x)\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{2 d (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}+\frac{3 (4 A+5 C) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos (c+d x) \left(A+C \sec ^2(c+d x)\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{2 d (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}","-\frac{(3 A+4 C) \tan ^3(c+d x)}{3 a d}-\frac{(3 A+4 C) \tan (c+d x)}{a d}+\frac{3 (4 A+5 C) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(4 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 (4 A+5 C) \tan (c+d x) \sec (c+d x)}{8 a d}",1,"(-3*(4*A + 5*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(A + C*Sec[c + d*x]^2))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (3*(4*A + 5*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(A + C*Sec[c + d*x]^2))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)*(-60*A*Sin[(d*x)/2] - 75*C*Sin[(d*x)/2] - 60*A*Sin[(3*d*x)/2] - 91*C*Sin[(3*d*x)/2] + 204*A*Sin[c - (d*x)/2] + 219*C*Sin[c - (d*x)/2] - 60*A*Sin[c + (d*x)/2] + 21*C*Sin[c + (d*x)/2] + 84*A*Sin[2*c + (d*x)/2] + 165*C*Sin[2*c + (d*x)/2] + 36*A*Sin[c + (3*d*x)/2] + 5*C*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] + 69*C*Sin[2*c + (3*d*x)/2] + 132*A*Sin[3*c + (3*d*x)/2] + 165*C*Sin[3*c + (3*d*x)/2] - 156*A*Sin[c + (5*d*x)/2] - 211*C*Sin[c + (5*d*x)/2] - 60*A*Sin[2*c + (5*d*x)/2] - 115*C*Sin[2*c + (5*d*x)/2] - 60*A*Sin[3*c + (5*d*x)/2] - 51*C*Sin[3*c + (5*d*x)/2] + 36*A*Sin[4*c + (5*d*x)/2] + 45*C*Sin[4*c + (5*d*x)/2] - 12*A*Sin[2*c + (7*d*x)/2] - 19*C*Sin[2*c + (7*d*x)/2] + 12*A*Sin[3*c + (7*d*x)/2] + 5*C*Sin[3*c + (7*d*x)/2] + 12*A*Sin[4*c + (7*d*x)/2] + 21*C*Sin[4*c + (7*d*x)/2] + 36*A*Sin[5*c + (7*d*x)/2] + 45*C*Sin[5*c + (7*d*x)/2] - 48*A*Sin[3*c + (9*d*x)/2] - 64*C*Sin[3*c + (9*d*x)/2] - 24*A*Sin[4*c + (9*d*x)/2] - 40*C*Sin[4*c + (9*d*x)/2] - 24*A*Sin[5*c + (9*d*x)/2] - 24*C*Sin[5*c + (9*d*x)/2]))/(192*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x]))","B",1
122,1,1090,133,6.5872765,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{2 (2 A+3 C) \cos (c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{2 (2 A+3 C) \cos (c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{4 \cos (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{4 \cos (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}-\frac{2 \cos (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(C \cos \left(\frac{c}{2}\right)-2 C \sin \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{2 \cos (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(C \cos \left(\frac{c}{2}\right)+2 C \sin \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{2 C \cos (c+d x) \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{2 C \cos (c+d x) \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{4 \cos (c+d x) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}","\frac{(3 A+4 C) \tan ^3(c+d x)}{3 a d}+\frac{(3 A+4 C) \tan (c+d x)}{a d}-\frac{(2 A+3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}-\frac{(2 A+3 C) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(2*(2*A + 3*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(A + C*Sec[c + d*x]^2))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*(2*A + 3*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(A + C*Sec[c + d*x]^2))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (4*Cos[c/2 + (d*x)/2]*Cos[c + d*x]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) - (2*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*(C*Cos[c/2] - 2*C*Sin[c/2]))/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (4*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2]))/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (2*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*(C*Cos[c/2] + 2*C*Sin[c/2]))/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (4*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2]))/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",0
123,1,316,107,3.0773603,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \left(A+C \sec ^2(c+d x)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(-2 (2 A+3 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{4 C \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-4 (A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{a d (\sec (c+d x)+1) (A \cos (2 (c+d x))+A+2 C)}","-\frac{(A+2 C) \tan (c+d x)}{a d}+\frac{(2 A+3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(2 A+3 C) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(Cos[(c + d*x)/2]*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*(-4*(A + C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(-2*(2*A + 3*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 6*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - C/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (4*C*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x]))","B",1
124,1,227,57,2.0379778,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \left(A+C \sec ^2(c+d x)\right) \left((A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+C \cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\sec (c+d x)+1) (A \cos (2 (c+d x))+A+2 C)}","\frac{(A+C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{C \tan (c+d x)}{a d}-\frac{C \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(4*Cos[(c + d*x)/2]*Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*((A + C)*Sec[c/2]*Sin[(d*x)/2] + C*Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x]))","B",1
125,1,143,49,0.4706548,"\int \frac{A+C \sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]),x]","-\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A \cos ^2(c+d x)+C\right) \left((A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-\cos \left(\frac{1}{2} (c+d x)\right) \left(A d x-C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\cos (c+d x)+1) (A \cos (2 (c+d x))+A+2 C)}","-\frac{(A+C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{A x}{a}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(-4*Cos[(c + d*x)/2]*(C + A*Cos[c + d*x]^2)*(-(Cos[(c + d*x)/2]*(A*d*x - C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])) + (A + C)*Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x])*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
126,1,108,52,0.3018886,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left(A \sin \left(c+\frac{d x}{2}\right)+A \sin \left(c+\frac{3 d x}{2}\right)+A \sin \left(2 c+\frac{3 d x}{2}\right)-2 A d x \cos \left(c+\frac{d x}{2}\right)+5 A \sin \left(\frac{d x}{2}\right)-2 A d x \cos \left(\frac{d x}{2}\right)+4 C \sin \left(\frac{d x}{2}\right)\right)}{4 a d}","\frac{(2 A+C) \sin (c+d x)}{a d}-\frac{(A+C) \sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{A x}{a}",1,"(Sec[c/2]*Sec[(c + d*x)/2]*(-2*A*d*x*Cos[(d*x)/2] - 2*A*d*x*Cos[c + (d*x)/2] + 5*A*Sin[(d*x)/2] + 4*C*Sin[(d*x)/2] + A*Sin[c + (d*x)/2] + A*Sin[c + (3*d*x)/2] + A*Sin[2*c + (3*d*x)/2]))/(4*a*d)","B",1
127,1,159,96,0.3613139,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(4 d x (3 A+2 C) \cos \left(c+\frac{d x}{2}\right)-4 A \sin \left(c+\frac{d x}{2}\right)-3 A \sin \left(c+\frac{3 d x}{2}\right)-3 A \sin \left(2 c+\frac{3 d x}{2}\right)+A \sin \left(2 c+\frac{5 d x}{2}\right)+A \sin \left(3 c+\frac{5 d x}{2}\right)+4 d x (3 A+2 C) \cos \left(\frac{d x}{2}\right)-20 A \sin \left(\frac{d x}{2}\right)-16 C \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","-\frac{(2 A+C) \sin (c+d x)}{a d}+\frac{(3 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A+C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{x (3 A+2 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(4*(3*A + 2*C)*d*x*Cos[(d*x)/2] + 4*(3*A + 2*C)*d*x*Cos[c + (d*x)/2] - 20*A*Sin[(d*x)/2] - 16*C*Sin[(d*x)/2] - 4*A*Sin[c + (d*x)/2] - 3*A*Sin[c + (3*d*x)/2] - 3*A*Sin[2*c + (3*d*x)/2] + A*Sin[2*c + (5*d*x)/2] + A*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","A",1
128,1,225,124,0.8091108,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-12 d x (3 A+2 C) \cos \left(c+\frac{d x}{2}\right)+21 A \sin \left(c+\frac{d x}{2}\right)+18 A \sin \left(c+\frac{3 d x}{2}\right)+18 A \sin \left(2 c+\frac{3 d x}{2}\right)-2 A \sin \left(2 c+\frac{5 d x}{2}\right)-2 A \sin \left(3 c+\frac{5 d x}{2}\right)+A \sin \left(3 c+\frac{7 d x}{2}\right)+A \sin \left(4 c+\frac{7 d x}{2}\right)-12 d x (3 A+2 C) \cos \left(\frac{d x}{2}\right)+69 A \sin \left(\frac{d x}{2}\right)+12 C \sin \left(c+\frac{d x}{2}\right)+12 C \sin \left(c+\frac{3 d x}{2}\right)+12 C \sin \left(2 c+\frac{3 d x}{2}\right)+60 C \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","-\frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}+\frac{(4 A+3 C) \sin (c+d x)}{a d}-\frac{(3 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{x (3 A+2 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-12*(3*A + 2*C)*d*x*Cos[(d*x)/2] - 12*(3*A + 2*C)*d*x*Cos[c + (d*x)/2] + 69*A*Sin[(d*x)/2] + 60*C*Sin[(d*x)/2] + 21*A*Sin[c + (d*x)/2] + 12*C*Sin[c + (d*x)/2] + 18*A*Sin[c + (3*d*x)/2] + 12*C*Sin[c + (3*d*x)/2] + 18*A*Sin[2*c + (3*d*x)/2] + 12*C*Sin[2*c + (3*d*x)/2] - 2*A*Sin[2*c + (5*d*x)/2] - 2*A*Sin[3*c + (5*d*x)/2] + A*Sin[3*c + (7*d*x)/2] + A*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","A",1
129,1,283,156,0.6873658,"\int \frac{\cos ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(72 d x (5 A+4 C) \cos \left(c+\frac{d x}{2}\right)-168 A \sin \left(c+\frac{d x}{2}\right)-120 A \sin \left(c+\frac{3 d x}{2}\right)-120 A \sin \left(2 c+\frac{3 d x}{2}\right)+40 A \sin \left(2 c+\frac{5 d x}{2}\right)+40 A \sin \left(3 c+\frac{5 d x}{2}\right)-5 A \sin \left(3 c+\frac{7 d x}{2}\right)-5 A \sin \left(4 c+\frac{7 d x}{2}\right)+3 A \sin \left(4 c+\frac{9 d x}{2}\right)+3 A \sin \left(5 c+\frac{9 d x}{2}\right)+72 d x (5 A+4 C) \cos \left(\frac{d x}{2}\right)-552 A \sin \left(\frac{d x}{2}\right)-96 C \sin \left(c+\frac{d x}{2}\right)-72 C \sin \left(c+\frac{3 d x}{2}\right)-72 C \sin \left(2 c+\frac{3 d x}{2}\right)+24 C \sin \left(2 c+\frac{5 d x}{2}\right)+24 C \sin \left(3 c+\frac{5 d x}{2}\right)-480 C \sin \left(\frac{d x}{2}\right)\right)}{192 a d (\cos (c+d x)+1)}","\frac{(4 A+3 C) \sin ^3(c+d x)}{3 a d}-\frac{(4 A+3 C) \sin (c+d x)}{a d}+\frac{(5 A+4 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (5 A+4 C) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{(A+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x (5 A+4 C)}{8 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(72*(5*A + 4*C)*d*x*Cos[(d*x)/2] + 72*(5*A + 4*C)*d*x*Cos[c + (d*x)/2] - 552*A*Sin[(d*x)/2] - 480*C*Sin[(d*x)/2] - 168*A*Sin[c + (d*x)/2] - 96*C*Sin[c + (d*x)/2] - 120*A*Sin[c + (3*d*x)/2] - 72*C*Sin[c + (3*d*x)/2] - 120*A*Sin[2*c + (3*d*x)/2] - 72*C*Sin[2*c + (3*d*x)/2] + 40*A*Sin[2*c + (5*d*x)/2] + 24*C*Sin[2*c + (5*d*x)/2] + 40*A*Sin[3*c + (5*d*x)/2] + 24*C*Sin[3*c + (5*d*x)/2] - 5*A*Sin[3*c + (7*d*x)/2] - 5*A*Sin[4*c + (7*d*x)/2] + 3*A*Sin[4*c + (9*d*x)/2] + 3*A*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x]))","A",1
130,1,623,172,3.0407008,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(A+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sec (c) \sec ^3(c+d x) \left(-60 A \sin \left(c-\frac{d x}{2}\right)+24 A \sin \left(c+\frac{d x}{2}\right)-60 A \sin \left(2 c+\frac{d x}{2}\right)-4 A \sin \left(c+\frac{3 d x}{2}\right)+36 A \sin \left(2 c+\frac{3 d x}{2}\right)-34 A \sin \left(3 c+\frac{3 d x}{2}\right)+42 A \sin \left(c+\frac{5 d x}{2}\right)+24 A \sin \left(3 c+\frac{5 d x}{2}\right)-18 A \sin \left(4 c+\frac{5 d x}{2}\right)+24 A \sin \left(2 c+\frac{7 d x}{2}\right)+3 A \sin \left(3 c+\frac{7 d x}{2}\right)+15 A \sin \left(4 c+\frac{7 d x}{2}\right)-6 A \sin \left(5 c+\frac{7 d x}{2}\right)+10 A \sin \left(3 c+\frac{9 d x}{2}\right)+3 A \sin \left(4 c+\frac{9 d x}{2}\right)+7 A \sin \left(5 c+\frac{9 d x}{2}\right)-3 (8 A+C) \sin \left(\frac{d x}{2}\right)+(66 A+155 C) \sin \left(\frac{3 d x}{2}\right)-153 C \sin \left(c-\frac{d x}{2}\right)+21 C \sin \left(c+\frac{d x}{2}\right)-135 C \sin \left(2 c+\frac{d x}{2}\right)+25 C \sin \left(c+\frac{3 d x}{2}\right)+45 C \sin \left(2 c+\frac{3 d x}{2}\right)-85 C \sin \left(3 c+\frac{3 d x}{2}\right)+99 C \sin \left(c+\frac{5 d x}{2}\right)+21 C \sin \left(2 c+\frac{5 d x}{2}\right)+33 C \sin \left(3 c+\frac{5 d x}{2}\right)-45 C \sin \left(4 c+\frac{5 d x}{2}\right)+57 C \sin \left(2 c+\frac{7 d x}{2}\right)+18 C \sin \left(3 c+\frac{7 d x}{2}\right)+24 C \sin \left(4 c+\frac{7 d x}{2}\right)-15 C \sin \left(5 c+\frac{7 d x}{2}\right)+24 C \sin \left(3 c+\frac{9 d x}{2}\right)+11 C \sin \left(4 c+\frac{9 d x}{2}\right)+13 C \sin \left(5 c+\frac{9 d x}{2}\right)\right)+192 (2 A+5 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{24 a^2 d (\sec (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 C)}","\frac{(5 A+12 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(5 A+12 C) \tan (c+d x)}{a^2 d}-\frac{(2 A+5 C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{2 (2 A+5 C) \tan (c+d x) \sec ^3(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{(2 A+5 C) \tan (c+d x) \sec (c+d x)}{a^2 d}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*(A + C*Sec[c + d*x]^2)*(192*(2*A + 5*C)*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(-3*(8*A + C)*Sin[(d*x)/2] + (66*A + 155*C)*Sin[(3*d*x)/2] - 60*A*Sin[c - (d*x)/2] - 153*C*Sin[c - (d*x)/2] + 24*A*Sin[c + (d*x)/2] + 21*C*Sin[c + (d*x)/2] - 60*A*Sin[2*c + (d*x)/2] - 135*C*Sin[2*c + (d*x)/2] - 4*A*Sin[c + (3*d*x)/2] + 25*C*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] + 45*C*Sin[2*c + (3*d*x)/2] - 34*A*Sin[3*c + (3*d*x)/2] - 85*C*Sin[3*c + (3*d*x)/2] + 42*A*Sin[c + (5*d*x)/2] + 99*C*Sin[c + (5*d*x)/2] + 21*C*Sin[2*c + (5*d*x)/2] + 24*A*Sin[3*c + (5*d*x)/2] + 33*C*Sin[3*c + (5*d*x)/2] - 18*A*Sin[4*c + (5*d*x)/2] - 45*C*Sin[4*c + (5*d*x)/2] + 24*A*Sin[2*c + (7*d*x)/2] + 57*C*Sin[2*c + (7*d*x)/2] + 3*A*Sin[3*c + (7*d*x)/2] + 18*C*Sin[3*c + (7*d*x)/2] + 15*A*Sin[4*c + (7*d*x)/2] + 24*C*Sin[4*c + (7*d*x)/2] - 6*A*Sin[5*c + (7*d*x)/2] - 15*C*Sin[5*c + (7*d*x)/2] + 10*A*Sin[3*c + (9*d*x)/2] + 24*C*Sin[3*c + (9*d*x)/2] + 3*A*Sin[4*c + (9*d*x)/2] + 11*C*Sin[4*c + (9*d*x)/2] + 7*A*Sin[5*c + (9*d*x)/2] + 13*C*Sin[5*c + (9*d*x)/2])))/(24*a^2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)","B",1
131,1,513,150,2.1432276,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(A+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sec (c) \sec ^2(c+d x) \left(-36 A \sin \left(c-\frac{d x}{2}\right)+36 A \sin \left(c+\frac{d x}{2}\right)-20 A \sin \left(2 c+\frac{d x}{2}\right)-18 A \sin \left(c+\frac{3 d x}{2}\right)+22 A \sin \left(2 c+\frac{3 d x}{2}\right)-18 A \sin \left(3 c+\frac{3 d x}{2}\right)+18 A \sin \left(c+\frac{5 d x}{2}\right)-6 A \sin \left(2 c+\frac{5 d x}{2}\right)+18 A \sin \left(3 c+\frac{5 d x}{2}\right)-6 A \sin \left(4 c+\frac{5 d x}{2}\right)+8 A \sin \left(2 c+\frac{7 d x}{2}\right)+8 A \sin \left(4 c+\frac{7 d x}{2}\right)-2 (10 A+7 C) \sin \left(\frac{d x}{2}\right)+(22 A+97 C) \sin \left(\frac{3 d x}{2}\right)-126 C \sin \left(c-\frac{d x}{2}\right)+42 C \sin \left(c+\frac{d x}{2}\right)-98 C \sin \left(2 c+\frac{d x}{2}\right)-3 C \sin \left(c+\frac{3 d x}{2}\right)+37 C \sin \left(2 c+\frac{3 d x}{2}\right)-63 C \sin \left(3 c+\frac{3 d x}{2}\right)+75 C \sin \left(c+\frac{5 d x}{2}\right)+15 C \sin \left(2 c+\frac{5 d x}{2}\right)+39 C \sin \left(3 c+\frac{5 d x}{2}\right)-21 C \sin \left(4 c+\frac{5 d x}{2}\right)+32 C \sin \left(2 c+\frac{7 d x}{2}\right)+12 C \sin \left(3 c+\frac{7 d x}{2}\right)+20 C \sin \left(4 c+\frac{7 d x}{2}\right)\right)+96 (2 A+7 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{24 a^2 d (\sec (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 C)}","-\frac{4 (A+4 C) \tan (c+d x)}{3 a^2 d}+\frac{(2 A+7 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{2 (A+4 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(2 A+7 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-1/24*(Cos[(c + d*x)/2]*(A + C*Sec[c + d*x]^2)*(96*(2*A + 7*C)*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-2*(10*A + 7*C)*Sin[(d*x)/2] + (22*A + 97*C)*Sin[(3*d*x)/2] - 36*A*Sin[c - (d*x)/2] - 126*C*Sin[c - (d*x)/2] + 36*A*Sin[c + (d*x)/2] + 42*C*Sin[c + (d*x)/2] - 20*A*Sin[2*c + (d*x)/2] - 98*C*Sin[2*c + (d*x)/2] - 18*A*Sin[c + (3*d*x)/2] - 3*C*Sin[c + (3*d*x)/2] + 22*A*Sin[2*c + (3*d*x)/2] + 37*C*Sin[2*c + (3*d*x)/2] - 18*A*Sin[3*c + (3*d*x)/2] - 63*C*Sin[3*c + (3*d*x)/2] + 18*A*Sin[c + (5*d*x)/2] + 75*C*Sin[c + (5*d*x)/2] - 6*A*Sin[2*c + (5*d*x)/2] + 15*C*Sin[2*c + (5*d*x)/2] + 18*A*Sin[3*c + (5*d*x)/2] + 39*C*Sin[3*c + (5*d*x)/2] - 6*A*Sin[4*c + (5*d*x)/2] - 21*C*Sin[4*c + (5*d*x)/2] + 8*A*Sin[2*c + (7*d*x)/2] + 32*C*Sin[2*c + (7*d*x)/2] + 12*C*Sin[3*c + (7*d*x)/2] + 8*A*Sin[4*c + (7*d*x)/2] + 20*C*Sin[4*c + (7*d*x)/2])))/(a^2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)","B",1
132,1,280,99,1.596694,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A+C \sec ^2(c+d x)\right) \left((A+C) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(A+C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+2 (A+7 C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 C \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{\sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 C)}","\frac{(A+4 C) \tan (c+d x)}{3 a^2 d}-\frac{2 C \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{2 C \tan (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(4*Cos[(c + d*x)/2]*(A + C*Sec[c + d*x]^2)*((A + C)*Sec[c/2]*Sin[(d*x)/2] + 2*(A + 7*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 6*C*Cos[(c + d*x)/2]^3*(2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Sin[d*x]/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (A + C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)","B",1
133,1,377,75,0.9371938,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(6 A \sin \left(c+\frac{d x}{2}\right)-4 A \sin \left(c+\frac{3 d x}{2}\right)-6 A \sin \left(\frac{d x}{2}\right)-6 C \sin \left(c+\frac{d x}{2}\right)+8 C \sin \left(c+\frac{3 d x}{2}\right)+3 C \cos \left(c+\frac{3 d x}{2}\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 C \cos \left(2 c+\frac{3 d x}{2}\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 C \cos \left(\frac{d x}{2}\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+9 C \cos \left(c+\frac{d x}{2}\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-3 C \cos \left(c+\frac{3 d x}{2}\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-3 C \cos \left(2 c+\frac{3 d x}{2}\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+18 C \sin \left(\frac{d x}{2}\right)\right)}{6 a^2 d (\sec (c+d x)+1)^2}","\frac{(A-5 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{(A+C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"-1/6*(Cos[(c + d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*(3*C*Cos[c + (3*d*x)/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 3*C*Cos[2*c + (3*d*x)/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*C*Cos[(d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 9*C*Cos[c + (d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 3*C*Cos[c + (3*d*x)/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 3*C*Cos[2*c + (3*d*x)/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 6*A*Sin[(d*x)/2] + 18*C*Sin[(d*x)/2] + 6*A*Sin[c + (d*x)/2] - 6*C*Sin[c + (d*x)/2] - 4*A*Sin[c + (3*d*x)/2] + 8*C*Sin[c + (3*d*x)/2]))/(a^2*d*(1 + Sec[c + d*x])^2)","B",1
134,1,141,68,0.6341933,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(12 A \sin \left(c+\frac{d x}{2}\right)-10 A \sin \left(c+\frac{3 d x}{2}\right)+9 A d x \cos \left(c+\frac{d x}{2}\right)+3 A d x \cos \left(c+\frac{3 d x}{2}\right)+3 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 A \sin \left(\frac{d x}{2}\right)+9 A d x \cos \left(\frac{d x}{2}\right)+2 C \sin \left(c+\frac{3 d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","-\frac{2 (2 A-C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{A x}{a^2}-\frac{(A+C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*A*d*x*Cos[(d*x)/2] + 9*A*d*x*Cos[c + (d*x)/2] + 3*A*d*x*Cos[c + (3*d*x)/2] + 3*A*d*x*Cos[2*c + (3*d*x)/2] - 18*A*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2] + 12*A*Sin[c + (d*x)/2] - 10*A*Sin[c + (3*d*x)/2] + 2*C*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
135,1,195,82,0.9949418,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-30 A \sin \left(c+\frac{d x}{2}\right)+41 A \sin \left(c+\frac{3 d x}{2}\right)+9 A \sin \left(2 c+\frac{3 d x}{2}\right)+3 A \sin \left(2 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{5 d x}{2}\right)-36 A d x \cos \left(c+\frac{d x}{2}\right)-12 A d x \cos \left(c+\frac{3 d x}{2}\right)-12 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+66 A \sin \left(\frac{d x}{2}\right)-36 A d x \cos \left(\frac{d x}{2}\right)-12 C \sin \left(c+\frac{d x}{2}\right)+8 C \sin \left(c+\frac{3 d x}{2}\right)+12 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d}","\frac{(10 A+C) \sin (c+d x)}{3 a^2 d}-\frac{2 A \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{2 A x}{a^2}-\frac{(A+C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(-36*A*d*x*Cos[(d*x)/2] - 36*A*d*x*Cos[c + (d*x)/2] - 12*A*d*x*Cos[c + (3*d*x)/2] - 12*A*d*x*Cos[2*c + (3*d*x)/2] + 66*A*Sin[(d*x)/2] + 12*C*Sin[(d*x)/2] - 30*A*Sin[c + (d*x)/2] - 12*C*Sin[c + (d*x)/2] + 41*A*Sin[c + (3*d*x)/2] + 8*C*Sin[c + (3*d*x)/2] + 9*A*Sin[2*c + (3*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] + 3*A*Sin[3*c + (5*d*x)/2]))/(48*a^2*d)","B",1
136,1,281,137,1.2651681,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(36 d x (7 A+2 C) \cos \left(c+\frac{d x}{2}\right)+147 A \sin \left(c+\frac{d x}{2}\right)-239 A \sin \left(c+\frac{3 d x}{2}\right)-63 A \sin \left(2 c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{5 d x}{2}\right)-15 A \sin \left(3 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{7 d x}{2}\right)+3 A \sin \left(4 c+\frac{7 d x}{2}\right)+84 A d x \cos \left(c+\frac{3 d x}{2}\right)+84 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+36 d x (7 A+2 C) \cos \left(\frac{d x}{2}\right)-381 A \sin \left(\frac{d x}{2}\right)+96 C \sin \left(c+\frac{d x}{2}\right)-80 C \sin \left(c+\frac{3 d x}{2}\right)+24 C d x \cos \left(c+\frac{3 d x}{2}\right)+24 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-144 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\sec (c+d x)+1)^2}","-\frac{4 (4 A+C) \sin (c+d x)}{3 a^2 d}+\frac{(7 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{2 (4 A+C) \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{x (7 A+2 C)}{2 a^2}-\frac{(A+C) \sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*(36*(7*A + 2*C)*d*x*Cos[(d*x)/2] + 36*(7*A + 2*C)*d*x*Cos[c + (d*x)/2] + 84*A*d*x*Cos[c + (3*d*x)/2] + 24*C*d*x*Cos[c + (3*d*x)/2] + 84*A*d*x*Cos[2*c + (3*d*x)/2] + 24*C*d*x*Cos[2*c + (3*d*x)/2] - 381*A*Sin[(d*x)/2] - 144*C*Sin[(d*x)/2] + 147*A*Sin[c + (d*x)/2] + 96*C*Sin[c + (d*x)/2] - 239*A*Sin[c + (3*d*x)/2] - 80*C*Sin[c + (3*d*x)/2] - 63*A*Sin[2*c + (3*d*x)/2] - 15*A*Sin[2*c + (5*d*x)/2] - 15*A*Sin[3*c + (5*d*x)/2] + 3*A*Sin[3*c + (7*d*x)/2] + 3*A*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Sec[c + d*x])^2)","B",1
137,1,349,163,1.0381694,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-72 d x (5 A+2 C) \cos \left(c+\frac{d x}{2}\right)-156 A \sin \left(c+\frac{d x}{2}\right)+342 A \sin \left(c+\frac{3 d x}{2}\right)+118 A \sin \left(2 c+\frac{3 d x}{2}\right)+30 A \sin \left(2 c+\frac{5 d x}{2}\right)+30 A \sin \left(3 c+\frac{5 d x}{2}\right)-3 A \sin \left(3 c+\frac{7 d x}{2}\right)-3 A \sin \left(4 c+\frac{7 d x}{2}\right)+A \sin \left(4 c+\frac{9 d x}{2}\right)+A \sin \left(5 c+\frac{9 d x}{2}\right)-120 A d x \cos \left(c+\frac{3 d x}{2}\right)-120 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-72 d x (5 A+2 C) \cos \left(\frac{d x}{2}\right)+516 A \sin \left(\frac{d x}{2}\right)-120 C \sin \left(c+\frac{d x}{2}\right)+164 C \sin \left(c+\frac{3 d x}{2}\right)+36 C \sin \left(2 c+\frac{3 d x}{2}\right)+12 C \sin \left(2 c+\frac{5 d x}{2}\right)+12 C \sin \left(3 c+\frac{5 d x}{2}\right)-48 C d x \cos \left(c+\frac{3 d x}{2}\right)-48 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+264 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\sec (c+d x)+1)^2}","-\frac{(12 A+5 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(12 A+5 C) \sin (c+d x)}{a^2 d}-\frac{(5 A+2 C) \sin (c+d x) \cos (c+d x)}{a^2 d}-\frac{2 (5 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{x (5 A+2 C)}{a^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*(-72*(5*A + 2*C)*d*x*Cos[(d*x)/2] - 72*(5*A + 2*C)*d*x*Cos[c + (d*x)/2] - 120*A*d*x*Cos[c + (3*d*x)/2] - 48*C*d*x*Cos[c + (3*d*x)/2] - 120*A*d*x*Cos[2*c + (3*d*x)/2] - 48*C*d*x*Cos[2*c + (3*d*x)/2] + 516*A*Sin[(d*x)/2] + 264*C*Sin[(d*x)/2] - 156*A*Sin[c + (d*x)/2] - 120*C*Sin[c + (d*x)/2] + 342*A*Sin[c + (3*d*x)/2] + 164*C*Sin[c + (3*d*x)/2] + 118*A*Sin[2*c + (3*d*x)/2] + 36*C*Sin[2*c + (3*d*x)/2] + 30*A*Sin[2*c + (5*d*x)/2] + 12*C*Sin[2*c + (5*d*x)/2] + 30*A*Sin[3*c + (5*d*x)/2] + 12*C*Sin[3*c + (5*d*x)/2] - 3*A*Sin[3*c + (7*d*x)/2] - 3*A*Sin[4*c + (7*d*x)/2] + A*Sin[4*c + (9*d*x)/2] + A*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Sec[c + d*x])^2)","B",1
138,1,632,198,3.0689315,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(A+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sec (c) \sec ^2(c+d x) \left(-654 A \sin \left(c-\frac{d x}{2}\right)+654 A \sin \left(c+\frac{d x}{2}\right)-490 A \sin \left(2 c+\frac{d x}{2}\right)-350 A \sin \left(c+\frac{3 d x}{2}\right)+530 A \sin \left(2 c+\frac{3 d x}{2}\right)-350 A \sin \left(3 c+\frac{3 d x}{2}\right)+378 A \sin \left(c+\frac{5 d x}{2}\right)-150 A \sin \left(2 c+\frac{5 d x}{2}\right)+378 A \sin \left(3 c+\frac{5 d x}{2}\right)-150 A \sin \left(4 c+\frac{5 d x}{2}\right)+190 A \sin \left(2 c+\frac{7 d x}{2}\right)-30 A \sin \left(3 c+\frac{7 d x}{2}\right)+190 A \sin \left(4 c+\frac{7 d x}{2}\right)-30 A \sin \left(5 c+\frac{7 d x}{2}\right)+44 A \sin \left(3 c+\frac{9 d x}{2}\right)+44 A \sin \left(5 c+\frac{9 d x}{2}\right)-5 (98 A+247 C) \sin \left(\frac{d x}{2}\right)+5 (106 A+761 C) \sin \left(\frac{3 d x}{2}\right)-4329 C \sin \left(c-\frac{d x}{2}\right)+1989 C \sin \left(c+\frac{d x}{2}\right)-3575 C \sin \left(2 c+\frac{d x}{2}\right)-475 C \sin \left(c+\frac{3 d x}{2}\right)+2005 C \sin \left(2 c+\frac{3 d x}{2}\right)-2275 C \sin \left(3 c+\frac{3 d x}{2}\right)+2673 C \sin \left(c+\frac{5 d x}{2}\right)+105 C \sin \left(2 c+\frac{5 d x}{2}\right)+1593 C \sin \left(3 c+\frac{5 d x}{2}\right)-975 C \sin \left(4 c+\frac{5 d x}{2}\right)+1325 C \sin \left(2 c+\frac{7 d x}{2}\right)+255 C \sin \left(3 c+\frac{7 d x}{2}\right)+875 C \sin \left(4 c+\frac{7 d x}{2}\right)-195 C \sin \left(5 c+\frac{7 d x}{2}\right)+304 C \sin \left(3 c+\frac{9 d x}{2}\right)+90 C \sin \left(4 c+\frac{9 d x}{2}\right)+214 C \sin \left(5 c+\frac{9 d x}{2}\right)\right)+1920 (2 A+13 C) \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{240 a^3 d (\sec (c+d x)+1)^3 (A \cos (2 (c+d x))+A+2 C)}","-\frac{2 (11 A+76 C) \tan (c+d x)}{15 a^3 d}+\frac{(2 A+13 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(11 A+76 C) \tan (c+d x) \sec ^2(c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(2 A+13 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(A+11 C) \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"-1/240*(Cos[(c + d*x)/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(1920*(2*A + 13*C)*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-5*(98*A + 247*C)*Sin[(d*x)/2] + 5*(106*A + 761*C)*Sin[(3*d*x)/2] - 654*A*Sin[c - (d*x)/2] - 4329*C*Sin[c - (d*x)/2] + 654*A*Sin[c + (d*x)/2] + 1989*C*Sin[c + (d*x)/2] - 490*A*Sin[2*c + (d*x)/2] - 3575*C*Sin[2*c + (d*x)/2] - 350*A*Sin[c + (3*d*x)/2] - 475*C*Sin[c + (3*d*x)/2] + 530*A*Sin[2*c + (3*d*x)/2] + 2005*C*Sin[2*c + (3*d*x)/2] - 350*A*Sin[3*c + (3*d*x)/2] - 2275*C*Sin[3*c + (3*d*x)/2] + 378*A*Sin[c + (5*d*x)/2] + 2673*C*Sin[c + (5*d*x)/2] - 150*A*Sin[2*c + (5*d*x)/2] + 105*C*Sin[2*c + (5*d*x)/2] + 378*A*Sin[3*c + (5*d*x)/2] + 1593*C*Sin[3*c + (5*d*x)/2] - 150*A*Sin[4*c + (5*d*x)/2] - 975*C*Sin[4*c + (5*d*x)/2] + 190*A*Sin[2*c + (7*d*x)/2] + 1325*C*Sin[2*c + (7*d*x)/2] - 30*A*Sin[3*c + (7*d*x)/2] + 255*C*Sin[3*c + (7*d*x)/2] + 190*A*Sin[4*c + (7*d*x)/2] + 875*C*Sin[4*c + (7*d*x)/2] - 30*A*Sin[5*c + (7*d*x)/2] - 195*C*Sin[5*c + (7*d*x)/2] + 44*A*Sin[3*c + (9*d*x)/2] + 304*C*Sin[3*c + (9*d*x)/2] + 90*C*Sin[4*c + (9*d*x)/2] + 44*A*Sin[5*c + (9*d*x)/2] + 214*C*Sin[5*c + (9*d*x)/2])))/(a^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^3)","B",1
139,1,457,145,3.0582584,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(A+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sec (c) \sec (c+d x) \left(-10 A \sin \left(c-\frac{d x}{2}\right)+10 A \sin \left(c+\frac{d x}{2}\right)-20 A \sin \left(2 c+\frac{d x}{2}\right)+22 A \sin \left(2 c+\frac{3 d x}{2}\right)+10 A \sin \left(c+\frac{5 d x}{2}\right)+10 A \sin \left(3 c+\frac{5 d x}{2}\right)+2 A \sin \left(2 c+\frac{7 d x}{2}\right)+2 A \sin \left(4 c+\frac{7 d x}{2}\right)-5 (4 A+51 C) \sin \left(\frac{d x}{2}\right)+(22 A+567 C) \sin \left(\frac{3 d x}{2}\right)-600 C \sin \left(c-\frac{d x}{2}\right)+375 C \sin \left(c+\frac{d x}{2}\right)-480 C \sin \left(2 c+\frac{d x}{2}\right)-60 C \sin \left(c+\frac{3 d x}{2}\right)+402 C \sin \left(2 c+\frac{3 d x}{2}\right)-225 C \sin \left(3 c+\frac{3 d x}{2}\right)+315 C \sin \left(c+\frac{5 d x}{2}\right)+30 C \sin \left(2 c+\frac{5 d x}{2}\right)+240 C \sin \left(3 c+\frac{5 d x}{2}\right)-45 C \sin \left(4 c+\frac{5 d x}{2}\right)+72 C \sin \left(2 c+\frac{7 d x}{2}\right)+15 C \sin \left(3 c+\frac{7 d x}{2}\right)+57 C \sin \left(4 c+\frac{7 d x}{2}\right)\right)+2880 C \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{60 a^3 d (\sec (c+d x)+1)^3 (A \cos (2 (c+d x))+A+2 C)}","\frac{(2 A+27 C) \tan (c+d x)}{15 a^3 d}-\frac{3 C \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{3 C \tan (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(A-9 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(2880*C*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]*(-5*(4*A + 51*C)*Sin[(d*x)/2] + (22*A + 567*C)*Sin[(3*d*x)/2] - 10*A*Sin[c - (d*x)/2] - 600*C*Sin[c - (d*x)/2] + 10*A*Sin[c + (d*x)/2] + 375*C*Sin[c + (d*x)/2] - 20*A*Sin[2*c + (d*x)/2] - 480*C*Sin[2*c + (d*x)/2] - 60*C*Sin[c + (3*d*x)/2] + 22*A*Sin[2*c + (3*d*x)/2] + 402*C*Sin[2*c + (3*d*x)/2] - 225*C*Sin[3*c + (3*d*x)/2] + 10*A*Sin[c + (5*d*x)/2] + 315*C*Sin[c + (5*d*x)/2] + 30*C*Sin[2*c + (5*d*x)/2] + 10*A*Sin[3*c + (5*d*x)/2] + 240*C*Sin[3*c + (5*d*x)/2] - 45*C*Sin[4*c + (5*d*x)/2] + 2*A*Sin[2*c + (7*d*x)/2] + 72*C*Sin[2*c + (7*d*x)/2] + 15*C*Sin[3*c + (7*d*x)/2] + 2*A*Sin[4*c + (7*d*x)/2] + 57*C*Sin[4*c + (7*d*x)/2])))/(60*a^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^3)","B",1
140,1,236,123,1.6159665,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(A+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \left(15 (A-5 C) \sin \left(c+\frac{d x}{2}\right)-15 A \sin \left(c+\frac{3 d x}{2}\right)-3 A \sin \left(2 c+\frac{5 d x}{2}\right)-5 (3 A-29 C) \sin \left(\frac{d x}{2}\right)+95 C \sin \left(c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{3 d x}{2}\right)+22 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)+240 C \cos ^5\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{15 a^3 d (\sec (c+d x)+1)^3 (A \cos (2 (c+d x))+A+2 C)}","\frac{(6 A-29 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(3 A-7 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"-1/15*(Cos[(c + d*x)/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(240*C*Cos[(c + d*x)/2]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*(-5*(3*A - 29*C)*Sin[(d*x)/2] + 15*(A - 5*C)*Sin[c + (d*x)/2] - 15*A*Sin[c + (3*d*x)/2] + 95*C*Sin[c + (3*d*x)/2] - 15*C*Sin[2*c + (3*d*x)/2] - 3*A*Sin[2*c + (5*d*x)/2] + 22*C*Sin[2*c + (5*d*x)/2])))/(a^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^3)","A",1
141,1,121,104,0.5719058,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(-30 A \sin \left(c+\frac{d x}{2}\right)+20 A \sin \left(c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{3 d x}{2}\right)+7 A \sin \left(2 c+\frac{5 d x}{2}\right)+20 (2 A+C) \sin \left(\frac{d x}{2}\right)+10 C \sin \left(c+\frac{3 d x}{2}\right)+2 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{240 a^3 d}","\frac{(2 A+7 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A-C) \tan (c+d x)}{3 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(20*(2*A + C)*Sin[(d*x)/2] - 30*A*Sin[c + (d*x)/2] + 20*A*Sin[c + (3*d*x)/2] + 10*C*Sin[c + (3*d*x)/2] - 15*A*Sin[2*c + (3*d*x)/2] + 7*A*Sin[2*c + (5*d*x)/2] + 2*C*Sin[2*c + (5*d*x)/2]))/(240*a^3*d)","A",1
142,1,227,106,0.8699316,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(270 A \sin \left(c+\frac{d x}{2}\right)-230 A \sin \left(c+\frac{3 d x}{2}\right)+90 A \sin \left(2 c+\frac{3 d x}{2}\right)-64 A \sin \left(2 c+\frac{5 d x}{2}\right)+150 A d x \cos \left(c+\frac{d x}{2}\right)+75 A d x \cos \left(c+\frac{3 d x}{2}\right)+75 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 A \sin \left(\frac{d x}{2}\right)+150 A d x \cos \left(\frac{d x}{2}\right)-30 C \sin \left(c+\frac{d x}{2}\right)+30 C \sin \left(c+\frac{3 d x}{2}\right)+6 C \sin \left(2 c+\frac{5 d x}{2}\right)+30 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","-\frac{(22 A-3 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{A x}{a^3}-\frac{(7 A-3 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(150*A*d*x*Cos[(d*x)/2] + 150*A*d*x*Cos[c + (d*x)/2] + 75*A*d*x*Cos[c + (3*d*x)/2] + 75*A*d*x*Cos[2*c + (3*d*x)/2] + 15*A*d*x*Cos[2*c + (5*d*x)/2] + 15*A*d*x*Cos[3*c + (5*d*x)/2] - 370*A*Sin[(d*x)/2] + 30*C*Sin[(d*x)/2] + 270*A*Sin[c + (d*x)/2] - 30*C*Sin[c + (d*x)/2] - 230*A*Sin[c + (3*d*x)/2] + 30*C*Sin[c + (3*d*x)/2] + 90*A*Sin[2*c + (3*d*x)/2] - 64*A*Sin[2*c + (5*d*x)/2] + 6*C*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","B",1
143,1,283,120,1.9891055,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(1125 A \sin \left(c+\frac{d x}{2}\right)-1215 A \sin \left(c+\frac{3 d x}{2}\right)+225 A \sin \left(2 c+\frac{3 d x}{2}\right)-363 A \sin \left(2 c+\frac{5 d x}{2}\right)-75 A \sin \left(3 c+\frac{5 d x}{2}\right)-15 A \sin \left(3 c+\frac{7 d x}{2}\right)-15 A \sin \left(4 c+\frac{7 d x}{2}\right)+900 A d x \cos \left(c+\frac{d x}{2}\right)+450 A d x \cos \left(c+\frac{3 d x}{2}\right)+450 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+90 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+90 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-1755 A \sin \left(\frac{d x}{2}\right)+900 A d x \cos \left(\frac{d x}{2}\right)+120 C \sin \left(c+\frac{d x}{2}\right)-80 C \sin \left(c+\frac{3 d x}{2}\right)+60 C \sin \left(2 c+\frac{3 d x}{2}\right)-28 C \sin \left(2 c+\frac{5 d x}{2}\right)-160 C \sin \left(\frac{d x}{2}\right)\right)}{960 a^3 d}","\frac{2 (36 A+C) \sin (c+d x)}{15 a^3 d}-\frac{3 A \sin (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{3 A x}{a^3}-\frac{(9 A-C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"-1/960*(Sec[c/2]*Sec[(c + d*x)/2]^5*(900*A*d*x*Cos[(d*x)/2] + 900*A*d*x*Cos[c + (d*x)/2] + 450*A*d*x*Cos[c + (3*d*x)/2] + 450*A*d*x*Cos[2*c + (3*d*x)/2] + 90*A*d*x*Cos[2*c + (5*d*x)/2] + 90*A*d*x*Cos[3*c + (5*d*x)/2] - 1755*A*Sin[(d*x)/2] - 160*C*Sin[(d*x)/2] + 1125*A*Sin[c + (d*x)/2] + 120*C*Sin[c + (d*x)/2] - 1215*A*Sin[c + (3*d*x)/2] - 80*C*Sin[c + (3*d*x)/2] + 225*A*Sin[2*c + (3*d*x)/2] + 60*C*Sin[2*c + (3*d*x)/2] - 363*A*Sin[2*c + (5*d*x)/2] - 28*C*Sin[2*c + (5*d*x)/2] - 75*A*Sin[3*c + (5*d*x)/2] - 15*A*Sin[3*c + (7*d*x)/2] - 15*A*Sin[4*c + (7*d*x)/2]))/(a^3*d)","B",1
144,1,385,183,1.5477087,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(600 d x (13 A+2 C) \cos \left(c+\frac{d x}{2}\right)+7560 A \sin \left(c+\frac{d x}{2}\right)-9230 A \sin \left(c+\frac{3 d x}{2}\right)+930 A \sin \left(2 c+\frac{3 d x}{2}\right)-2782 A \sin \left(2 c+\frac{5 d x}{2}\right)-750 A \sin \left(3 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{7 d x}{2}\right)-105 A \sin \left(4 c+\frac{7 d x}{2}\right)+15 A \sin \left(4 c+\frac{9 d x}{2}\right)+15 A \sin \left(5 c+\frac{9 d x}{2}\right)+3900 A d x \cos \left(c+\frac{3 d x}{2}\right)+3900 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+600 d x (13 A+2 C) \cos \left(\frac{d x}{2}\right)-12760 A \sin \left(\frac{d x}{2}\right)+2160 C \sin \left(c+\frac{d x}{2}\right)-1840 C \sin \left(c+\frac{3 d x}{2}\right)+720 C \sin \left(2 c+\frac{3 d x}{2}\right)-512 C \sin \left(2 c+\frac{5 d x}{2}\right)+600 C d x \cos \left(c+\frac{3 d x}{2}\right)+600 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+120 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+120 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-2960 C \sin \left(\frac{d x}{2}\right)\right)}{3840 a^3 d}","-\frac{2 (76 A+11 C) \sin (c+d x)}{15 a^3 d}+\frac{(13 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(76 A+11 C) \sin (c+d x) \cos (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{x (13 A+2 C)}{2 a^3}-\frac{(11 A+C) \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(600*(13*A + 2*C)*d*x*Cos[(d*x)/2] + 600*(13*A + 2*C)*d*x*Cos[c + (d*x)/2] + 3900*A*d*x*Cos[c + (3*d*x)/2] + 600*C*d*x*Cos[c + (3*d*x)/2] + 3900*A*d*x*Cos[2*c + (3*d*x)/2] + 600*C*d*x*Cos[2*c + (3*d*x)/2] + 780*A*d*x*Cos[2*c + (5*d*x)/2] + 120*C*d*x*Cos[2*c + (5*d*x)/2] + 780*A*d*x*Cos[3*c + (5*d*x)/2] + 120*C*d*x*Cos[3*c + (5*d*x)/2] - 12760*A*Sin[(d*x)/2] - 2960*C*Sin[(d*x)/2] + 7560*A*Sin[c + (d*x)/2] + 2160*C*Sin[c + (d*x)/2] - 9230*A*Sin[c + (3*d*x)/2] - 1840*C*Sin[c + (3*d*x)/2] + 930*A*Sin[2*c + (3*d*x)/2] + 720*C*Sin[2*c + (3*d*x)/2] - 2782*A*Sin[2*c + (5*d*x)/2] - 512*C*Sin[2*c + (5*d*x)/2] - 750*A*Sin[3*c + (5*d*x)/2] - 105*A*Sin[3*c + (7*d*x)/2] - 105*A*Sin[4*c + (7*d*x)/2] + 15*A*Sin[4*c + (9*d*x)/2] + 15*A*Sin[5*c + (9*d*x)/2]))/(3840*a^3*d)","B",1
145,1,455,216,1.8448998,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(600 d x (23 A+6 C) \cos \left(c+\frac{d x}{2}\right)+11110 A \sin \left(c+\frac{d x}{2}\right)-15380 A \sin \left(c+\frac{3 d x}{2}\right)+380 A \sin \left(2 c+\frac{3 d x}{2}\right)-4777 A \sin \left(2 c+\frac{5 d x}{2}\right)-1625 A \sin \left(3 c+\frac{5 d x}{2}\right)-230 A \sin \left(3 c+\frac{7 d x}{2}\right)-230 A \sin \left(4 c+\frac{7 d x}{2}\right)+20 A \sin \left(4 c+\frac{9 d x}{2}\right)+20 A \sin \left(5 c+\frac{9 d x}{2}\right)-5 A \sin \left(5 c+\frac{11 d x}{2}\right)-5 A \sin \left(6 c+\frac{11 d x}{2}\right)+6900 A d x \cos \left(c+\frac{3 d x}{2}\right)+6900 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+1380 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+1380 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+600 d x (23 A+6 C) \cos \left(\frac{d x}{2}\right)-20410 A \sin \left(\frac{d x}{2}\right)+4500 C \sin \left(c+\frac{d x}{2}\right)-4860 C \sin \left(c+\frac{3 d x}{2}\right)+900 C \sin \left(2 c+\frac{3 d x}{2}\right)-1452 C \sin \left(2 c+\frac{5 d x}{2}\right)-300 C \sin \left(3 c+\frac{5 d x}{2}\right)-60 C \sin \left(3 c+\frac{7 d x}{2}\right)-60 C \sin \left(4 c+\frac{7 d x}{2}\right)+1800 C d x \cos \left(c+\frac{3 d x}{2}\right)+1800 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+360 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+360 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-7020 C \sin \left(\frac{d x}{2}\right)\right)}{3840 a^3 d}","-\frac{4 (34 A+9 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A+9 C) \sin (c+d x)}{5 a^3 d}-\frac{(23 A+6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(23 A+6 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{x (23 A+6 C)}{2 a^3}-\frac{(13 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"-1/3840*(Sec[c/2]*Sec[(c + d*x)/2]^5*(600*(23*A + 6*C)*d*x*Cos[(d*x)/2] + 600*(23*A + 6*C)*d*x*Cos[c + (d*x)/2] + 6900*A*d*x*Cos[c + (3*d*x)/2] + 1800*C*d*x*Cos[c + (3*d*x)/2] + 6900*A*d*x*Cos[2*c + (3*d*x)/2] + 1800*C*d*x*Cos[2*c + (3*d*x)/2] + 1380*A*d*x*Cos[2*c + (5*d*x)/2] + 360*C*d*x*Cos[2*c + (5*d*x)/2] + 1380*A*d*x*Cos[3*c + (5*d*x)/2] + 360*C*d*x*Cos[3*c + (5*d*x)/2] - 20410*A*Sin[(d*x)/2] - 7020*C*Sin[(d*x)/2] + 11110*A*Sin[c + (d*x)/2] + 4500*C*Sin[c + (d*x)/2] - 15380*A*Sin[c + (3*d*x)/2] - 4860*C*Sin[c + (3*d*x)/2] + 380*A*Sin[2*c + (3*d*x)/2] + 900*C*Sin[2*c + (3*d*x)/2] - 4777*A*Sin[2*c + (5*d*x)/2] - 1452*C*Sin[2*c + (5*d*x)/2] - 1625*A*Sin[3*c + (5*d*x)/2] - 300*C*Sin[3*c + (5*d*x)/2] - 230*A*Sin[3*c + (7*d*x)/2] - 60*C*Sin[3*c + (7*d*x)/2] - 230*A*Sin[4*c + (7*d*x)/2] - 60*C*Sin[4*c + (7*d*x)/2] + 20*A*Sin[4*c + (9*d*x)/2] + 20*A*Sin[5*c + (9*d*x)/2] - 5*A*Sin[5*c + (11*d*x)/2] - 5*A*Sin[6*c + (11*d*x)/2]))/(a^3*d)","B",1
146,1,746,232,4.3983148,"\int \frac{\sec ^5(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^5*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sec (c) \sec ^2(c+d x) \left(-17220 A \sin \left(c-\frac{d x}{2}\right)+17220 A \sin \left(c+\frac{d x}{2}\right)-14140 A \sin \left(2 c+\frac{d x}{2}\right)-9800 A \sin \left(c+\frac{3 d x}{2}\right)+15160 A \sin \left(2 c+\frac{3 d x}{2}\right)-9800 A \sin \left(3 c+\frac{3 d x}{2}\right)+10920 A \sin \left(c+\frac{5 d x}{2}\right)-4760 A \sin \left(2 c+\frac{5 d x}{2}\right)+10920 A \sin \left(3 c+\frac{5 d x}{2}\right)-4760 A \sin \left(4 c+\frac{5 d x}{2}\right)+5890 A \sin \left(2 c+\frac{7 d x}{2}\right)-1470 A \sin \left(3 c+\frac{7 d x}{2}\right)+5890 A \sin \left(4 c+\frac{7 d x}{2}\right)-1470 A \sin \left(5 c+\frac{7 d x}{2}\right)+2030 A \sin \left(3 c+\frac{9 d x}{2}\right)-210 A \sin \left(4 c+\frac{9 d x}{2}\right)+2030 A \sin \left(5 c+\frac{9 d x}{2}\right)-210 A \sin \left(6 c+\frac{9 d x}{2}\right)+320 A \sin \left(4 c+\frac{11 d x}{2}\right)+320 A \sin \left(6 c+\frac{11 d x}{2}\right)-14 (1010 A+5229 C) \sin \left(\frac{d x}{2}\right)+4 (3790 A+41667 C) \sin \left(\frac{3 d x}{2}\right)-183162 C \sin \left(c-\frac{d x}{2}\right)+100842 C \sin \left(c+\frac{d x}{2}\right)-155526 C \sin \left(2 c+\frac{d x}{2}\right)-37380 C \sin \left(c+\frac{3 d x}{2}\right)+101148 C \sin \left(2 c+\frac{3 d x}{2}\right)-102900 C \sin \left(3 c+\frac{3 d x}{2}\right)+119364 C \sin \left(c+\frac{5 d x}{2}\right)-8820 C \sin \left(2 c+\frac{5 d x}{2}\right)+78204 C \sin \left(3 c+\frac{5 d x}{2}\right)-49980 C \sin \left(4 c+\frac{5 d x}{2}\right)+64053 C \sin \left(2 c+\frac{7 d x}{2}\right)+3885 C \sin \left(3 c+\frac{7 d x}{2}\right)+44733 C \sin \left(4 c+\frac{7 d x}{2}\right)-15435 C \sin \left(5 c+\frac{7 d x}{2}\right)+21987 C \sin \left(3 c+\frac{9 d x}{2}\right)+3675 C \sin \left(4 c+\frac{9 d x}{2}\right)+16107 C \sin \left(5 c+\frac{9 d x}{2}\right)-2205 C \sin \left(6 c+\frac{9 d x}{2}\right)+3456 C \sin \left(4 c+\frac{11 d x}{2}\right)+840 C \sin \left(5 c+\frac{11 d x}{2}\right)+2616 C \sin \left(6 c+\frac{11 d x}{2}\right)\right)+53760 (2 A+21 C) \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3360 a^4 d (\sec (c+d x)+1)^4 (A \cos (2 (c+d x))+A+2 C)}","-\frac{32 (5 A+54 C) \tan (c+d x)}{105 a^4 d}+\frac{(2 A+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(10 A+129 C) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{16 (5 A+54 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(2 A+21 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(A+C) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3}",1,"-1/3360*(Cos[(c + d*x)/2]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(53760*(2*A + 21*C)*Cos[(c + d*x)/2]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-14*(1010*A + 5229*C)*Sin[(d*x)/2] + 4*(3790*A + 41667*C)*Sin[(3*d*x)/2] - 17220*A*Sin[c - (d*x)/2] - 183162*C*Sin[c - (d*x)/2] + 17220*A*Sin[c + (d*x)/2] + 100842*C*Sin[c + (d*x)/2] - 14140*A*Sin[2*c + (d*x)/2] - 155526*C*Sin[2*c + (d*x)/2] - 9800*A*Sin[c + (3*d*x)/2] - 37380*C*Sin[c + (3*d*x)/2] + 15160*A*Sin[2*c + (3*d*x)/2] + 101148*C*Sin[2*c + (3*d*x)/2] - 9800*A*Sin[3*c + (3*d*x)/2] - 102900*C*Sin[3*c + (3*d*x)/2] + 10920*A*Sin[c + (5*d*x)/2] + 119364*C*Sin[c + (5*d*x)/2] - 4760*A*Sin[2*c + (5*d*x)/2] - 8820*C*Sin[2*c + (5*d*x)/2] + 10920*A*Sin[3*c + (5*d*x)/2] + 78204*C*Sin[3*c + (5*d*x)/2] - 4760*A*Sin[4*c + (5*d*x)/2] - 49980*C*Sin[4*c + (5*d*x)/2] + 5890*A*Sin[2*c + (7*d*x)/2] + 64053*C*Sin[2*c + (7*d*x)/2] - 1470*A*Sin[3*c + (7*d*x)/2] + 3885*C*Sin[3*c + (7*d*x)/2] + 5890*A*Sin[4*c + (7*d*x)/2] + 44733*C*Sin[4*c + (7*d*x)/2] - 1470*A*Sin[5*c + (7*d*x)/2] - 15435*C*Sin[5*c + (7*d*x)/2] + 2030*A*Sin[3*c + (9*d*x)/2] + 21987*C*Sin[3*c + (9*d*x)/2] - 210*A*Sin[4*c + (9*d*x)/2] + 3675*C*Sin[4*c + (9*d*x)/2] + 2030*A*Sin[5*c + (9*d*x)/2] + 16107*C*Sin[5*c + (9*d*x)/2] - 210*A*Sin[6*c + (9*d*x)/2] - 2205*C*Sin[6*c + (9*d*x)/2] + 320*A*Sin[4*c + (11*d*x)/2] + 3456*C*Sin[4*c + (11*d*x)/2] + 840*C*Sin[5*c + (11*d*x)/2] + 320*A*Sin[6*c + (11*d*x)/2] + 2616*C*Sin[6*c + (11*d*x)/2])))/(a^4*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^4)","B",1
147,1,544,183,2.6618948,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) \sec (c) \sec (c+d x) \left(-126 A \sin \left(c-\frac{d x}{2}\right)+126 A \sin \left(c+\frac{d x}{2}\right)-210 A \sin \left(2 c+\frac{d x}{2}\right)+252 A \sin \left(2 c+\frac{3 d x}{2}\right)+132 A \sin \left(c+\frac{5 d x}{2}\right)+132 A \sin \left(3 c+\frac{5 d x}{2}\right)+42 A \sin \left(2 c+\frac{7 d x}{2}\right)+42 A \sin \left(4 c+\frac{7 d x}{2}\right)+6 A \sin \left(3 c+\frac{9 d x}{2}\right)+6 A \sin \left(5 c+\frac{9 d x}{2}\right)-70 (3 A+154 C) \sin \left(\frac{d x}{2}\right)+28 (9 A+671 C) \sin \left(\frac{3 d x}{2}\right)-20524 C \sin \left(c-\frac{d x}{2}\right)+14644 C \sin \left(c+\frac{d x}{2}\right)-16660 C \sin \left(2 c+\frac{d x}{2}\right)-4690 C \sin \left(c+\frac{3 d x}{2}\right)+14378 C \sin \left(2 c+\frac{3 d x}{2}\right)-9100 C \sin \left(3 c+\frac{3 d x}{2}\right)+11668 C \sin \left(c+\frac{5 d x}{2}\right)-630 C \sin \left(2 c+\frac{5 d x}{2}\right)+9358 C \sin \left(3 c+\frac{5 d x}{2}\right)-2940 C \sin \left(4 c+\frac{5 d x}{2}\right)+4228 C \sin \left(2 c+\frac{7 d x}{2}\right)+315 C \sin \left(3 c+\frac{7 d x}{2}\right)+3493 C \sin \left(4 c+\frac{7 d x}{2}\right)-420 C \sin \left(5 c+\frac{7 d x}{2}\right)+664 C \sin \left(3 c+\frac{9 d x}{2}\right)+105 C \sin \left(4 c+\frac{9 d x}{2}\right)+559 C \sin \left(5 c+\frac{9 d x}{2}\right)\right)+107520 C \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{840 a^4 d (\sec (c+d x)+1)^4 (A \cos (2 (c+d x))+A+2 C)}","\frac{2 (3 A+122 C) \tan (c+d x)}{105 a^4 d}+\frac{(3 A-88 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{4 C \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{4 C \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{2 (A-6 C) \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(107520*C*Cos[(c + d*x)/2]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c/2]*Sec[c]*Sec[c + d*x]*(-70*(3*A + 154*C)*Sin[(d*x)/2] + 28*(9*A + 671*C)*Sin[(3*d*x)/2] - 126*A*Sin[c - (d*x)/2] - 20524*C*Sin[c - (d*x)/2] + 126*A*Sin[c + (d*x)/2] + 14644*C*Sin[c + (d*x)/2] - 210*A*Sin[2*c + (d*x)/2] - 16660*C*Sin[2*c + (d*x)/2] - 4690*C*Sin[c + (3*d*x)/2] + 252*A*Sin[2*c + (3*d*x)/2] + 14378*C*Sin[2*c + (3*d*x)/2] - 9100*C*Sin[3*c + (3*d*x)/2] + 132*A*Sin[c + (5*d*x)/2] + 11668*C*Sin[c + (5*d*x)/2] - 630*C*Sin[2*c + (5*d*x)/2] + 132*A*Sin[3*c + (5*d*x)/2] + 9358*C*Sin[3*c + (5*d*x)/2] - 2940*C*Sin[4*c + (5*d*x)/2] + 42*A*Sin[2*c + (7*d*x)/2] + 4228*C*Sin[2*c + (7*d*x)/2] + 315*C*Sin[3*c + (7*d*x)/2] + 42*A*Sin[4*c + (7*d*x)/2] + 3493*C*Sin[4*c + (7*d*x)/2] - 420*C*Sin[5*c + (7*d*x)/2] + 6*A*Sin[3*c + (9*d*x)/2] + 664*C*Sin[3*c + (9*d*x)/2] + 105*C*Sin[4*c + (9*d*x)/2] + 6*A*Sin[5*c + (9*d*x)/2] + 559*C*Sin[5*c + (9*d*x)/2])))/(840*a^4*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^4)","B",1
148,1,283,161,2.4412333,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \left(6720 C \cos ^7\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec \left(\frac{c}{2}\right) \left(-70 (2 A-31 C) \sin \left(c+\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)+56 A \sin \left(2 c+\frac{5 d x}{2}\right)+8 A \sin \left(3 c+\frac{7 d x}{2}\right)+70 (2 A-49 C) \sin \left(\frac{d x}{2}\right)-2625 C \sin \left(c+\frac{3 d x}{2}\right)+735 C \sin \left(2 c+\frac{3 d x}{2}\right)-1015 C \sin \left(2 c+\frac{5 d x}{2}\right)+105 C \sin \left(3 c+\frac{5 d x}{2}\right)-160 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)\right)}{210 a^4 d (\sec (c+d x)+1)^4 (A \cos (2 (c+d x))+A+2 C)}","\frac{(16 A-215 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(8 A-55 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{2 (2 A-5 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"-1/210*(Cos[(c + d*x)/2]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(6720*C*Cos[(c + d*x)/2]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c/2]*(70*(2*A - 49*C)*Sin[(d*x)/2] - 70*(2*A - 31*C)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] - 2625*C*Sin[c + (3*d*x)/2] + 735*C*Sin[2*c + (3*d*x)/2] + 56*A*Sin[2*c + (5*d*x)/2] - 1015*C*Sin[2*c + (5*d*x)/2] + 105*C*Sin[3*c + (5*d*x)/2] + 8*A*Sin[3*c + (7*d*x)/2] - 160*C*Sin[3*c + (7*d*x)/2])))/(a^4*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^4)","A",1
149,1,151,138,0.6644115,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(-175 A \sin \left(c+\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)-105 A \sin \left(2 c+\frac{3 d x}{2}\right)+91 A \sin \left(2 c+\frac{5 d x}{2}\right)+13 A \sin \left(3 c+\frac{7 d x}{2}\right)+70 (4 A+3 C) \sin \left(\frac{d x}{2}\right)+126 C \sin \left(c+\frac{3 d x}{2}\right)+42 C \sin \left(2 c+\frac{5 d x}{2}\right)+6 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{6720 a^4 d}","\frac{4 (2 A+9 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(23 A-54 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 (3 A-4 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(70*(4*A + 3*C)*Sin[(d*x)/2] - 175*A*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 126*C*Sin[c + (3*d*x)/2] - 105*A*Sin[2*c + (3*d*x)/2] + 91*A*Sin[2*c + (5*d*x)/2] + 42*C*Sin[2*c + (5*d*x)/2] + 13*A*Sin[3*c + (7*d*x)/2] + 6*C*Sin[3*c + (7*d*x)/2]))/(6720*a^4*d)","A",1
150,1,171,142,0.7377997,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(-70 (9 A+2 C) \sin \left(c+\frac{d x}{2}\right)+441 A \sin \left(c+\frac{3 d x}{2}\right)-315 A \sin \left(2 c+\frac{3 d x}{2}\right)+147 A \sin \left(2 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{5 d x}{2}\right)+36 A \sin \left(3 c+\frac{7 d x}{2}\right)+70 (9 A+2 C) \sin \left(\frac{d x}{2}\right)+168 C \sin \left(c+\frac{3 d x}{2}\right)+56 C \sin \left(2 c+\frac{5 d x}{2}\right)+8 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{6720 a^4 d}","\frac{(6 A+13 C) \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{(6 A+13 C) \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{2 (4 A-3 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(70*(9*A + 2*C)*Sin[(d*x)/2] - 70*(9*A + 2*C)*Sin[c + (d*x)/2] + 441*A*Sin[c + (3*d*x)/2] + 168*C*Sin[c + (3*d*x)/2] - 315*A*Sin[2*c + (3*d*x)/2] + 147*A*Sin[2*c + (5*d*x)/2] + 56*C*Sin[2*c + (5*d*x)/2] - 105*A*Sin[3*c + (5*d*x)/2] + 36*A*Sin[3*c + (7*d*x)/2] + 8*C*Sin[3*c + (7*d*x)/2]))/(6720*a^4*d)","A",1
151,1,315,136,1.1452657,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(8260 A \sin \left(c+\frac{d x}{2}\right)-7140 A \sin \left(c+\frac{3 d x}{2}\right)+3780 A \sin \left(2 c+\frac{3 d x}{2}\right)-2800 A \sin \left(2 c+\frac{5 d x}{2}\right)+840 A \sin \left(3 c+\frac{5 d x}{2}\right)-520 A \sin \left(3 c+\frac{7 d x}{2}\right)+3675 A d x \cos \left(c+\frac{d x}{2}\right)+2205 A d x \cos \left(c+\frac{3 d x}{2}\right)+2205 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+735 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+735 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+105 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+105 A d x \cos \left(4 c+\frac{7 d x}{2}\right)-9940 A \sin \left(\frac{d x}{2}\right)+3675 A d x \cos \left(\frac{d x}{2}\right)-350 C \sin \left(c+\frac{d x}{2}\right)+336 C \sin \left(c+\frac{3 d x}{2}\right)-210 C \sin \left(2 c+\frac{3 d x}{2}\right)+182 C \sin \left(2 c+\frac{5 d x}{2}\right)+26 C \sin \left(3 c+\frac{7 d x}{2}\right)+560 C \sin \left(\frac{d x}{2}\right)\right)}{13440 a^4 d}","-\frac{8 (20 A-C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(55 A-8 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{A x}{a^4}-\frac{2 (5 A-2 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A+C) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(3675*A*d*x*Cos[(d*x)/2] + 3675*A*d*x*Cos[c + (d*x)/2] + 2205*A*d*x*Cos[c + (3*d*x)/2] + 2205*A*d*x*Cos[2*c + (3*d*x)/2] + 735*A*d*x*Cos[2*c + (5*d*x)/2] + 735*A*d*x*Cos[3*c + (5*d*x)/2] + 105*A*d*x*Cos[3*c + (7*d*x)/2] + 105*A*d*x*Cos[4*c + (7*d*x)/2] - 9940*A*Sin[(d*x)/2] + 560*C*Sin[(d*x)/2] + 8260*A*Sin[c + (d*x)/2] - 350*C*Sin[c + (d*x)/2] - 7140*A*Sin[c + (3*d*x)/2] + 336*C*Sin[c + (3*d*x)/2] + 3780*A*Sin[2*c + (3*d*x)/2] - 210*C*Sin[2*c + (3*d*x)/2] - 2800*A*Sin[2*c + (5*d*x)/2] + 182*C*Sin[2*c + (5*d*x)/2] + 840*A*Sin[3*c + (5*d*x)/2] - 520*A*Sin[3*c + (7*d*x)/2] + 26*C*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)","B",1
152,1,371,152,1.3103508,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(46130 A \sin \left(c+\frac{d x}{2}\right)-46116 A \sin \left(c+\frac{3 d x}{2}\right)+18060 A \sin \left(2 c+\frac{3 d x}{2}\right)-19292 A \sin \left(2 c+\frac{5 d x}{2}\right)+2100 A \sin \left(3 c+\frac{5 d x}{2}\right)-3791 A \sin \left(3 c+\frac{7 d x}{2}\right)-735 A \sin \left(4 c+\frac{7 d x}{2}\right)-105 A \sin \left(4 c+\frac{9 d x}{2}\right)-105 A \sin \left(5 c+\frac{9 d x}{2}\right)+29400 A d x \cos \left(c+\frac{d x}{2}\right)+17640 A d x \cos \left(c+\frac{3 d x}{2}\right)+17640 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+5880 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+5880 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+840 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+840 A d x \cos \left(4 c+\frac{7 d x}{2}\right)-60830 A \sin \left(\frac{d x}{2}\right)+29400 A d x \cos \left(\frac{d x}{2}\right)+2520 C \sin \left(c+\frac{d x}{2}\right)-1764 C \sin \left(c+\frac{3 d x}{2}\right)+1260 C \sin \left(2 c+\frac{3 d x}{2}\right)-588 C \sin \left(2 c+\frac{5 d x}{2}\right)+420 C \sin \left(3 c+\frac{5 d x}{2}\right)-144 C \sin \left(3 c+\frac{7 d x}{2}\right)-2520 C \sin \left(\frac{d x}{2}\right)\right)}{26880 a^4 d}","\frac{2 (332 A+3 C) \sin (c+d x)}{105 a^4 d}-\frac{(88 A-3 C) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{4 A \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{4 A x}{a^4}-\frac{2 (6 A-C) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A+C) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"-1/26880*(Sec[c/2]*Sec[(c + d*x)/2]^7*(29400*A*d*x*Cos[(d*x)/2] + 29400*A*d*x*Cos[c + (d*x)/2] + 17640*A*d*x*Cos[c + (3*d*x)/2] + 17640*A*d*x*Cos[2*c + (3*d*x)/2] + 5880*A*d*x*Cos[2*c + (5*d*x)/2] + 5880*A*d*x*Cos[3*c + (5*d*x)/2] + 840*A*d*x*Cos[3*c + (7*d*x)/2] + 840*A*d*x*Cos[4*c + (7*d*x)/2] - 60830*A*Sin[(d*x)/2] - 2520*C*Sin[(d*x)/2] + 46130*A*Sin[c + (d*x)/2] + 2520*C*Sin[c + (d*x)/2] - 46116*A*Sin[c + (3*d*x)/2] - 1764*C*Sin[c + (3*d*x)/2] + 18060*A*Sin[2*c + (3*d*x)/2] + 1260*C*Sin[2*c + (3*d*x)/2] - 19292*A*Sin[2*c + (5*d*x)/2] - 588*C*Sin[2*c + (5*d*x)/2] + 2100*A*Sin[3*c + (5*d*x)/2] + 420*C*Sin[3*c + (5*d*x)/2] - 3791*A*Sin[3*c + (7*d*x)/2] - 144*C*Sin[3*c + (7*d*x)/2] - 735*A*Sin[4*c + (7*d*x)/2] - 105*A*Sin[4*c + (9*d*x)/2] - 105*A*Sin[5*c + (9*d*x)/2]))/(a^4*d)","B",1
153,1,505,215,2.4497398,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(14700 d x (21 A+2 C) \cos \left(c+\frac{d x}{2}\right)+386190 A \sin \left(c+\frac{d x}{2}\right)-422478 A \sin \left(c+\frac{3 d x}{2}\right)+132930 A \sin \left(2 c+\frac{3 d x}{2}\right)-181461 A \sin \left(2 c+\frac{5 d x}{2}\right)+3675 A \sin \left(3 c+\frac{5 d x}{2}\right)-36003 A \sin \left(3 c+\frac{7 d x}{2}\right)-9555 A \sin \left(4 c+\frac{7 d x}{2}\right)-945 A \sin \left(4 c+\frac{9 d x}{2}\right)-945 A \sin \left(5 c+\frac{9 d x}{2}\right)+105 A \sin \left(5 c+\frac{11 d x}{2}\right)+105 A \sin \left(6 c+\frac{11 d x}{2}\right)+185220 A d x \cos \left(c+\frac{3 d x}{2}\right)+185220 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+61740 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+61740 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+8820 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+8820 A d x \cos \left(4 c+\frac{7 d x}{2}\right)+14700 d x (21 A+2 C) \cos \left(\frac{d x}{2}\right)-539490 A \sin \left(\frac{d x}{2}\right)+66080 C \sin \left(c+\frac{d x}{2}\right)-57120 C \sin \left(c+\frac{3 d x}{2}\right)+30240 C \sin \left(2 c+\frac{3 d x}{2}\right)-22400 C \sin \left(2 c+\frac{5 d x}{2}\right)+6720 C \sin \left(3 c+\frac{5 d x}{2}\right)-4160 C \sin \left(3 c+\frac{7 d x}{2}\right)+17640 C d x \cos \left(c+\frac{3 d x}{2}\right)+17640 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+5880 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+5880 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+840 C d x \cos \left(3 c+\frac{7 d x}{2}\right)+840 C d x \cos \left(4 c+\frac{7 d x}{2}\right)-79520 C \sin \left(\frac{d x}{2}\right)\right)}{107520 a^4 d}","-\frac{32 (54 A+5 C) \sin (c+d x)}{105 a^4 d}+\frac{(21 A+2 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{16 (54 A+5 C) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(129 A+10 C) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{x (21 A+2 C)}{2 a^4}-\frac{(A+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{2 A \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(14700*(21*A + 2*C)*d*x*Cos[(d*x)/2] + 14700*(21*A + 2*C)*d*x*Cos[c + (d*x)/2] + 185220*A*d*x*Cos[c + (3*d*x)/2] + 17640*C*d*x*Cos[c + (3*d*x)/2] + 185220*A*d*x*Cos[2*c + (3*d*x)/2] + 17640*C*d*x*Cos[2*c + (3*d*x)/2] + 61740*A*d*x*Cos[2*c + (5*d*x)/2] + 5880*C*d*x*Cos[2*c + (5*d*x)/2] + 61740*A*d*x*Cos[3*c + (5*d*x)/2] + 5880*C*d*x*Cos[3*c + (5*d*x)/2] + 8820*A*d*x*Cos[3*c + (7*d*x)/2] + 840*C*d*x*Cos[3*c + (7*d*x)/2] + 8820*A*d*x*Cos[4*c + (7*d*x)/2] + 840*C*d*x*Cos[4*c + (7*d*x)/2] - 539490*A*Sin[(d*x)/2] - 79520*C*Sin[(d*x)/2] + 386190*A*Sin[c + (d*x)/2] + 66080*C*Sin[c + (d*x)/2] - 422478*A*Sin[c + (3*d*x)/2] - 57120*C*Sin[c + (3*d*x)/2] + 132930*A*Sin[2*c + (3*d*x)/2] + 30240*C*Sin[2*c + (3*d*x)/2] - 181461*A*Sin[2*c + (5*d*x)/2] - 22400*C*Sin[2*c + (5*d*x)/2] + 3675*A*Sin[3*c + (5*d*x)/2] + 6720*C*Sin[3*c + (5*d*x)/2] - 36003*A*Sin[3*c + (7*d*x)/2] - 4160*C*Sin[3*c + (7*d*x)/2] - 9555*A*Sin[4*c + (7*d*x)/2] - 945*A*Sin[4*c + (9*d*x)/2] - 945*A*Sin[5*c + (9*d*x)/2] + 105*A*Sin[5*c + (11*d*x)/2] + 105*A*Sin[6*c + (11*d*x)/2]))/(107520*a^4*d)","B",1
154,1,575,248,2.9523523,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(58800 d x (11 A+2 C) \cos \left(c+\frac{d x}{2}\right)+687260 A \sin \left(c+\frac{d x}{2}\right)-814107 A \sin \left(c+\frac{3 d x}{2}\right)+204645 A \sin \left(2 c+\frac{3 d x}{2}\right)-357609 A \sin \left(2 c+\frac{5 d x}{2}\right)-18025 A \sin \left(3 c+\frac{5 d x}{2}\right)-72522 A \sin \left(3 c+\frac{7 d x}{2}\right)-24010 A \sin \left(4 c+\frac{7 d x}{2}\right)-2310 A \sin \left(4 c+\frac{9 d x}{2}\right)-2310 A \sin \left(5 c+\frac{9 d x}{2}\right)+175 A \sin \left(5 c+\frac{11 d x}{2}\right)+175 A \sin \left(6 c+\frac{11 d x}{2}\right)-35 A \sin \left(6 c+\frac{13 d x}{2}\right)-35 A \sin \left(7 c+\frac{13 d x}{2}\right)+388080 A d x \cos \left(c+\frac{3 d x}{2}\right)+388080 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+129360 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+129360 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+18480 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+18480 A d x \cos \left(4 c+\frac{7 d x}{2}\right)+58800 d x (11 A+2 C) \cos \left(\frac{d x}{2}\right)-1010660 A \sin \left(\frac{d x}{2}\right)+184520 C \sin \left(c+\frac{d x}{2}\right)-184464 C \sin \left(c+\frac{3 d x}{2}\right)+72240 C \sin \left(2 c+\frac{3 d x}{2}\right)-77168 C \sin \left(2 c+\frac{5 d x}{2}\right)+8400 C \sin \left(3 c+\frac{5 d x}{2}\right)-15164 C \sin \left(3 c+\frac{7 d x}{2}\right)-2940 C \sin \left(4 c+\frac{7 d x}{2}\right)-420 C \sin \left(4 c+\frac{9 d x}{2}\right)-420 C \sin \left(5 c+\frac{9 d x}{2}\right)+70560 C d x \cos \left(c+\frac{3 d x}{2}\right)+70560 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+23520 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+23520 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+3360 C d x \cos \left(3 c+\frac{7 d x}{2}\right)+3360 C d x \cos \left(4 c+\frac{7 d x}{2}\right)-243320 C \sin \left(\frac{d x}{2}\right)\right)}{107520 a^4 d}","-\frac{4 (454 A+83 C) \sin ^3(c+d x)}{105 a^4 d}+\frac{4 (454 A+83 C) \sin (c+d x)}{35 a^4 d}-\frac{2 (11 A+2 C) \sin (c+d x) \cos (c+d x)}{a^4 d}-\frac{4 (11 A+2 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac{(178 A+31 C) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{2 x (11 A+2 C)}{a^4}-\frac{2 (8 A+C) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"-1/107520*(Sec[c/2]*Sec[(c + d*x)/2]^7*(58800*(11*A + 2*C)*d*x*Cos[(d*x)/2] + 58800*(11*A + 2*C)*d*x*Cos[c + (d*x)/2] + 388080*A*d*x*Cos[c + (3*d*x)/2] + 70560*C*d*x*Cos[c + (3*d*x)/2] + 388080*A*d*x*Cos[2*c + (3*d*x)/2] + 70560*C*d*x*Cos[2*c + (3*d*x)/2] + 129360*A*d*x*Cos[2*c + (5*d*x)/2] + 23520*C*d*x*Cos[2*c + (5*d*x)/2] + 129360*A*d*x*Cos[3*c + (5*d*x)/2] + 23520*C*d*x*Cos[3*c + (5*d*x)/2] + 18480*A*d*x*Cos[3*c + (7*d*x)/2] + 3360*C*d*x*Cos[3*c + (7*d*x)/2] + 18480*A*d*x*Cos[4*c + (7*d*x)/2] + 3360*C*d*x*Cos[4*c + (7*d*x)/2] - 1010660*A*Sin[(d*x)/2] - 243320*C*Sin[(d*x)/2] + 687260*A*Sin[c + (d*x)/2] + 184520*C*Sin[c + (d*x)/2] - 814107*A*Sin[c + (3*d*x)/2] - 184464*C*Sin[c + (3*d*x)/2] + 204645*A*Sin[2*c + (3*d*x)/2] + 72240*C*Sin[2*c + (3*d*x)/2] - 357609*A*Sin[2*c + (5*d*x)/2] - 77168*C*Sin[2*c + (5*d*x)/2] - 18025*A*Sin[3*c + (5*d*x)/2] + 8400*C*Sin[3*c + (5*d*x)/2] - 72522*A*Sin[3*c + (7*d*x)/2] - 15164*C*Sin[3*c + (7*d*x)/2] - 24010*A*Sin[4*c + (7*d*x)/2] - 2940*C*Sin[4*c + (7*d*x)/2] - 2310*A*Sin[4*c + (9*d*x)/2] - 420*C*Sin[4*c + (9*d*x)/2] - 2310*A*Sin[5*c + (9*d*x)/2] - 420*C*Sin[5*c + (9*d*x)/2] + 175*A*Sin[5*c + (11*d*x)/2] + 175*A*Sin[6*c + (11*d*x)/2] - 35*A*Sin[6*c + (13*d*x)/2] - 35*A*Sin[7*c + (13*d*x)/2]))/(a^4*d)","B",1
155,1,143,223,1.1945428,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} ((2871 A+3020 C) \cos (c+d x)+13 (99 A+80 C) \cos (2 (c+d x))+1287 A \cos (3 (c+d x))+198 A \cos (4 (c+d x))+198 A \cos (5 (c+d x))+1089 A+1040 C \cos (3 (c+d x))+160 C \cos (4 (c+d x))+160 C \cos (5 (c+d x))+1510 C)}{3465 d}","\frac{2 a (99 A+80 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (99 A+80 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+80 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{4 a (99 A+80 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}",1,"((1089*A + 1510*C + (2871*A + 3020*C)*Cos[c + d*x] + 13*(99*A + 80*C)*Cos[2*(c + d*x)] + 1287*A*Cos[3*(c + d*x)] + 1040*C*Cos[3*(c + d*x)] + 198*A*Cos[4*(c + d*x)] + 160*C*Cos[4*(c + d*x)] + 198*A*Cos[5*(c + d*x)] + 160*C*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3465*d)","A",1
156,1,122,180,1.0807666,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} (2 (63 A+88 C) \cos (c+d x)+11 (21 A+16 C) \cos (2 (c+d x))+42 A \cos (3 (c+d x))+42 A \cos (4 (c+d x))+189 A+32 C \cos (3 (c+d x))+32 C \cos (4 (c+d x))+214 C)}{315 d}","\frac{2 (21 A+16 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+16 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a (21 A+16 C) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}",1,"((189*A + 214*C + 2*(63*A + 88*C)*Cos[c + d*x] + 11*(21*A + 16*C)*Cos[2*(c + d*x)] + 42*A*Cos[3*(c + d*x)] + 32*C*Cos[3*(c + d*x)] + 42*A*Cos[4*(c + d*x)] + 32*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(315*d)","A",1
157,1,99,137,0.8742095,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} (3 (35 A+36 C) \cos (c+d x)+(35 A+24 C) \cos (2 (c+d x))+35 A \cos (3 (c+d x))+35 A+24 C \cos (3 (c+d x))+54 C)}{105 d}","\frac{2 (35 A+18 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (35 A+27 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{7 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}",1,"((35*A + 54*C + 3*(35*A + 36*C)*Cos[c + d*x] + (35*A + 24*C)*Cos[2*(c + d*x)] + 35*A*Cos[3*(c + d*x)] + 24*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(105*d)","A",1
158,1,71,95,1.0653844,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} ((15 A+8 C) \cos (2 (c+d x))+15 A+8 C \cos (c+d x)+14 C)}{15 d}","\frac{2 a (15 A+7 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}-\frac{4 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}",1,"((15*A + 14*C + 8*C*Cos[c + d*x] + (15*A + 8*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d)","A",1
159,1,96,96,0.6829012,"\int \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(3 A \cos (c+d x) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)+C (2 \cos (c+d x)+1) \sqrt{\sec (c+d x)-1}\right)}{3 d \sqrt{\sec (c+d x)-1}}","\frac{2 \sqrt{a} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a C \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(2*(3*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[c + d*x] + C*(1 + 2*Cos[c + d*x])*Sqrt[-1 + Sec[c + d*x]])*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3*d*Sqrt[-1 + Sec[c + d*x]])","A",1
160,1,84,94,0.3710858,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{a \tan (c+d x) \left(\sqrt{1-\sec (c+d x)} (A \cos (c+d x)+2 C)+A \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{a (A-2 C) \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{\sqrt{a} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(a*(A*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + (2*C + A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
161,1,108,110,0.5432765,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (3 A+8 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 A \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{8 d}","\frac{\sqrt{a} (3 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a A \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(3*A + 8*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*A*Sqrt[Cos[c + d*x]]*(2*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(8*d)","A",1
162,1,117,153,0.3909795,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 A \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+C \left(\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{a (5 A+8 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (5 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{a A \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"((C*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]]) + 2*A*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","C",1
163,1,70,196,0.1996966,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(A \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+C \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)\right)}{d}","\frac{a (35 A+48 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (35 A+48 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+48 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}",1,"(2*(C*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]] + A*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
164,1,144,225,1.4592146,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} ((4147 A+4228 C) \cos (c+d x)+2 (737 A+728 C) \cos (2 (c+d x))+1859 A \cos (3 (c+d x))+286 A \cos (4 (c+d x))+286 A \cos (5 (c+d x))+1188 A+1456 C \cos (3 (c+d x))+224 C \cos (4 (c+d x))+224 C \cos (5 (c+d x))+1652 C)}{2310 d}","\frac{2 a^2 (33 A+28 C) \tan (c+d x) \sec ^3(c+d x)}{231 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (143 A+112 C) \tan (c+d x)}{165 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (143 A+112 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{385 d}-\frac{4 a (143 A+112 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{1155 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{33 d}",1,"(a*(1188*A + 1652*C + (4147*A + 4228*C)*Cos[c + d*x] + 2*(737*A + 728*C)*Cos[2*(c + d*x)] + 1859*A*Cos[3*(c + d*x)] + 1456*C*Cos[3*(c + d*x)] + 286*A*Cos[4*(c + d*x)] + 224*C*Cos[4*(c + d*x)] + 286*A*Cos[5*(c + d*x)] + 224*C*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(2310*d)","A",1
165,1,121,174,1.3093102,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} ((567 A+748 C) \cos (c+d x)+(882 A+748 C) \cos (2 (c+d x))+189 A \cos (3 (c+d x))+189 A \cos (4 (c+d x))+693 A+136 C \cos (3 (c+d x))+136 C \cos (4 (c+d x))+752 C)}{630 d}","\frac{8 a^2 (63 A+47 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (63 A+22 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+47 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{21 a d}",1,"(a*(693*A + 752*C + (567*A + 748*C)*Cos[c + d*x] + (882*A + 748*C)*Cos[2*(c + d*x)] + 189*A*Cos[3*(c + d*x)] + 136*C*Cos[3*(c + d*x)] + 189*A*Cos[4*(c + d*x)] + 136*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(630*d)","A",1
166,1,100,132,1.2421238,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} ((525 A+468 C) \cos (c+d x)+2 (35 A+52 C) \cos (2 (c+d x))+175 A \cos (3 (c+d x))+70 A+104 C \cos (3 (c+d x))+164 C)}{210 d}","\frac{8 a^2 (35 A+19 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (35 A+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}-\frac{4 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}",1,"(a*(70*A + 164*C + (525*A + 468*C)*Cos[c + d*x] + 2*(35*A + 52*C)*Cos[2*(c + d*x)] + 175*A*Cos[3*(c + d*x)] + 104*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d)","A",1
167,1,122,133,1.2224083,"\int (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)-1} ((5 A+6 C) \cos (2 (c+d x))+5 A+6 C \cos (c+d x)+8 C)+10 A \cos ^2(c+d x) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{5 d \sqrt{\sec (c+d x)-1}}","\frac{2 a^{3/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (5 A+4 C) \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a*(10*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[c + d*x]^2 + (5*A + 8*C + 6*C*Cos[c + d*x] + (5*A + 6*C)*Cos[2*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sec[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(5*d*Sqrt[-1 + Sec[c + d*x]])","A",1
168,1,113,136,1.2472996,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \tan (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)-1} (3 A \cos (2 (c+d x))+3 A+20 C \cos (c+d x)+4 C)+18 A \cos (c+d x) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{6 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)-1}}","\frac{3 a^{3/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a^2 (3 A-8 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{a (3 A-2 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}",1,"(a*(18*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[c + d*x] + (3*A + 4*C + 20*C*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(6*d*(1 + Cos[c + d*x])*Sqrt[-1 + Sec[c + d*x]])","A",1
169,1,109,151,0.7652117,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (7 A+8 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (7 A \cos (c+d x)+A \cos (2 (c+d x))+A+8 C)\right)}{8 d}","\frac{a^{3/2} (7 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}-\frac{a (A-4 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(7*A + 8*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(A + 8*C + 7*A*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
170,1,118,155,1.2161969,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\cos (c+d x) \sqrt{\sec (c+d x)-1} (22 A \cos (c+d x)+4 A \cos (2 (c+d x))+37 A+24 C)+(33 A+72 C) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{24 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)-1}}","\frac{a^{3/2} (11 A+24 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (19 A+24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{a A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"(a*((33*A + 72*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]] + Cos[c + d*x]*(37*A + 24*C + 22*A*Cos[c + d*x] + 4*A*Cos[2*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(24*d*(1 + Cos[c + d*x])*Sqrt[-1 + Sec[c + d*x]])","A",1
171,1,140,200,1.5900629,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (75 A+112 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((62 A+32 C) \cos (c+d x)+20 A \cos (2 (c+d x))+4 A \cos (3 (c+d x))+95 A+112 C)\right)}{128 d}","\frac{a^{3/2} (75 A+112 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (75 A+112 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (13 A+16 C) \sin (c+d x) \cos (c+d x)}{32 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{8 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(75*A + 112*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (95*A + 112*C + (62*A + 32*C)*Cos[c + d*x] + 20*A*Cos[2*(c + d*x)] + 4*A*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(128*d)","A",1
172,1,159,245,2.1961788,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(15 \sqrt{2} (133 A+176 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (1007 A+880 C) \cos (c+d x)+4 (181 A+80 C) \cos (2 (c+d x))+228 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+2671 A+2960 C)\right)}{3840 d}","\frac{a^{3/2} (133 A+176 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (67 A+80 C) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (133 A+176 C) \sin (c+d x) \cos (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{3 a A \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(133*A + 176*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (2671*A + 2960*C + 2*(1007*A + 880*C)*Cos[c + d*x] + 4*(181*A + 80*C)*Cos[2*(c + d*x)] + 228*A*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
173,1,169,273,2.0089787,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \sqrt{a (\sec (c+d x)+1)} (1120 (286 A+347 C) \cos (c+d x)+14 (32747 A+30334 C) \cos (2 (c+d x))+141570 A \cos (3 (c+d x))+156585 A \cos (4 (c+d x))+20878 A \cos (5 (c+d x))+20878 A \cos (6 (c+d x))+322751 A+125520 C \cos (3 (c+d x))+125520 C \cos (4 (c+d x))+16736 C \cos (5 (c+d x))+16736 C \cos (6 (c+d x))+343612 C)}{180180 d}","\frac{2 a^3 (2717 A+2224 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (10439 A+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (143 A+136 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}-\frac{4 a^2 (10439 A+8368 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+8368 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac{10 a C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d}",1,"(a^2*(322751*A + 343612*C + 1120*(286*A + 347*C)*Cos[c + d*x] + 14*(32747*A + 30334*C)*Cos[2*(c + d*x)] + 141570*A*Cos[3*(c + d*x)] + 125520*C*Cos[3*(c + d*x)] + 156585*A*Cos[4*(c + d*x)] + 125520*C*Cos[4*(c + d*x)] + 20878*A*Cos[5*(c + d*x)] + 16736*C*Cos[5*(c + d*x)] + 20878*A*Cos[6*(c + d*x)] + 16736*C*Cos[6*(c + d*x)])*Sec[c + d*x]^6*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(180180*d)","A",1
174,1,147,211,1.55776,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} (2 (4983 A+5014 C) \cos (c+d x)+52 (66 A+71 C) \cos (2 (c+d x))+4587 A \cos (3 (c+d x))+759 A \cos (4 (c+d x))+759 A \cos (5 (c+d x))+2673 A+3692 C \cos (3 (c+d x))+568 C \cos (4 (c+d x))+568 C \cos (5 (c+d x))+3628 C)}{2772 d}","\frac{64 a^3 (33 A+25 C) \tan (c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (33 A+25 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{693 d}+\frac{2 (99 A+26 C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (33 A+25 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{231 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d}+\frac{10 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{99 a d}",1,"(a^2*(2673*A + 3628*C + 2*(4983*A + 5014*C)*Cos[c + d*x] + 52*(66*A + 71*C)*Cos[2*(c + d*x)] + 4587*A*Cos[3*(c + d*x)] + 3692*C*Cos[3*(c + d*x)] + 759*A*Cos[4*(c + d*x)] + 568*C*Cos[4*(c + d*x)] + 759*A*Cos[5*(c + d*x)] + 568*C*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(2772*d)","A",1
175,1,125,169,1.5405911,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} (4 (441 A+698 C) \cos (c+d x)+4 (966 A+803 C) \cos (2 (c+d x))+588 A \cos (3 (c+d x))+903 A \cos (4 (c+d x))+2961 A+584 C \cos (3 (c+d x))+584 C \cos (4 (c+d x))+2908 C)}{1260 d}","\frac{64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (21 A+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a (21 A+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}-\frac{4 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}",1,"(a^2*(2961*A + 2908*C + 4*(441*A + 698*C)*Cos[c + d*x] + 4*(966*A + 803*C)*Cos[2*(c + d*x)] + 588*A*Cos[3*(c + d*x)] + 584*C*Cos[3*(c + d*x)] + 903*A*Cos[4*(c + d*x)] + 584*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d)","A",1
176,1,151,170,1.9887216,"\int (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)-1} ((84 A+93 C) \cos (c+d x)+(7 A+23 C) \cos (2 (c+d x))+28 A \cos (3 (c+d x))+7 A+23 C \cos (3 (c+d x))+29 C)+42 A \cos ^3(c+d x) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{21 d \sqrt{\sec (c+d x)-1}}","\frac{2 a^{5/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (49 A+32 C) \tan (c+d x)}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (7 A+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*(42*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[c + d*x]^3 + (7*A + 29*C + (84*A + 93*C)*Cos[c + d*x] + (7*A + 23*C)*Cos[2*(c + d*x)] + 28*A*Cos[3*(c + d*x)] + 23*C*Cos[3*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(21*d*Sqrt[-1 + Sec[c + d*x]])","A",1
177,1,145,173,1.8446107,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan (c+d x) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)-1} ((45 A+112 C) \cos (c+d x)+4 (15 A+43 C) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+60 A+196 C)+300 A \cos ^2(c+d x) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{60 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)-1}}","\frac{5 a^{5/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 (15 A+64 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (15 A-16 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}-\frac{a (5 A-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{d}",1,"(a^2*(300*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[c + d*x]^2 + (60*A + 196*C + (45*A + 112*C)*Cos[c + d*x] + 4*(15*A + 43*C)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(60*d*(1 + Cos[c + d*x])*Sqrt[-1 + Sec[c + d*x]])","A",1
178,1,137,188,0.8882114,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(6 \sqrt{2} (19 A+8 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((9 A+128 C) \cos (c+d x)+33 A \cos (2 (c+d x))+3 A \cos (3 (c+d x))+33 A+16 C)\right)}{48 d}","\frac{a^{5/2} (19 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (27 A-56 C) \sin (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (A-8 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}-\frac{a (3 A-4 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{6 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{5/2}}{2 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(19*A + 8*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(33*A + 16*C + (9*A + 128*C)*Cos[c + d*x] + 33*A*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
179,1,132,192,1.5870028,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)-1} (3 (27 A+8 C) \cos (c+d x)+17 A \cos (2 (c+d x))+2 A \cos (3 (c+d x))+17 A+48 C)+15 (5 A+8 C) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{24 d (\cos (c+d x)+1) \sqrt{\sec (c+d x)-1}}","\frac{5 a^{5/2} (5 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (49 A-24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (3 A-8 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d}+\frac{5 a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d}",1,"(a^2*(15*(5*A + 8*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]] + (17*A + 48*C + 3*(27*A + 8*C)*Cos[c + d*x] + 17*A*Cos[2*(c + d*x)] + 2*A*Cos[3*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(24*d*(1 + Cos[c + d*x])*Sqrt[-1 + Sec[c + d*x]])","A",1
180,1,143,200,1.6177593,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (163 A+304 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((362 A+96 C) \cos (c+d x)+92 A \cos (2 (c+d x))+12 A \cos (3 (c+d x))+581 A+528 C)\right)}{384 d}","\frac{a^{5/2} (163 A+304 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (299 A+432 C) \sin (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (17 A+16 C) \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}+\frac{5 a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(163*A + 304*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (581*A + 528*C + (362*A + 96*C)*Cos[c + d*x] + 92*A*Cos[2*(c + d*x)] + 12*A*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(384*d)","A",1
181,1,160,245,2.5193637,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(15 \sqrt{2} (283 A+400 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((3874 A+2720 C) \cos (c+d x)+4 (331 A+80 C) \cos (2 (c+d x))+348 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+5521 A+6320 C)\right)}{3840 d}","\frac{a^{5/2} (283 A+400 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (787 A+1040 C) \sin (c+d x) \cos (c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (79 A+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{240 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(283*A + 400*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (5521*A + 6320*C + (3874*A + 2720*C)*Cos[c + d*x] + 4*(331*A + 80*C)*Cos[2*(c + d*x)] + 348*A*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
182,1,182,290,2.2279554,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (1015 A+1304 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((3234 A+2896 C) \cos (c+d x)+4 (315 A+184 C) \cos (2 (c+d x))+428 A \cos (3 (c+d x))+112 A \cos (4 (c+d x))+16 A \cos (5 (c+d x))+4193 A+96 C \cos (3 (c+d x))+4648 C)\right)}{3072 d}","\frac{a^{5/2} (1015 A+1304 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1015 A+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (109 A+136 C) \sin (c+d x) \cos ^2(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1015 A+1304 C) \sin (c+d x) \cos (c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (23 A+24 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{96 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(1015*A + 1304*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (4193*A + 4648*C + (3234*A + 2896*C)*Cos[c + d*x] + 4*(315*A + 184*C)*Cos[2*(c + d*x)] + 428*A*Cos[3*(c + d*x)] + 96*C*Cos[3*(c + d*x)] + 112*A*Cos[4*(c + d*x)] + 16*A*Cos[5*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3072*d)","A",1
183,1,474,236,6.732414,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos ^2(c+d x) \sqrt{\sec (c+d x)+1} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(-\frac{4 \sec (c) \sec ^2(c+d x) (-63 A \sin (d x)+40 C \sin (c)-97 C \sin (d x))}{315 d}+\frac{4 \sec (c) \sec (c+d x) (63 A \sin (c)-84 A \sin (d x)+97 C \sin (c)-126 C \sin (d x))}{315 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(357 A \sin \left(\frac{d x}{2}\right)+383 C \sin \left(\frac{d x}{2}\right)\right)}{315 d}+\frac{8 \sin \left(\frac{c}{2}\right) (273 A \cos (c)-84 A+257 C \cos (c)-126 C)}{315 d \left(\cos \left(\frac{c}{2}\right)+\cos \left(\frac{3 c}{2}\right)\right)}+\frac{4 C \sec (c) \sin (d x) \sec ^4(c+d x)}{9 d}+\frac{4 \sec (c) \sec ^3(c+d x) (7 C \sin (c)-8 C \sin (d x))}{63 d}\right)}{\sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}-\frac{2 \sqrt{2} (A+C) \sin (c+d x) \cos ^4(c+d x) \sqrt{\sec (c+d x)-1} (\sec (c+d x)+1)^2 \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right) \left(A+C \sec ^2(c+d x)\right)}{d (\cos (c+d x)+1) \sqrt{1-\cos ^2(c+d x)} \sqrt{a (\sec (c+d x)+1)} \sqrt{\cos ^2(c+d x) (\sec (c+d x)-1) (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 (21 A+19 C) \tan (c+d x) \sec ^2(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (21 A+29 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 a d}+\frac{4 (147 A+143 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}-\frac{2 C \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^2*Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((8*(-84*A - 126*C + 273*A*Cos[c] + 257*C*Cos[c])*Sin[c/2])/(315*d*(Cos[c/2] + Cos[(3*c)/2])) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(357*A*Sin[(d*x)/2] + 383*C*Sin[(d*x)/2]))/(315*d) + (4*C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(9*d) + (4*Sec[c]*Sec[c + d*x]*(63*A*Sin[c] + 97*C*Sin[c] - 84*A*Sin[d*x] - 126*C*Sin[d*x]))/(315*d) - (4*Sec[c]*Sec[c + d*x]^2*(40*C*Sin[c] - 63*A*Sin[d*x] - 97*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]^3*(7*C*Sin[c] - 8*C*Sin[d*x]))/(63*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a*(1 + Sec[c + d*x])]) - (2*Sqrt[2]*(A + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[c + d*x]^4*Sqrt[-1 + Sec[c + d*x]]*(1 + Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])*Sqrt[1 - Cos[c + d*x]^2]*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Sqrt[Cos[c + d*x]^2*(-1 + Sec[c + d*x])*(1 + Sec[c + d*x])])","B",1
184,1,173,193,6.52764,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(A+C \sec ^2(c+d x)\right) \left(105 \sqrt{2} (A+C) \cot (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)+2 \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) ((35 A+43 C) \cos (2 (c+d x))+35 A+24 C \cos (c+d x)+73 C)\right)}{105 a d (A \cos (2 (c+d x))+A+2 C)}","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (35 A+31 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 a d}-\frac{4 (35 A+37 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}-\frac{2 C \tan (c+d x) \sec ^2(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Cos[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*(A + C*Sec[c + d*x]^2)*(105*Sqrt[2]*(A + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cot[c + d*x]*Sqrt[-1 + Sec[c + d*x]] + 2*(35*A + 73*C + 24*C*Cos[c + d*x] + (35*A + 43*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*a*d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
185,1,160,152,4.4657502,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) ((15 A+13 C) \cos (2 (c+d x))+15 A-2 C \cos (c+d x)+19 C)-15 \sqrt{2} (A+C) \cot (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)\right)}{15 a d (A \cos (2 (c+d x))+A+2 C)}","-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (15 A+14 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^2(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 a d}",1,"(2*Cos[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*(A + C*Sec[c + d*x]^2)*(-15*Sqrt[2]*(A + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cot[c + d*x]*Sqrt[-1 + Sec[c + d*x]] + (15*A + 19*C - 2*C*Cos[c + d*x] + (15*A + 13*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Tan[(c + d*x)/2]))/(15*a*d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
186,1,125,109,1.554068,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{2 \cos \left(\frac{c}{2}\right) \cos (c) \sin (c+d x) \left(3 \sqrt{2} (A+C) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)+8 C \sin ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x)\right)}{3 d \left(\cos \left(\frac{c}{2}\right)+\cos \left(\frac{3 c}{2}\right)\right) (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 a d}-\frac{4 C \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(-2*Cos[c/2]*Cos[c]*(3*Sqrt[2]*(A + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]] + 8*C*Sec[c + d*x]^2*Sin[(c + d*x)/2]^4)*Sin[c + d*x])/(3*d*(Cos[c/2] + Cos[(3*c)/2])*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
187,1,126,115,0.9827699,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\sqrt{2} (A+C) \cos (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)-2 A \cos (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)+2 C (\cos (c+d x)-1)\right)}{d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"((2*C*(-1 + Cos[c + d*x]) - 2*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[c + d*x]*Sqrt[-1 + Sec[c + d*x]] + Sqrt[2]*(A + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[c + d*x]*Sqrt[-1 + Sec[c + d*x]])*Tan[c + d*x])/(d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
188,1,113,113,0.8103222,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \left(-\sqrt{2} (A+C) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)+A (\cos (c+d x)-1)+A \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((A*(-1 + Cos[c + d*x]) + A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Sqrt[-1 + Sec[c + d*x]] - Sqrt[2]*(A + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]])*Sin[c + d*x])/(d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
189,1,11895,159,26.6674698,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{(7 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{A \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"Result too large to show","C",0
190,1,145,200,0.6335383,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} \left(8 A \cos ^2(c+d x)-2 A \cos (c+d x)+21 A+24 C\right)-3 (9 A+8 C) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+24 \sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{24 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(7 A+8 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{(9 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{A \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"((-3*(9*A + 8*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 24*Sqrt[2]*(A + C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(21*A + 24*C - 2*A*Cos[c + d*x] + 8*A*Cos[c + d*x]^2)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(24*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
191,1,160,243,0.810226,"\int \frac{\cos ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} \left((86 A+96 C) \cos (c+d x)+48 A \cos ^3(c+d x)-8 A \cos ^2(c+d x)-63 A-48 C\right)+(321 A+336 C) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-192 \sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{192 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{(21 A+16 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{(107 A+112 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A+48 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}-\frac{A \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}",1,"(((321*A + 336*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 192*Sqrt[2]*(A + C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-63*A - 48*C + (86*A + 96*C)*Cos[c + d*x] - 8*A*Cos[c + d*x]^2 + 48*A*Cos[c + d*x]^3)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(192*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
192,1,527,259,6.912007,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\cos ^2(c+d x) (\sec (c+d x)+1)^{3/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{2 d}+\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{c}{2}\right)+C \sin \left(\frac{c}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(-805 A \sin \left(\frac{d x}{2}\right)-1649 C \sin \left(\frac{d x}{2}\right)\right)}{105 d}-\frac{4 \sec (c) \sec (c+d x) (-35 A \sin (d x)+39 C \sin (c)-112 C \sin (d x))}{105 d}-\frac{2 \sin \left(\frac{c}{2}\right) (665 A \cos (c)-140 A+1201 C \cos (c)-448 C)}{105 d \left(\cos \left(\frac{c}{2}\right)+\cos \left(\frac{3 c}{2}\right)\right)}+\frac{4 C \sec (c) \sin (d x) \sec ^3(c+d x)}{7 d}+\frac{4 \sec (c) \sec ^2(c+d x) (5 C \sin (c)-13 C \sin (d x))}{35 d}\right)}{(a (\sec (c+d x)+1))^{3/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(11 A+19 C) \sin (c+d x) \cos ^4(c+d x) \sqrt{\sec (c+d x)-1} (\sec (c+d x)+1)^3 \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{2} d (\cos (c+d x)+1) \sqrt{1-\cos ^2(c+d x)} (a (\sec (c+d x)+1))^{3/2} \sqrt{\cos ^2(c+d x) (\sec (c+d x)-1) (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}","\frac{(11 A+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(245 A+397 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{210 a^2 d}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(7 A+11 C) \tan (c+d x) \sec ^3(c+d x)}{14 a d \sqrt{a \sec (c+d x)+a}}-\frac{(35 A+67 C) \tan (c+d x) \sec ^2(c+d x)}{70 a d \sqrt{a \sec (c+d x)+a}}-\frac{(455 A+799 C) \tan (c+d x)}{105 a d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^2*Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((-2*(-140*A - 448*C + 665*A*Cos[c] + 1201*C*Cos[c])*Sin[c/2])/(105*d*(Cos[c/2] + Cos[(3*c)/2])) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(A*Sin[c/2] + C*Sin[c/2]))/(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(-805*A*Sin[(d*x)/2] - 1649*C*Sin[(d*x)/2]))/(105*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(2*d) + (4*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(7*d) - (4*Sec[c]*Sec[c + d*x]*(39*C*Sin[c] - 35*A*Sin[d*x] - 112*C*Sin[d*x]))/(105*d) + (4*Sec[c]*Sec[c + d*x]^2*(5*C*Sin[c] - 13*C*Sin[d*x]))/(35*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(3/2)) + ((11*A + 19*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[c + d*x]^4*Sqrt[-1 + Sec[c + d*x]]*(1 + Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(Sqrt[2]*d*(1 + Cos[c + d*x])*Sqrt[1 - Cos[c + d*x]^2]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(3/2)*Sqrt[Cos[c + d*x]^2*(-1 + Sec[c + d*x])*(1 + Sec[c + d*x])])","B",1
193,1,189,214,5.0635363,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \cos (c+d x) \left(A+C \sec ^2(c+d x)\right) \left(\sec ^3(c+d x) ((75 A+131 C) \cos (c+d x)+8 (5 A+9 C) \cos (2 (c+d x))+25 A \cos (3 (c+d x))+40 A+49 C \cos (3 (c+d x))+88 C)-10 \sqrt{2} (7 A+15 C) \cot ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)\right)}{20 d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 C)}","-\frac{(7 A+15 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(5 A+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{10 a^2 d}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(5 A+9 C) \tan (c+d x) \sec ^2(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{(15 A+31 C) \tan (c+d x)}{5 a d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*(-10*Sqrt[2]*(7*A + 15*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cot[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + (40*A + 88*C + (75*A + 131*C)*Cos[c + d*x] + 8*(5*A + 9*C)*Cos[2*(c + d*x)] + 25*A*Cos[3*(c + d*x)] + 49*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3)*Sin[c + d*x])/(20*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
194,1,162,169,3.7693994,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \cos (c+d x) \left(A+C \sec ^2(c+d x)\right) \left(3 \sqrt{2} (3 A+11 C) \cot ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)-\sec ^2(c+d x) ((3 A+19 C) \cos (2 (c+d x))+3 A+24 C \cos (c+d x)+11 C)\right)}{6 d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(3 A+11 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(3 A+7 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{6 a^2 d}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(3 A+13 C) \tan (c+d x)}{3 a d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]*(A + C*Sec[c + d*x]^2)*(3*Sqrt[2]*(3*A + 11*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cot[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] - (3*A + 11*C + 24*C*Cos[c + d*x] + (3*A + 19*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2)*Sin[c + d*x])/(6*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
195,1,127,126,1.9595197,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(2 \sin ^2\left(\frac{1}{2} (c+d x)\right) (A+4 C \sec (c+d x)+5 C)+\sqrt{2} (A-7 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)\right)}{2 a d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(A-7 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A+5 C) \tan (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"-1/2*((Sqrt[2]*(A - 7*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + 2*(A + 5*C + 4*C*Sec[c + d*x])*Sin[(c + d*x)/2]^2)*Tan[(c + d*x)/2])/(a*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
196,1,154,125,1.464271,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left((A+C) (\cos (c+d x)-1)-\sqrt{2} (5 A-3 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)+8 A \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{2 a d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","-\frac{(5 A-3 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"-1/2*(((A + C)*(-1 + Cos[c + d*x]) + 8*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] - Sqrt[2]*(5*A - 3*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]])*Tan[(c + d*x)/2])/(a*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
197,1,167,158,1.745577,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(2 \sin ^2\left(\frac{1}{2} (c+d x)\right) (2 A \cos (c+d x)+3 A+C)+\sqrt{2} (9 A+C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)-12 A \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{2 a d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(9 A+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A+C) \sin (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"-1/2*((-12*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + Sqrt[2]*(9*A + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + 2*(3*A + C + 2*A*Cos[c + d*x])*Sin[(c + d*x)/2]^2)*Tan[(c + d*x)/2])/(a*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
198,1,11995,217,27.0361435,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{(19 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A+5 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A+2 C) \sin (c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}+\frac{(2 A+C) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
199,1,204,266,3.1078052,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(\sin ^2\left(\frac{1}{2} (c+d x)\right) ((43 A+24 C) \cos (c+d x)-3 A \cos (2 (c+d x))+2 A \cos (3 (c+d x))+60 A+36 C)-3 (47 A+24 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)+6 \sqrt{2} (17 A+9 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)\right)}{12 a d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","-\frac{(47 A+24 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{3 (7 A+4 C) \sin (c+d x)}{8 a d \sqrt{a \sec (c+d x)+a}}+\frac{(5 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(13 A+6 C) \sin (c+d x) \cos (c+d x)}{12 a d \sqrt{a \sec (c+d x)+a}}",1,"-1/12*((-3*(47*A + 24*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + 6*Sqrt[2]*(17*A + 9*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + (60*A + 36*C + (43*A + 24*C)*Cos[c + d*x] - 3*A*Cos[2*(c + d*x)] + 2*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]^2)*Tan[(c + d*x)/2])/(a*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
200,1,220,259,3.8979276,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \left(50 (153 A+521 C) \cos (c+d x)+108 (45 A+157 C) \cos (2 (c+d x))+\frac{60 \sqrt{2} (75 A+283 C) (\cos (c+d x)+1)^2 \cos ^3(c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)}{\cos (c+d x)-1}+2550 A \cos (3 (c+d x))+735 A \cos (4 (c+d x))+4125 A+9110 C \cos (3 (c+d x))+2671 C \cos (4 (c+d x))+15053 C\right)}{960 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","-\frac{(75 A+283 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(195 A+787 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{240 a^3 d}+\frac{(45 A+157 C) \tan (c+d x) \sec ^2(c+d x)}{80 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(465 A+1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^4(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(5 A+21 C) \tan (c+d x) \sec ^3(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((4125*A + 15053*C + 50*(153*A + 521*C)*Cos[c + d*x] + 108*(45*A + 157*C)*Cos[2*(c + d*x)] + 2550*A*Cos[3*(c + d*x)] + 9110*C*Cos[3*(c + d*x)] + 735*A*Cos[4*(c + d*x)] + 2671*C*Cos[4*(c + d*x)] + (60*Sqrt[2]*(75*A + 283*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[c + d*x]^3*(1 + Cos[c + d*x])^2*Sqrt[-1 + Sec[c + d*x]])/(-1 + Cos[c + d*x]))*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*Tan[c + d*x])/(960*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
201,1,196,212,2.8743781,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \sec (c+d x) \left(A+C \sec ^2(c+d x)\right) \left(-(81 A+1537 C) \cos (c+d x)-2 (39 A+503 C) \cos (2 (c+d x))-\frac{6 \sqrt{2} (19 A+163 C) \cos ^2(c+d x) (\cos (c+d x)+1)^2 \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)}{\cos (c+d x)-1}-27 A \cos (3 (c+d x))-78 A-299 C \cos (3 (c+d x))-878 C\right)}{96 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(19 A+163 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 (3 A+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{(21 A+197 C) \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(A+17 C) \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((-78*A - 878*C - (81*A + 1537*C)*Cos[c + d*x] - 2*(39*A + 503*C)*Cos[2*(c + d*x)] - 27*A*Cos[3*(c + d*x)] - 299*C*Cos[3*(c + d*x)] - (6*Sqrt[2]*(19*A + 163*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^2*Sqrt[-1 + Sec[c + d*x]])/(-1 + Cos[c + d*x]))*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Tan[c + d*x])/(96*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
202,1,136,165,3.8855712,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{-\left(\tan ^3\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (10 (A+17 C) \cos (c+d x)+(A+49 C) \cos (2 (c+d x))+A+113 C)\right)-5 \sqrt{2} (A-15 C) \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)}{32 a^2 d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{5 (A-15 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(A+9 C) \tan (c+d x)}{4 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(3 A-13 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(-5*Sqrt[2]*(A - 15*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x] - (A + 113*C + 10*(A + 17*C)*Cos[c + d*x] + (A + 49*C)*Cos[2*(c + d*x)])*Sec[c + d*x]*Tan[(c + d*x)/2]^3)/(32*a^2*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
203,1,120,130,2.5947492,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan ^3\left(\frac{1}{2} (c+d x)\right) ((9 C-7 A) \cos (c+d x)-3 A+13 C)-\frac{(3 A+19 C) \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)}{\sqrt{2}}}{16 a^2 d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(3 A+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(7 A-9 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x) \sec (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(-(((3*A + 19*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x])/Sqrt[2]) + (-3*A + 13*C + (-7*A + 9*C)*Cos[c + d*x])*Tan[(c + d*x)/2]^3)/(16*a^2*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
204,1,153,162,3.6795403,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan ^3\left(\frac{1}{2} (c+d x)\right) ((15 A-C) \cos (c+d x)+11 A-5 C)+\frac{(43 A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)}{\sqrt{2}}-32 A \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)}{16 a^2 d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","-\frac{(43 A-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-5 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(-32*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x] + ((43*A - 5*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x])/Sqrt[2] + (11*A - 5*C + (15*A - C)*Cos[c + d*x])*Tan[(c + d*x)/2]^3)/(16*a^2*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
205,1,166,199,4.0458496,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{-\left(\tan ^3\left(\frac{1}{2} (c+d x)\right) ((55 A+7 C) \cos (c+d x)+8 A \cos (2 (c+d x))+43 A+3 C)\right)-\frac{(115 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)}{\sqrt{2}}+80 A \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)}{16 a^2 d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(115 A+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(35 A+3 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(15 A-C) \sin (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(80*A*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x] - ((115*A + 3*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x])/Sqrt[2] - (43*A + 3*C + (55*A + 7*C)*Cos[c + d*x] + 8*A*Cos[2*(c + d*x)])*Tan[(c + d*x)/2]^3)/(16*a^2*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
206,1,12039,262,27.2321755,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{(39 A+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A+43 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(63 A+11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(31 A+7 C) \sin (c+d x) \cos (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(19 A+3 C) \sin (c+d x) \cos (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \cos (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
207,1,409,205,5.4822262,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{2 a \csc (c) e^{-i d x} \cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) (5 A+3 C) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left(35 A \left(3 e^{i (c+d x)}+e^{2 i (c+d x)}+3 e^{3 i (c+d x)}-1\right) \left(1+e^{2 i (c+d x)}\right)^2+C \left(21 e^{i (c+d x)}-85 e^{2 i (c+d x)}+189 e^{3 i (c+d x)}+85 e^{4 i (c+d x)}+231 e^{5 i (c+d x)}+25 e^{6 i (c+d x)}+63 e^{7 i (c+d x)}-25\right)\right)}{\left(1+e^{2 i (c+d x)}\right)^3}+10 (7 A+5 C) \sin (c) e^{i d x} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d (A \cos (2 (c+d x))+A+2 C)}","\frac{2 a (7 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (5 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(2*a*Cos[c + d*x]^2*Csc[c]*(A + C*Sec[c + d*x]^2)*(7*Sqrt[2]*(5*A + 3*C)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(35*A*(1 + E^((2*I)*(c + d*x)))^2*(-1 + 3*E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))) + C*(-25 + 21*E^(I*(c + d*x)) - 85*E^((2*I)*(c + d*x)) + 189*E^((3*I)*(c + d*x)) + 85*E^((4*I)*(c + d*x)) + 231*E^((5*I)*(c + d*x)) + 25*E^((6*I)*(c + d*x)) + 63*E^((7*I)*(c + d*x))))*Sqrt[Sec[c + d*x]])/(E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3) + 10*(7*A + 5*C)*E^(I*d*x)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*Sin[c]))/(105*d*E^(I*d*x)*(A + 2*C + A*Cos[2*(c + d*x)]))","C",1
208,1,286,172,2.3691063,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{2 a e^{-i c} \left(-1+e^{2 i c}\right) \csc (c) \left(A+C \sec ^2(c+d x)\right) \left((5 A+3 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-5 i (3 A+C) \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-15 A e^{i (c+d x)}-30 A e^{3 i (c+d x)}-15 A e^{5 i (c+d x)}-3 C e^{i (c+d x)}-24 C e^{3 i (c+d x)}-5 C e^{4 i (c+d x)}-9 C e^{5 i (c+d x)}+5 C\right)}{15 d \left(1+e^{2 i (c+d x)}\right)^2 \sec ^{\frac{3}{2}}(c+d x) (A \cos (2 (c+d x))+A+2 C)}","\frac{2 a (5 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(2*a*(-1 + E^((2*I)*c))*Csc[c]*(5*C - 15*A*E^(I*(c + d*x)) - 3*C*E^(I*(c + d*x)) - 30*A*E^((3*I)*(c + d*x)) - 24*C*E^((3*I)*(c + d*x)) - 5*C*E^((4*I)*(c + d*x)) - 15*A*E^((5*I)*(c + d*x)) - 9*C*E^((5*I)*(c + d*x)) - (5*I)*(3*A + C)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 3*C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*c)*(1 + E^((2*I)*(c + d*x)))^2*(A + 2*C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(3/2))","C",1
209,1,168,135,1.4206003,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\sin (d x)-i \cos (d x)) \left((A-C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 i (3 A+C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 A \cos (2 (c+d x))-3 A+2 i C \sin (c+d x)+3 i C \sin (2 (c+d x))+3 C \cos (2 (c+d x))+3 C\right)}{3 d}","\frac{2 a (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a C \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(a*Sec[c + d*x]^(3/2)*((-I)*Cos[d*x] + Sin[d*x])*(-3*A + 3*C - 3*A*Cos[2*(c + d*x)] + 3*C*Cos[2*(c + d*x)] + (2*I)*(3*A + C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + (A - C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + (2*I)*C*Sin[c + d*x] + (3*I)*C*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
210,1,169,135,1.3727028,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2 i (A-C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \sin (2 (c+d x))+6 i A \cos (c+d x)+6 C \sin (c+d x)-6 i C \cos (c+d x)\right)}{3 d}","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((6*I)*A*Cos[c + d*x] - (6*I)*C*Cos[c + d*x] + 2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*I)*(A - C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 6*C*Sin[c + d*x] + A*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
211,1,169,141,1.7480886,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2 i (3 A+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (10 A \sin (c+d x)+3 A \sin (2 (c+d x))+6 i (3 A+5 C))+10 (A+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(10*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*I)*(3*A + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((6*I)*(3*A + 5*C) + 10*A*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)])))/(15*d*E^(I*d*x))","C",1
212,1,188,174,2.4667393,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-28 i (3 A+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (5 (23 A+28 C) \sin (c+d x)+42 A \sin (2 (c+d x))+15 A \sin (3 (c+d x))+84 i (3 A+5 C))+20 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(20*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (28*I)*(3*A + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((84*I)*(3*A + 5*C) + 5*(23*A + 28*C)*Sin[c + d*x] + 42*A*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
213,1,204,205,2.9155659,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i (7 A+9 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (23 A+28 C) \sin (c+d x)+14 (19 A+18 C) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+1176 i A+1512 i C)+120 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 a (7 A+9 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 A+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(120*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(7*A + 9*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((1176*I)*A + (1512*I)*C + 30*(23*A + 28*C)*Sin[c + d*x] + 14*(19*A + 18*C)*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
214,1,821,270,6.8906931,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{4 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{8 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{45 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{7}{2}}(c+d x)}+\frac{10 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{9 d}+\frac{\sec (c) (7 C \sin (c)+18 C \sin (d x)) \sec ^3(c+d x)}{63 d}+\frac{\sec (c) (90 C \sin (c)+63 A \sin (d x)+112 C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac{\sec (c) (63 A \sin (c)+112 C \sin (c)+210 A \sin (d x)+150 C \sin (d x)) \sec (c+d x)}{315 d}+\frac{8 (3 A+2 C) \cos (d x) \csc (c)}{15 d}+\frac{2 (7 A+5 C) \tan (c)}{21 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (21 A+19 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{16 a^2 (3 A+2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (3 A+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d}",1,"(4*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (8*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (10*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((8*(3*A + 2*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(9*d) + (Sec[c]*Sec[c + d*x]^3*(7*C*Sin[c] + 18*C*Sin[d*x]))/(63*d) + (Sec[c]*Sec[c + d*x]^2*(90*C*Sin[c] + 63*A*Sin[d*x] + 112*C*Sin[d*x]))/(315*d) + (Sec[c]*Sec[c + d*x]*(63*A*Sin[c] + 112*C*Sin[c] + 210*A*Sin[d*x] + 150*C*Sin[d*x]))/(315*d) + (2*(7*A + 5*C)*Tan[c])/(21*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",1
215,1,436,237,5.970007,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \csc (c) e^{-i d x} \cos ^4(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(A+C \sec ^2(c+d x)\right) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) (5 A+3 C) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left(35 A \left(6 e^{i (c+d x)}+e^{2 i (c+d x)}+6 e^{3 i (c+d x)}-1\right) \left(1+e^{2 i (c+d x)}\right)^2+6 C \left(7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}-10\right)\right)}{2 \left(1+e^{2 i (c+d x)}\right)^3}+20 (7 A+3 C) \sin (c) e^{i d x} \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d (A \cos (2 (c+d x))+A+2 C)}","\frac{2 a^2 (35 A+33 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{8 a^2 (7 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d}",1,"(a^2*Cos[c + d*x]^4*Csc[c]*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(7*Sqrt[2]*(5*A + 3*C)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(35*A*(1 + E^((2*I)*(c + d*x)))^2*(-1 + 6*E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + 6*E^((3*I)*(c + d*x))) + 6*C*(-10 + 7*E^(I*(c + d*x)) - 20*E^((2*I)*(c + d*x)) + 63*E^((3*I)*(c + d*x)) + 20*E^((4*I)*(c + d*x)) + 77*E^((5*I)*(c + d*x)) + 10*E^((6*I)*(c + d*x)) + 21*E^((7*I)*(c + d*x))))*Sqrt[Sec[c + d*x]])/(2*E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3) + 20*(7*A + 3*C)*E^(I*d*x)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*Sin[c]))/(105*d*E^(I*d*x)*(A + 2*C + A*Cos[2*(c + d*x)]))","C",1
216,1,312,196,6.0072993,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(A+C \sec ^2(c+d x)\right) \left(\frac{-3 \csc (c) \cos (d x) (5 A \cos (2 c)-5 A-16 C)+30 A \cos (c) \sin (d x)+2 C \tan (c+d x) (3 \sec (c+d x)+10)}{2 d \sec ^{\frac{7}{2}}(c+d x)}-\frac{2 i \sqrt{2} \cos ^4(c+d x) \left(5 \left(-1+e^{2 i c}\right) (3 A+C) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 \left(-1+e^{2 i c}\right) C \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+12 C \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)}{15 (A \cos (2 (c+d x))+A+2 C)}","\frac{2 a^2 (15 A+17 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 C \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)}{15 d}-\frac{16 a^2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}{5 d}",1,"(a^2*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(((-2*I)*Sqrt[2]*Cos[c + d*x]^4*(12*C*Sqrt[1 + E^((2*I)*(c + d*x))] + 12*C*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + 5*(3*A + C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (-3*(-5*A - 16*C + 5*A*Cos[2*c])*Cos[d*x]*Csc[c] + 30*A*Cos[c]*Sin[d*x] + 2*C*(10 + 3*Sec[c + d*x])*Tan[c + d*x])/(2*d*Sec[c + d*x]^(7/2))))/(15*(A + 2*C + A*Cos[2*(c + d*x)]))","C",0
217,1,191,198,2.1533657,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-4 i (A-C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+16 (A+C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \sin (c+d x)+A \sin (3 (c+d x))+12 i A \cos (2 (c+d x))+12 i A+4 C \sin (c+d x)+12 C \sin (2 (c+d x))-12 i C \cos (2 (c+d x))-12 i C\right)}{6 d}","-\frac{2 a^2 (A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}-\frac{2 (A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)}{3 d}+\frac{8 a^2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}}",1,"(a^2*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((12*I)*A - (12*I)*C + (12*I)*A*Cos[2*(c + d*x)] - (12*I)*C*Cos[2*(c + d*x)] + 16*(A + C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - (4*I)*(A - C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + A*Sin[c + d*x] + 4*C*Sin[c + d*x] + 12*C*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(6*d*E^(I*d*x))","C",1
218,1,318,196,4.7399637,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(A+C \sec ^2(c+d x)\right) \left(-\frac{\csc (c) ((99 A+60 C) \cos (2 c+d x)-2 A \sin (c) (20 \sin (2 (c+d x))+3 \sin (3 (c+d x)))+(93 A-60 C) \cos (d x))}{8 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 i \sqrt{2} \cos ^4(c+d x) \left(-5 \left(-1+e^{2 i c}\right) (A+3 C) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+12 A \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+12 A \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)}{15 (A \cos (2 (c+d x))+A+2 C)}","-\frac{2 a^2 (7 A-15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{8 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{16 a^2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(((2*I)*Sqrt[2]*Cos[c + d*x]^4*(12*A*Sqrt[1 + E^((2*I)*(c + d*x))] + 12*A*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - 5*(A + 3*C)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) - (Csc[c]*((93*A - 60*C)*Cos[d*x] + (99*A + 60*C)*Cos[2*c + d*x] - 2*A*Sin[c]*(20*Sin[2*(c + d*x)] + 3*Sin[3*(c + d*x)])))/(8*d*Sec[c + d*x]^(7/2))))/(15*(A + 2*C + A*Cos[2*(c + d*x)]))","C",0
219,1,189,204,2.4263109,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i (3 A+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (5 (51 A+28 C) \sin (c+d x)+84 A \sin (2 (c+d x))+15 A \sin (3 (c+d x))+504 i A+840 i C)+80 (3 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a^2 (33 A+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (3 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(80*(3*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(3*A + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((504*I)*A + (840*I)*C + 5*(51*A + 28*C)*Sin[c + d*x] + 84*A*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
220,1,206,237,2.9690772,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-448 i (2 A+3 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (60 (23 A+28 C) \sin (c+d x)+14 (37 A+18 C) \sin (2 (c+d x))+180 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+2688 i A+4032 i C)+240 (5 A+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 a^2 (19 A+21 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (2 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (448*I)*(2*A + 3*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((2688*I)*A + (4032*I)*C + 60*(23*A + 28*C)*Sin[c + d*x] + 14*(37*A + 18*C)*Sin[2*(c + d*x)] + 180*A*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
221,1,228,270,3.293547,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2464 i (7 A+9 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (941 A+1122 C) \sin (c+d x)+616 (19 A+18 C) \sin (2 (c+d x))+4545 A \sin (3 (c+d x))+1540 A \sin (4 (c+d x))+315 A \sin (5 (c+d x))+51744 i A+1980 C \sin (3 (c+d x))+66528 i C)+960 (25 A+33 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{27720 d}","\frac{4 a^2 (7 A+9 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (89 A+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{8 a^2 (25 A+33 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{8 a^2 (25 A+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (7 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(960*(25*A + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(7*A + 9*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((51744*I)*A + (66528*I)*C + 30*(941*A + 1122*C)*Sin[c + d*x] + 616*(19*A + 18*C)*Sin[2*(c + d*x)] + 4545*A*Sin[3*(c + d*x)] + 1980*C*Sin[3*(c + d*x)] + 1540*A*Sin[4*(c + d*x)] + 315*A*Sin[5*(c + d*x)])))/(27720*d*E^(I*d*x))","C",1
222,1,863,319,7.088215,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{7 A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C)}+\frac{C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C)}+\frac{13 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}+\frac{5 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{11 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{22 d}+\frac{\sec (c) (3 C \sin (c)+11 C \sin (d x)) \sec ^4(c+d x)}{66 d}+\frac{\sec (c) (77 C \sin (c)+33 A \sin (d x)+126 C \sin (d x)) \sec ^3(c+d x)}{462 d}+\frac{\sec (c) (165 A \sin (c)+630 C \sin (c)+693 A \sin (d x)+770 C \sin (d x)) \sec ^2(c+d x)}{2310 d}+\frac{\sec (c) (693 A \sin (c)+770 C \sin (c)+1430 A \sin (d x)+1050 C \sin (d x)) \sec (c+d x)}{2310 d}+\frac{(7 A+5 C) \cos (d x) \csc (c)}{5 d}+\frac{(143 A+105 C) \tan (c)}{231 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}","\frac{8 a^3 (44 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{385 d}+\frac{4 a^3 (143 A+105 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{2 (33 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (7 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (143 A+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{33 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d}",1,"(7*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(15*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(3*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (13*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (5*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(11*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(((7*A + 5*C)*Cos[d*x]*Csc[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(22*d) + (Sec[c]*Sec[c + d*x]^4*(3*C*Sin[c] + 11*C*Sin[d*x]))/(66*d) + (Sec[c]*Sec[c + d*x]^3*(77*C*Sin[c] + 33*A*Sin[d*x] + 126*C*Sin[d*x]))/(462*d) + (Sec[c]*Sec[c + d*x]^2*(165*A*Sin[c] + 630*C*Sin[c] + 693*A*Sin[d*x] + 770*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]*(693*A*Sin[c] + 770*C*Sin[c] + 1430*A*Sin[d*x] + 1050*C*Sin[d*x]))/(2310*d) + ((143*A + 105*C)*Tan[c])/(231*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",0
223,1,818,286,6.924844,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{3 A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C)}+\frac{17 C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{45 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C)}+\frac{A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}+\frac{11 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{18 d}+\frac{\sec (c) (7 C \sin (c)+27 C \sin (d x)) \sec ^3(c+d x)}{126 d}+\frac{\sec (c) (135 C \sin (c)+63 A \sin (d x)+238 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac{\sec (c) (63 A \sin (c)+238 C \sin (c)+315 A \sin (d x)+330 C \sin (d x)) \sec (c+d x)}{630 d}+\frac{(27 A+17 C) \cos (d x) \csc (c)}{15 d}+\frac{(21 A+22 C) \tan (c)}{42 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x)}","\frac{8 a^3 (21 A+16 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (63 A+73 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (27 A+17 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (21 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (27 A+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}",1,"(3*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(5*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (17*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(45*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])) + (A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (11*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(((27*A + 17*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^3*(7*C*Sin[c] + 27*C*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^2*(135*C*Sin[c] + 63*A*Sin[d*x] + 238*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(63*A*Sin[c] + 238*C*Sin[c] + 315*A*Sin[d*x] + 330*C*Sin[d*x]))/(630*d) + ((21*A + 22*C)*Tan[c])/(42*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",1
224,1,280,253,3.8227311,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 e^{-i d x} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(14 i (5 A+7 C) e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{7/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (35 A+13 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+70 A \sin (c+d x)+630 A \sin (2 (c+d x))+70 A \sin (3 (c+d x))+315 A \sin (4 (c+d x))-840 i A \cos (2 (c+d x))-210 i A \cos (4 (c+d x))-630 i A+380 C \sin (c+d x)+840 C \sin (2 (c+d x))+260 C \sin (3 (c+d x))+294 C \sin (4 (c+d x))-1176 i C \cos (2 (c+d x))-294 i C \cos (4 (c+d x))-882 i C\right)}{420 d}","\frac{8 a^3 (70 A+53 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (35 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{12 C \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)^2}{35 a d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}",1,"(a^3*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])*((-630*I)*A - (882*I)*C - (840*I)*A*Cos[2*(c + d*x)] - (1176*I)*C*Cos[2*(c + d*x)] - (210*I)*A*Cos[4*(c + d*x)] - (294*I)*C*Cos[4*(c + d*x)] + 80*(35*A + 13*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + ((14*I)*(5*A + 7*C)*(1 + E^((2*I)*(c + d*x)))^(7/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 70*A*Sin[c + d*x] + 380*C*Sin[c + d*x] + 630*A*Sin[2*(c + d*x)] + 840*C*Sin[2*(c + d*x)] + 70*A*Sin[3*(c + d*x)] + 260*C*Sin[3*(c + d*x)] + 315*A*Sin[4*(c + d*x)] + 294*C*Sin[4*(c + d*x)]))/(420*d*E^(I*d*x))","C",1
225,1,255,259,2.9623715,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^3 e^{-i d x} \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-4 i (5 A-9 C) e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (5 A+3 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+30 A \sin (c+d x)+10 A \sin (2 (c+d x))+30 A \sin (3 (c+d x))+5 A \sin (4 (c+d x))+180 i A \cos (c+d x)+60 i A \cos (3 (c+d x))+132 C \sin (c+d x)+60 C \sin (2 (c+d x))+108 C \sin (3 (c+d x))-324 i C \cos (c+d x)-108 i C \cos (3 (c+d x))\right)}{60 d}","\frac{4 a^3 (5 A+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 (5 A-9 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 (5 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)^2}{15 a d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d \sqrt{\sec (c+d x)}}",1,"(a^3*Sec[c + d*x]^(5/2)*(Cos[d*x] + I*Sin[d*x])*((180*I)*A*Cos[c + d*x] - (324*I)*C*Cos[c + d*x] + (60*I)*A*Cos[3*(c + d*x)] - (108*I)*C*Cos[3*(c + d*x)] + 80*(5*A + 3*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] - ((4*I)*(5*A - 9*C)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 30*A*Sin[c + d*x] + 132*C*Sin[c + d*x] + 10*A*Sin[2*(c + d*x)] + 60*C*Sin[2*(c + d*x)] + 30*A*Sin[3*(c + d*x)] + 108*C*Sin[3*(c + d*x)] + 5*A*Sin[4*(c + d*x)]))/(60*d*E^(I*d*x))","C",1
226,1,221,253,2.3692021,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{a^3 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-8 i (9 A-5 C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (3 A+5 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+30 A \sin (c+d x)+6 A \sin (2 (c+d x))+30 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+216 i A \cos (2 (c+d x))+216 i A+40 C \sin (c+d x)+180 C \sin (2 (c+d x))-120 i C \cos (2 (c+d x))-120 i C\right)}{60 d}","-\frac{8 a^3 (3 A-10 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 (9 A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (9 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{5 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^3*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((216*I)*A - (120*I)*C + (216*I)*A*Cos[2*(c + d*x)] - (120*I)*C*Cos[2*(c + d*x)] + 80*(3*A + 5*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - (8*I)*(9*A - 5*C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 30*A*Sin[c + d*x] + 40*C*Sin[c + d*x] + 6*A*Sin[2*(c + d*x)] + 180*C*Sin[2*(c + d*x)] + 30*A*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(60*d*E^(I*d*x))","C",1
227,1,218,253,2.4097497,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (7 A+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (13 A+35 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 A \sin (c+d x)+550 A \sin (2 (c+d x))+126 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+2352 i A \cos (c+d x)+840 C \sin (c+d x)+140 C \sin (2 (c+d x))+1680 i C \cos (c+d x)\right)}{420 d}","-\frac{4 a^3 (41 A-35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+5 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{12 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((2352*I)*A*Cos[c + d*x] + (1680*I)*C*Cos[c + d*x] + 80*(13*A + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(7*A + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 126*A*Sin[c + d*x] + 840*C*Sin[c + d*x] + 550*A*Sin[2*(c + d*x)] + 140*C*Sin[2*(c + d*x)] + 126*A*Sin[3*(c + d*x)] + 15*A*Sin[4*(c + d*x)]))/(420*d*E^(I*d*x))","C",1
228,1,206,253,2.92465,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (17 A+27 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (97 A+84 C) \sin (c+d x)+14 (73 A+18 C) \sin (2 (c+d x))+270 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+5712 i A+9072 i C)+240 (11 A+21 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 (73 A+63 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (16 A+21 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (11 A+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (17 A+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{21 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(11*A + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(17*A + 27*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((5712*I)*A + (9072*I)*C + 30*(97*A + 84*C)*Sin[c + d*x] + 14*(73*A + 18*C)*Sin[2*(c + d*x)] + 270*A*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
229,1,228,286,3.495454,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2464 i (5 A+7 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (10 (1953 A+2354 C) \sin (c+d x)+308 (25 A+18 C) \sin (2 (c+d x))+2835 A \sin (3 (c+d x))+770 A \sin (4 (c+d x))+105 A \sin (5 (c+d x))+36960 i A+660 C \sin (3 (c+d x))+51744 i C)+160 (105 A+143 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{9240 d}","\frac{8 a^3 (35 A+44 C) \sin (c+d x)}{385 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (35 A+33 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (105 A+143 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (105 A+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{33 a d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(160*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(5*A + 7*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((36960*I)*A + (51744*I)*C + 10*(1953*A + 2354*C)*Sin[c + d*x] + 308*(25*A + 18*C)*Sin[2*(c + d*x)] + 2835*A*Sin[3*(c + d*x)] + 660*C*Sin[3*(c + d*x)] + 770*A*Sin[4*(c + d*x)] + 105*A*Sin[5*(c + d*x)])))/(9240*d*E^(I*d*x))","C",1
230,1,250,319,4.2110077,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-4928 i (175 A+221 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (780 (1811 A+2134 C) \sin (c+d x)+77 (7825 A+7592 C) \sin (2 (c+d x))+251550 A \sin (3 (c+d x))+90860 A \sin (4 (c+d x))+24570 A \sin (5 (c+d x))+3465 A \sin (6 (c+d x))+2587200 i A+154440 C \sin (3 (c+d x))+20020 C \sin (4 (c+d x))+3267264 i C)+12480 (95 A+121 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{720720 d}","\frac{4 a^3 (175 A+221 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{40 a^3 (118 A+143 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (145 A+143 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (95 A+121 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (95 A+121 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (175 A+221 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{12 A \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac{11}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(12480*(95*A + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4928*I)*(175*A + 221*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((2587200*I)*A + (3267264*I)*C + 780*(1811*A + 2134*C)*Sin[c + d*x] + 77*(7825*A + 7592*C)*Sin[2*(c + d*x)] + 251550*A*Sin[3*(c + d*x)] + 154440*C*Sin[3*(c + d*x)] + 90860*A*Sin[4*(c + d*x)] + 20020*C*Sin[4*(c + d*x)] + 24570*A*Sin[5*(c + d*x)] + 3465*A*Sin[6*(c + d*x)])))/(720720*d*E^(I*d*x))","C",1
231,1,342,232,6.4306701,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(-3 i (5 A+7 C) e^{-2 i (c+d x)} \left(1+e^{i (c+d x)}\right) \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 i (2 (45 A+56 C) \cos (c+d x)+6 (5 A+7 C) \cos (2 (c+d x))+15 i A \sin (c+d x)+15 i A \sin (3 (c+d x))+30 A \cos (3 (c+d x))+30 A+31 i C \sin (c+d x)-4 i C \sin (2 (c+d x))+19 i C \sin (3 (c+d x))+44 C \cos (3 (c+d x))+54 C)+40 (3 A+5 C) \cos \left(\frac{1}{2} (c+d x)\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{60 a d (\sec (c+d x)+1)}","-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(5 A+7 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(3 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (5 A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"-1/60*(Cos[(c + d*x)/2]*Sec[c + d*x]^(7/2)*(((-3*I)*(5*A + 7*C)*(1 + E^(I*(c + d*x)))*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 40*(3*A + 5*C)*Cos[(c + d*x)/2]*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + (2*I)*(30*A + 54*C + 2*(45*A + 56*C)*Cos[c + d*x] + 6*(5*A + 7*C)*Cos[2*(c + d*x)] + 30*A*Cos[3*(c + d*x)] + 44*C*Cos[3*(c + d*x)] + (15*I)*A*Sin[c + d*x] + (31*I)*C*Sin[c + d*x] - (4*I)*C*Sin[2*(c + d*x)] + (15*I)*A*Sin[3*(c + d*x)] + (19*I)*C*Sin[3*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(a*d*E^(I*d*x)*(1 + Sec[c + d*x]))","C",1
232,1,324,190,4.5367421,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(-i (A+3 C) e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(e^{i (c+d x)}+e^{2 i (c+d x)}+e^{3 i (c+d x)}+1\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 i ((3 A+7 C) \cos (2 (c+d x))+3 A-2 i C \sin (c+d x)+2 i C \sin (2 (c+d x))+6 C \cos (c+d x)+5 C)+2 (3 A+5 C) \sqrt{\cos (c+d x)} \left(\cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+i \sin \left(\frac{1}{2} (c+d x)\right)\right) (\cos (c+d x)-i \sin (c+d x))\right)}{6 a d (\sec (c+d x)+1)}","-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(3 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(3 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(((-I)*(A + 3*C)*Sqrt[1 + E^((2*I)*(c + d*x))]*(1 + E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x)))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] + I*Sin[(c + d*x)/2])*(Cos[c + d*x] - I*Sin[c + d*x]) + (2*I)*(3*A + 5*C + 6*C*Cos[c + d*x] + (3*A + 7*C)*Cos[2*(c + d*x)] - (2*I)*C*Sin[c + d*x] + (2*I)*C*Sin[2*(c + d*x)]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a*d*E^(I*d*x)*(1 + Sec[c + d*x]))","C",1
233,1,776,152,6.7129547,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sqrt{2} A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos (c+d x) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A+C \sec ^2(c+d x)\right)}{3 d (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}+\frac{\sqrt{2} C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos (c+d x) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A+C \sec ^2(c+d x)\right)}{d (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}+\frac{\cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \left(A+C \sec ^2(c+d x)\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{2 (A+3 C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)}{d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right)}{d}\right)}{\sqrt{\sec (c+d x)} (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}+\frac{2 A \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A+C \sec ^2(c+d x)\right)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}-\frac{2 C \sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{\cos (c+d x)} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A+C \sec ^2(c+d x)\right)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a) (A \cos (2 c+2 d x)+A+2 C)}","-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (2*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + C*Sec[c + d*x]^2)*((2*(A + 3*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (4*(A + C)*Tan[c/2])/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",1
234,1,795,124,6.4636917,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{2 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(-\frac{2 (\cos (2 c) A+2 A+C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{4 \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{8 A \cos (c) \sin (d x)}{d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}-\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-((Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x]))) - (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + C*Sec[c + d*x]^2)*((-2*(2*A + C + A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*A*Cos[c]*Sin[d*x])/d + (4*(A + C)*Tan[c/2])/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",1
235,1,232,162,2.7888879,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(i (3 A+C) e^{\frac{1}{2} i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right) (2 A \cos (c+d x)+5 A+3 C)-3 i (3 A+C) \cos \left(\frac{1}{2} (c+d x)\right)\right)+2 (5 A+3 C) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a d (\sec (c+d x)+1)}","\frac{(5 A+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}+\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*(2*(5*A + 3*C)*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + I*(3*A + C)*E^((I/2)*(c + d*x))*(1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((-3*I)*(3*A + C)*Cos[(c + d*x)/2] + (5*A + 3*C + 2*A*Cos[c + d*x])*Sin[(c + d*x)/2])))/(3*a*d*E^(I*d*x)*(1 + Sec[c + d*x]))","C",1
236,1,248,199,3.5027679,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-6 i (7 A+5 C) e^{\frac{1}{2} i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) \left(-2 \sin \left(\frac{1}{2} (c+d x)\right) (4 A \cos (c+d x)-3 A \cos (2 (c+d x))+22 A+15 C)+18 i (7 A+5 C) \cos \left(\frac{1}{2} (c+d x)\right)\right)-20 (5 A+3 C) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 a d (\sec (c+d x)+1)}","-\frac{(A+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{(7 A+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(5 A+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(5 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (7 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*(-20*(5*A + 3*C)*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (6*I)*(7*A + 5*C)*E^((I/2)*(c + d*x))*(1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((18*I)*(7*A + 5*C)*Cos[(c + d*x)/2] - 2*(22*A + 15*C + 4*A*Cos[c + d*x] - 3*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2])))/(30*a*d*E^(I*d*x)*(1 + Sec[c + d*x]))","C",1
237,1,884,229,7.6619902,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}-\frac{14 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{8 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{40 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+4 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A+7 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{16 C \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{16 (5 \cos (c) C+C+A \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}","-\frac{(A+7 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{2 (A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(A+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{2 (A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (14*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (40*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((-4*(A + 7*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 4*C*Sin[(d*x)/2]))/(3*d) + (16*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (16*(C + A*Cos[c] + 5*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",1
238,1,293,191,4.8680196,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(-2 i (A+25 C) \cos (c+d x)+8 (A-5 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)+A \sin (2 (c+d x))+i A \cos (2 (c+d x))+i A+4 i C e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+12 C \sin (c+d x)+7 C \sin (2 (c+d x))-17 i C \cos (2 (c+d x))-29 i C\right)}{6 a^2 d (\sec (c+d x)+1)^2}","\frac{(A-5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 C \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{4 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(I*A - (29*I)*C - (2*I)*(A + 25*C)*Cos[c + d*x] + I*A*Cos[2*(c + d*x)] - (17*I)*C*Cos[2*(c + d*x)] + ((4*I)*C*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 8*(A - 5*C)*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + 12*C*Sin[c + d*x] + A*Sin[2*(c + d*x)] + 7*C*Sin[2*(c + d*x)])*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
239,1,859,165,6.8054705,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}-\frac{2 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{8 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{8 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{16 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}-\frac{16 (2 A-C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}","\frac{(A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{2 (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((4*(A - C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (16*(2*A - C)*Tan[c/2])/(3*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",1
240,1,298,170,3.7327917,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \left(A+C \sec ^2(c+d x)\right) \left(\frac{8 i e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right) \sqrt{\sec (c+d x)} \left(A \left(16 e^{i (c+d x)}+20 e^{2 i (c+d x)}+9 e^{3 i (c+d x)}+3\right)-C e^{i (c+d x)} \left(-1+e^{i (c+d x)}\right)\right)}{d \left(1+e^{i (c+d x)}\right)^3}-\frac{8 (5 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{32 i A e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right) \sqrt{\sec (c+d x)}}{d}\right)}{3 a^2 (\sec (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 C)}","-\frac{(5 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{(5 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*(((8*I)*(1 + E^((2*I)*(c + d*x)))*(-(C*E^(I*(c + d*x))*(-1 + E^(I*(c + d*x)))) + A*(3 + 16*E^(I*(c + d*x)) + 20*E^((2*I)*(c + d*x)) + 9*E^((3*I)*(c + d*x))))*Sqrt[Sec[c + d*x]])/(d*E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))^3) - (8*(5*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d - ((32*I)*A*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]*Sqrt[Sec[c + d*x]])/d)*(A + C*Sec[c + d*x]^2))/(3*a^2*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)","C",1
241,1,912,201,6.8993386,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{14 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{2 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{40 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{8 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{16 \sec \left(\frac{c}{2}\right) \left(5 A \sin \left(\frac{d x}{2}\right)+2 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (2 \cos (2 c) A+5 A+C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{8 A \cos (2 d x) \sin (2 c)}{3 d}-\frac{32 A \cos (c) \sin (d x)}{d}+\frac{8 A \cos (2 c) \sin (2 d x)}{3 d}-\frac{16 (5 A+2 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}","\frac{2 (5 A+C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(7 A+C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}+\frac{2 (5 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(7 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(14*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (40*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((4*(5*A + C + 2*A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (8*A*Cos[2*d*x]*Sin[2*c])/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(5*A*Sin[(d*x)/2] + 2*C*Sin[(d*x)/2]))/(3*d) - (32*A*Cos[c]*Sin[d*x])/d + (8*A*Cos[2*c]*Sin[2*d*x])/(3*d) - (16*(5*A + 2*C)*Tan[c/2])/(3*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",1
242,1,301,236,6.6937359,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{\sin (c) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(8 i (14 A+5 C) e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+400 (3 A+C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \cos (c+d x) \left(-72 i (14 A+5 C) \cos \left(\frac{1}{2} (c+d x)\right)-24 i (14 A+5 C) \cos \left(\frac{3}{2} (c+d x)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((179 A+60 C) \cos (c+d x)+8 A \cos (2 (c+d x))-3 A \cos (3 (c+d x))+158 A+50 C)\right)\right)}{120 a^2 d (\sec (c+d x)+1)^2}","-\frac{(3 A+C) \sin (c+d x)}{a^2 d \sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}+\frac{4 (14 A+5 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (3 A+C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{5 (3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (14 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"-1/120*(Cos[(c + d*x)/2]*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^(5/2)*Sin[c]*(Cos[d*x] + I*Sin[d*x])*(400*(3*A + C)*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + ((8*I)*(14*A + 5*C)*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((I/2)*(c + d*x)) + 2*Cos[c + d*x]*((-72*I)*(14*A + 5*C)*Cos[(c + d*x)/2] - (24*I)*(14*A + 5*C)*Cos[(3*(c + d*x))/2] + 2*(158*A + 50*C + (179*A + 60*C)*Cos[c + d*x] + 8*A*Cos[2*(c + d*x)] - 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2])))/(a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
243,1,984,282,8.1118065,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{6 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{238 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{44 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+13 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (3 A+13 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+29 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (9 A+119 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{32 C \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{8 (33 \cos (c) C+4 C+3 A \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}","-\frac{(9 A+119 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a^3 d}-\frac{(9 A+119 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(9 A+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{2 C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2}",1,"(-6*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (238*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (44*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((-4*(9*A + 119*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(3*A*Sin[(d*x)/2] + 13*C*Sin[(d*x)/2]))/(15*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] + 29*C*Sin[(d*x)/2]))/(3*d) + (32*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (8*(4*C + 3*A*Cos[c] + 33*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (8*(3*A + 13*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",1
244,1,953,249,7.2024767,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{98 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{52 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-4 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{16 (A-4 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-13 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A-49 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{8 (13 C-A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}","\frac{(A-13 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(A-49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(A-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 (A-4 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (98*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((-4*(A - 49*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 13*C*Sin[(d*x)/2]))/(3*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - 4*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*(-A + 13*C)*Tan[c/2])/(3*d) + (16*(A - 4*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",1
245,1,952,220,6.9852991,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{6 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)-3 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (7 A-3 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-9 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{8 (A+3 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}","\frac{(A-9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 (2 A-3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (6*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((4*(A - 9*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(7*A*Sin[(d*x)/2] - 3*C*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(3*d) + (8*(A + 3*C)*Tan[c/2])/(3*d) - (8*(7*A - 3*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",1
246,1,954,222,7.0994637,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{6 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{2 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(6 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{16 (6 A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (9 A-C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{8 (9 A-C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}","\frac{(3 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{2 (3 A-2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(6*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((4*(9*A - C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(6*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(15*d) - (8*(9*A - C)*Tan[c/2])/(3*d) + (16*(6*A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
247,1,975,226,7.0411774,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\frac{98 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{2 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{52 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(17 A \sin \left(\frac{d x}{2}\right)+7 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (17 A+7 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(23 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (10 \cos (2 c) A+39 A-C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{32 A \cos (c) \sin (d x)}{d}+\frac{8 (23 A+C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}","-\frac{(13 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(13 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(49 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (4 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(-98*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((-4*(39*A - C + 10*A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(23*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(17*A*Sin[(d*x)/2] + 7*C*Sin[(d*x)/2]))/(15*d) + (32*A*Cos[c]*Sin[d*x])/d + (8*(23*A + C)*Tan[c/2])/(3*d) - (8*(17*A + 7*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
248,1,1008,249,7.1777234,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\frac{238 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{6 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{44 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(11 A \sin \left(\frac{d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{16 (11 A+6 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(43 A \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (30 \cos (2 c) A+89 A+9 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{16 A \cos (2 d x) \sin (2 c)}{3 d}-\frac{96 A \cos (c) \sin (d x)}{d}+\frac{16 A \cos (2 c) \sin (2 d x)}{3 d}-\frac{8 (43 A+9 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}","\frac{(11 A+C) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(119 A+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(11 A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(119 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}-\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(238*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (6*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (44*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((4*(89*A + 9*C + 30*A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (16*A*Cos[2*d*x]*Sin[2*c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(11*A*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2]))/(15*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(43*A*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(3*d) - (96*A*Cos[c]*Sin[d*x])/d + (16*A*Cos[2*c]*Sin[2*d*x])/(3*d) - (8*(43*A + 9*C)*Tan[c/2])/(3*d) + (16*(11*A + 6*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
249,1,1052,290,7.3484825,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{154 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{98 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{84 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{52 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(27 A \sin \left(\frac{d x}{2}\right)+17 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (27 A+17 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{184 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (133 \cos (2 c) A+329 A+78 C+20 C \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{16 A \cos (2 d x) \sin (2 c)}{d}+\frac{8 A \cos (3 d x) \sin (3 c)}{5 d}+\frac{8 (133 A+20 C) \cos (c) \sin (d x)}{5 d}-\frac{16 A \cos (2 c) \sin (2 d x)}{d}+\frac{8 A \cos (3 c) \sin (3 d x)}{5 d}+\frac{184 (3 A+C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}","-\frac{(63 A+13 C) \sin (c+d x)}{10 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}+\frac{7 (33 A+7 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(63 A+13 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(63 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (33 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{2 (6 A+C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"(-154*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (98*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (84*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((-2*(329*A + 78*C + 133*A*Cos[2*c] + 20*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (16*A*Cos[2*d*x]*Sin[2*c])/d + (8*A*Cos[3*d*x]*Sin[3*c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (184*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(27*A*Sin[(d*x)/2] + 17*C*Sin[(d*x)/2]))/(15*d) + (8*(133*A + 20*C)*Cos[c]*Sin[d*x])/(5*d) - (16*A*Cos[2*c]*Sin[2*d*x])/d + (8*A*Cos[3*c]*Sin[3*d*x])/(5*d) + (184*(3*A + C)*Tan[c/2])/(3*d) - (8*(27*A + 17*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
250,1,238,214,2.3942987,"\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) ((432 A+539 C) \cos (c+d x)+4 (48 A+35 C) \cos (2 (c+d x))+144 A \cos (3 (c+d x))+192 A+105 C \cos (3 (c+d x))+332 C)-\frac{12 (48 A+35 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)}{\sqrt{\sec (c+d x)+1}}\right)}{384 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a (48 A+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a (48 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(A + C*Sec[c + d*x]^2)*((192*A + 332*C + (432*A + 539*C)*Cos[c + d*x] + 4*(48*A + 35*C)*Cos[2*(c + d*x)] + 144*A*Cos[3*(c + d*x)] + 105*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(9/2)*Tan[(c + d*x)/2] - (12*(48*A + 35*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2])/Sqrt[1 + Sec[c + d*x]]))/(384*d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",0
251,1,211,169,1.8717095,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) (3 (8 A+5 C) \cos (2 (c+d x))+24 A+20 C \cos (c+d x)+31 C)-\frac{6 (8 A+5 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)}{\sqrt{\sec (c+d x)+1}}\right)}{24 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a (8 A+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (8 A+5 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{a C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(A + C*Sec[c + d*x]^2)*((24*A + 31*C + 20*C*Cos[c + d*x] + 3*(8*A + 5*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2)*Tan[(c + d*x)/2] - (6*(8*A + 5*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2])/Sqrt[1 + Sec[c + d*x]]))/(24*d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",0
252,1,202,124,2.5198799,"\int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(A+C \sec ^2(c+d x)\right) \left(C \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \sin \left(\frac{3}{2} (c+d x)\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x)-\frac{2 (8 A+3 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)}{\sqrt{\sec (c+d x)+1}}\right)}{4 d (A \cos (2 (c+d x))+A+2 C)}","\frac{\sqrt{a} (8 A+3 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(A + C*Sec[c + d*x]^2)*(C*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(Sin[(c + d*x)/2] + 3*Sin[(3*(c + d*x))/2]) - (2*(8*A + 3*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2])/Sqrt[1 + Sec[c + d*x]]))/(4*d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",0
253,1,177,115,2.6354434,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","-\frac{\cot (c+d x) \sqrt{a (\sec (c+d x)+1)} \left((C-2 A) \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+\sqrt{\sec (c+d x)+1} \sec ^{\frac{3}{2}}(c+d x) (A \cos (2 (c+d x))+A-C)+C \sqrt{\tan ^2(c+d x)} \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{d \sqrt{\sec (c+d x)+1}}","\frac{a (2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{d}+\frac{\sqrt{a} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"-((Cot[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*((-2*A + C)*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]] + (A - C + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(3/2)*Sqrt[1 + Sec[c + d*x]] + C*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(d*Sqrt[1 + Sec[c + d*x]]))","A",1
254,1,179,116,1.7561782,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","-\frac{\csc (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(2 A \sqrt{\sec (c+d x)+1}+A (\cos (2 (c+d x))-3) \sec (c+d x) \sqrt{\sec (c+d x)+1}+6 C \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{3 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{\sec (c+d x)+1}}","\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"-1/3*(Csc[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(2*A*Sqrt[1 + Sec[c + d*x]] + A*(-3 + Cos[2*(c + d*x)])*Sec[c + d*x]*Sqrt[1 + Sec[c + d*x]] + 6*C*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]))/(d*Sec[c + d*x]^(3/2)*Sqrt[1 + Sec[c + d*x]])","A",0
255,1,68,122,0.546781,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (8 A \cos (c+d x)+3 A \cos (2 (c+d x))+19 A+30 C)}{15 d \sqrt{\sec (c+d x)}}","\frac{2 a (7 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}",1,"((19*A + 30*C + 8*A*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d*Sqrt[Sec[c + d*x]])","A",1
256,1,84,168,0.7824967,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((141 A+140 C) \cos (c+d x)+36 A \cos (2 (c+d x))+15 A \cos (3 (c+d x))+228 A+280 C)}{210 d \sqrt{\sec (c+d x)}}","\frac{4 a (24 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (24 A+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"((228*A + 280*C + (141*A + 140*C)*Cos[c + d*x] + 36*A*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d*Sqrt[Sec[c + d*x]])","A",1
257,1,102,213,1.2186914,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (16 (47 A+42 C) \cos (c+d x)+4 (83 A+63 C) \cos (2 (c+d x))+80 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+1321 A+1596 C)}{1260 d \sqrt{\sec (c+d x)}}","\frac{2 a (16 A+21 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"((1321*A + 1596*C + 16*(47*A + 42*C)*Cos[c + d*x] + 4*(83*A + 63*C)*Cos[2*(c + d*x)] + 80*A*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d*Sqrt[Sec[c + d*x]])","A",1
258,1,273,265,3.4777464,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a (\sec (c+d x)+1))^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \sqrt{\sec (c+d x)+1} (12 (880 A+1273 C) \cos (c+d x)+4 (3280 A+3059 C) \cos (2 (c+d x))+3520 A \cos (3 (c+d x))+2640 A \cos (4 (c+d x))+10480 A+2660 C \cos (3 (c+d x))+1995 C \cos (4 (c+d x))+13313 C)-120 (176 A+133 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{7680 d (\sec (c+d x)+1)^{3/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{a^{3/2} (176 A+133 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^2 (80 A+67 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{3 a C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}",1,"(Cos[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(3/2)*(A + C*Sec[c + d*x]^2)*((10480*A + 13313*C + 12*(880*A + 1273*C)*Cos[c + d*x] + 4*(3280*A + 3059*C)*Cos[2*(c + d*x)] + 3520*A*Cos[3*(c + d*x)] + 2660*C*Cos[3*(c + d*x)] + 2640*A*Cos[4*(c + d*x)] + 1995*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(11/2)*Sqrt[1 + Sec[c + d*x]]*Tan[(c + d*x)/2] - 120*(176*A + 133*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(7680*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^(3/2))","A",0
259,1,251,218,3.1162297,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a (\sec (c+d x)+1))^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{\sec (c+d x)+1} (7 (48 A+55 C) \cos (c+d x)+4 (16 A+25 C) \cos (2 (c+d x))+112 A \cos (3 (c+d x))+64 A+75 C \cos (3 (c+d x))+164 C)-4 (112 A+75 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{128 d (\sec (c+d x)+1)^{3/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{a^{3/2} (112 A+75 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (16 A+13 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{32 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (112 A+75 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}+\frac{a C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{8 d}",1,"(Cos[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(3/2)*(A + C*Sec[c + d*x]^2)*((64*A + 164*C + 7*(48*A + 55*C)*Cos[c + d*x] + 4*(16*A + 25*C)*Cos[2*(c + d*x)] + 112*A*Cos[3*(c + d*x)] + 75*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(9/2)*Sqrt[1 + Sec[c + d*x]]*Tan[(c + d*x)/2] - 4*(112*A + 75*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(128*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^(3/2))","A",0
260,1,223,171,2.3392345,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a (\sec (c+d x)+1))^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{\sec (c+d x)+1} (3 (8 A+11 C) \cos (2 (c+d x))+24 A+44 C \cos (c+d x)+49 C)-6 (24 A+11 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{24 d (\sec (c+d x)+1)^{3/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{a^{3/2} (24 A+11 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(Cos[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(3/2)*(A + C*Sec[c + d*x]^2)*((24*A + 49*C + 44*C*Cos[c + d*x] + 3*(8*A + 11*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2)*Sqrt[1 + Sec[c + d*x]]*Tan[(c + d*x)/2] - 6*(24*A + 11*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(24*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^(3/2))","A",0
261,1,209,171,4.9806958,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{(a (\sec (c+d x)+1))^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\tan \left(\frac{1}{2} (c+d x)\right) (4 A \cos (2 (c+d x))+4 A+7 C \cos (c+d x)+2 C)}{\sqrt{\frac{1}{\cos (c+d x)+1}}}-(8 A+7 C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{2 d (\sec (c+d x)+1)^{3/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{a^{3/2} (8 A+7 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (8 A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{3 a C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"((a*(1 + Sec[c + d*x]))^(3/2)*(A + C*Sec[c + d*x]^2)*(((4*A + 2*C + 7*C*Cos[c + d*x] + 4*A*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/Sqrt[(1 + Cos[c + d*x])^(-1)] - (8*A + 7*C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^(3/2))","A",0
262,1,382,169,6.4664573,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{6 C \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{3/2} \left(\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-\log (\sec (c+d x)+1)\right) \left(A+C \sec ^2(c+d x)\right)}{d \left(1-\cos ^2(c+d x)\right) (\sec (c+d x)+1)^{3/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{3/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(8 A \sin \left(\frac{d x}{2}\right)-3 C \sin \left(\frac{d x}{2}\right)\right)}{3 d}-\frac{2 (8 A-3 C) \tan \left(\frac{c}{2}\right)}{3 d}+\frac{16 A \sin (c) \cos (d x)}{3 d}+\frac{2 A \sin (2 c) \cos (2 d x)}{3 d}+\frac{16 A \cos (c) \sin (d x)}{3 d}+\frac{2 A \cos (2 c) \sin (2 d x)}{3 d}\right)}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{3/2} (A \cos (2 c+2 d x)+A+2 C)}","\frac{3 a^{3/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 (8 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{a (2 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(6*C*Cos[c + d*x]^3*(-Log[1 + Sec[c + d*x]] + Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(a*(1 + Sec[c + d*x]))^(3/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(1 - Cos[c + d*x]^2)*(A + 2*C + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(3/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2)*(A + C*Sec[c + d*x]^2)*((16*A*Cos[d*x]*Sin[c])/(3*d) + (2*A*Cos[2*d*x]*Sin[2*c])/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(8*A*Sin[(d*x)/2] - 3*C*Sin[(d*x)/2]))/(3*d) + (16*A*Cos[c]*Sin[d*x])/(3*d) + (2*A*Cos[2*c]*Sin[2*d*x])/(3*d) - (2*(8*A - 3*C)*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^(3/2))","B",0
263,1,428,163,6.3386237,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{4 C \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{3/2} \left(\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-\log (\sec (c+d x)+1)\right) \left(A+C \sec ^2(c+d x)\right)}{d \left(1-\cos ^2(c+d x)\right) (\sec (c+d x)+1)^{3/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{3/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{(17 A+20 C) \sin (c) \cos (d x)}{5 d}+\frac{(17 A+20 C) \cos (c) \sin (d x)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(4 A \sin \left(\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right)}{5 d}-\frac{4 (4 A+5 C) \tan \left(\frac{c}{2}\right)}{5 d}+\frac{4 A \sin (2 c) \cos (2 d x)}{5 d}+\frac{A \sin (3 c) \cos (3 d x)}{5 d}+\frac{4 A \cos (2 c) \sin (2 d x)}{5 d}+\frac{A \cos (3 c) \sin (3 d x)}{5 d}\right)}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{3/2} (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 a^{3/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (4 A+5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d \sqrt{\sec (c+d x)}}",1,"(4*C*Cos[c + d*x]^3*(-Log[1 + Sec[c + d*x]] + Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(a*(1 + Sec[c + d*x]))^(3/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(1 - Cos[c + d*x]^2)*(A + 2*C + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(3/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2)*(A + C*Sec[c + d*x]^2)*(((17*A + 20*C)*Cos[d*x]*Sin[c])/(5*d) + (4*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (A*Cos[3*d*x]*Sin[3*c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(4*A*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2]))/(5*d) + ((17*A + 20*C)*Cos[c]*Sin[d*x])/(5*d) + (4*A*Cos[2*c]*Sin[2*d*x])/(5*d) + (A*Cos[3*c]*Sin[3*d*x])/(5*d) - (4*(4*A + 5*C)*Tan[c/2])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^(3/2))","B",0
264,1,85,169,1.1351421,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((253 A+140 C) \cos (c+d x)+78 A \cos (2 (c+d x))+15 A \cos (3 (c+d x))+494 A+700 C)}{210 d \sqrt{\sec (c+d x)}}","\frac{8 a^2 (19 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (19 A+35 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d \sqrt{\sec (c+d x)}}+\frac{6 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*(494*A + 700*C + (253*A + 140*C)*Cos[c + d*x] + 78*A*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d*Sqrt[Sec[c + d*x]])","A",1
265,1,103,219,1.4255604,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (799 A+756 C) \cos (c+d x)+4 (137 A+63 C) \cos (2 (c+d x))+170 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+2689 A+3276 C)}{1260 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 (52 A+63 C) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a*(2689*A + 3276*C + 2*(799*A + 756*C)*Cos[c + d*x] + 4*(137*A + 63*C)*Cos[2*(c + d*x)] + 170*A*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d*Sqrt[Sec[c + d*x]])","A",1
266,1,125,266,2.135445,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (5789 A+5566 C) \cos (c+d x)+8 (581 A+429 C) \cos (2 (c+d x))+1645 A \cos (3 (c+d x))+490 A \cos (4 (c+d x))+105 A \cos (5 (c+d x))+18494 A+660 C \cos (3 (c+d x))+21736 C)}{9240 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 (112 A+143 C) \sin (c+d x)}{385 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (28 A+33 C) \sin (c+d x)}{231 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{33 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a*(18494*A + 21736*C + 2*(5789*A + 5566*C)*Cos[c + d*x] + 8*(581*A + 429*C)*Cos[2*(c + d*x)] + 1645*A*Cos[3*(c + d*x)] + 660*C*Cos[3*(c + d*x)] + 490*A*Cos[4*(c + d*x)] + 105*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(9240*d*Sqrt[Sec[c + d*x]])","A",1
267,1,295,312,4.1019141,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a (\sec (c+d x)+1))^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{13}{2}}(c+d x) \sqrt{\sec (c+d x)+1} (14 (4056 A+4591 C) \cos (c+d x)+16 (1496 A+1711 C) \cos (2 (c+d x))+25448 A \cos (3 (c+d x))+5216 A \cos (4 (c+d x))+3912 A \cos (5 (c+d x))+18720 A+21721 C \cos (3 (c+d x))+4060 C \cos (4 (c+d x))+3045 C \cos (5 (c+d x))+27412 C)-48 (1304 A+1015 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{12288 d (\sec (c+d x)+1)^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{a^{5/2} (1304 A+1015 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{512 d}+\frac{a^3 (136 A+109 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{96 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}+\frac{a C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d}",1,"(Cos[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*((18720*A + 27412*C + 14*(4056*A + 4591*C)*Cos[c + d*x] + 16*(1496*A + 1711*C)*Cos[2*(c + d*x)] + 25448*A*Cos[3*(c + d*x)] + 21721*C*Cos[3*(c + d*x)] + 5216*A*Cos[4*(c + d*x)] + 4060*C*Cos[4*(c + d*x)] + 3912*A*Cos[5*(c + d*x)] + 3045*C*Cos[5*(c + d*x)])*Sec[c + d*x]^(13/2)*Sqrt[1 + Sec[c + d*x]]*Tan[(c + d*x)/2] - 48*(1304*A + 1015*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(12288*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^(5/2))","A",0
268,1,273,265,4.196133,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a (\sec (c+d x)+1))^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \sqrt{\sec (c+d x)+1} (12 (1360 A+2343 C) \cos (c+d x)+4 (6640 A+6509 C) \cos (2 (c+d x))+5440 A \cos (3 (c+d x))+6000 A \cos (4 (c+d x))+20560 A+5660 C \cos (3 (c+d x))+4245 C \cos (4 (c+d x))+24863 C)-120 (400 A+283 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{7680 d (\sec (c+d x)+1)^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{a^{5/2} (400 A+283 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (1040 A+787 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (400 A+283 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{240 d}+\frac{a C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}",1,"(Cos[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*((20560*A + 24863*C + 12*(1360*A + 2343*C)*Cos[c + d*x] + 4*(6640*A + 6509*C)*Cos[2*(c + d*x)] + 5440*A*Cos[3*(c + d*x)] + 5660*C*Cos[3*(c + d*x)] + 6000*A*Cos[4*(c + d*x)] + 4245*C*Cos[4*(c + d*x)])*Sec[c + d*x]^(11/2)*Sqrt[1 + Sec[c + d*x]]*Tan[(c + d*x)/2] - 120*(400*A + 283*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(7680*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^(5/2))","A",0
269,1,250,218,3.9633402,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a (\sec (c+d x)+1))^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{\sec (c+d x)+1} ((1584 A+2203 C) \cos (c+d x)+4 (48 A+163 C) \cos (2 (c+d x))+528 A \cos (3 (c+d x))+192 A+489 C \cos (3 (c+d x))+844 C)-12 (304 A+163 C) \sqrt{\tan ^2(c+d x)} \csc (c+d x) \left(\log (\sec (c+d x)+1)-\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)\right)}{384 d (\sec (c+d x)+1)^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{a^{5/2} (304 A+163 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{32 d}+\frac{5 a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}",1,"(Cos[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*((192*A + 844*C + (1584*A + 2203*C)*Cos[c + d*x] + 4*(48*A + 163*C)*Cos[2*(c + d*x)] + 528*A*Cos[3*(c + d*x)] + 489*C*Cos[3*(c + d*x)])*Sec[c + d*x]^(9/2)*Sqrt[1 + Sec[c + d*x]]*Tan[(c + d*x)/2] - 12*(304*A + 163*C)*Csc[c + d*x]*(Log[1 + Sec[c + d*x]] - Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(384*d*(A + 2*C + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^(5/2))","A",0
270,1,411,218,6.8261116,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{5 (8 A+5 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left(\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-\log (\sec (c+d x)+1)\right) \left(A+C \sec ^2(c+d x)\right)}{4 d \left(1-\cos ^2(c+d x)\right) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{5/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(49 C \sin \left(\frac{d x}{2}\right)-24 A \sin \left(\frac{d x}{2}\right)\right)}{12 d}-\frac{\tan \left(\frac{c}{2}\right) \sec (c) (24 A \cos (c)-75 C \cos (c)-26 C)}{12 d}+\frac{4 A \sin (c) \cos (d x)}{d}+\frac{4 A \cos (c) \sin (d x)}{d}+\frac{2 C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) \sec (c+d x) (4 C \sin (c)+13 C \sin (d x))}{6 d}\right)}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}","\frac{5 a^{5/2} (8 A+5 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (24 A-49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+31 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{5 a C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d}",1,"(5*(8*A + 5*C)*Cos[c + d*x]^3*(-Log[1 + Sec[c + d*x]] + Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(a*(1 + Sec[c + d*x]))^(5/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(4*d*(1 - Cos[c + d*x]^2)*(A + 2*C + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(5/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*((4*A*Cos[d*x]*Sin[c])/d + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(-24*A*Sin[(d*x)/2] + 49*C*Sin[(d*x)/2]))/(12*d) + (4*A*Cos[c]*Sin[d*x])/d + (2*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(4*C*Sin[c] + 13*C*Sin[d*x]))/(6*d) - ((-26*C + 24*A*Cos[c] - 75*C*Cos[c])*Sec[c]*Tan[c/2])/(12*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^(5/2))","A",0
271,1,416,224,7.033523,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{(8 A+19 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left(\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-\log (\sec (c+d x)+1)\right) \left(A+C \sec ^2(c+d x)\right)}{2 d \left(1-\cos ^2(c+d x)\right) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{5/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(27 C \sin \left(\frac{d x}{2}\right)-56 A \sin \left(\frac{d x}{2}\right)\right)}{6 d}-\frac{\tan \left(\frac{c}{2}\right) \sec (c) (56 A \cos (c)-33 C \cos (c)-6 C)}{6 d}+\frac{28 A \sin (c) \cos (d x)}{3 d}+\frac{2 A \sin (2 c) \cos (2 d x)}{3 d}+\frac{28 A \cos (c) \sin (d x)}{3 d}+\frac{2 A \cos (2 c) \sin (2 d x)}{3 d}+\frac{C \sec (c) \sin (d x) \sec (c+d x)}{d}\right)}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}","\frac{a^{5/2} (8 A+19 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (56 A-27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{12 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (8 A-21 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{12 d}-\frac{a (4 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{6 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt{\sec (c+d x)}}",1,"((8*A + 19*C)*Cos[c + d*x]^3*(-Log[1 + Sec[c + d*x]] + Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(a*(1 + Sec[c + d*x]))^(5/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(2*d*(1 - Cos[c + d*x]^2)*(A + 2*C + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(5/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*((28*A*Cos[d*x]*Sin[c])/(3*d) + (2*A*Cos[2*d*x]*Sin[2*c])/(3*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(-56*A*Sin[(d*x)/2] + 27*C*Sin[(d*x)/2]))/(6*d) + (28*A*Cos[c]*Sin[d*x])/(3*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*A*Cos[2*c]*Sin[2*d*x])/(3*d) - ((-6*C + 56*A*Cos[c] - 33*C*Cos[c])*Sec[c]*Tan[c/2])/(6*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^(5/2))","A",0
272,1,428,210,6.6567243,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{10 C \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left(\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-\log (\sec (c+d x)+1)\right) \left(A+C \sec ^2(c+d x)\right)}{d \left(1-\cos ^2(c+d x)\right) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{5/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{(131 A+60 C) \sin (c) \cos (d x)}{15 d}+\frac{(131 A+60 C) \cos (c) \sin (d x)}{15 d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(64 A \sin \left(\frac{d x}{2}\right)+15 C \sin \left(\frac{d x}{2}\right)\right)}{15 d}-\frac{2 (64 A+15 C) \tan \left(\frac{c}{2}\right)}{15 d}+\frac{22 A \sin (2 c) \cos (2 d x)}{15 d}+\frac{A \sin (3 c) \cos (3 d x)}{5 d}+\frac{22 A \cos (2 c) \sin (2 d x)}{15 d}+\frac{A \cos (3 c) \sin (3 d x)}{5 d}\right)}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}","\frac{5 a^{5/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 (64 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (16 A-15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(10*C*Cos[c + d*x]^3*(-Log[1 + Sec[c + d*x]] + Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(a*(1 + Sec[c + d*x]))^(5/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(1 - Cos[c + d*x]^2)*(A + 2*C + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(5/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*(((131*A + 60*C)*Cos[d*x]*Sin[c])/(15*d) + (22*A*Cos[2*d*x]*Sin[2*c])/(15*d) + (A*Cos[3*d*x]*Sin[3*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(64*A*Sin[(d*x)/2] + 15*C*Sin[(d*x)/2]))/(15*d) + ((131*A + 60*C)*Cos[c]*Sin[d*x])/(15*d) + (22*A*Cos[2*c]*Sin[2*d*x])/(15*d) + (A*Cos[3*c]*Sin[3*d*x])/(5*d) - (2*(64*A + 15*C)*Tan[c/2])/(15*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^(5/2))","B",0
273,1,474,210,6.4466318,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{4 C \sin (c+d x) \cos ^3(c+d x) \sqrt{\sec ^2(c+d x)-1} (a (\sec (c+d x)+1))^{5/2} \left(\log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-\log (\sec (c+d x)+1)\right) \left(A+C \sec ^2(c+d x)\right)}{d \left(1-\cos ^2(c+d x)\right) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{(a (\sec (c+d x)+1))^{5/2} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{(137 A+196 C) \sin (c) \cos (d x)}{21 d}+\frac{(31 A+14 C) \sin (2 c) \cos (2 d x)}{21 d}+\frac{(137 A+196 C) \cos (c) \sin (d x)}{21 d}+\frac{(31 A+14 C) \cos (2 c) \sin (2 d x)}{21 d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(32 A \sin \left(\frac{d x}{2}\right)+49 C \sin \left(\frac{d x}{2}\right)\right)}{21 d}-\frac{4 (32 A+49 C) \tan \left(\frac{c}{2}\right)}{21 d}+\frac{3 A \sin (3 c) \cos (3 d x)}{7 d}+\frac{A \sin (4 c) \cos (4 d x)}{14 d}+\frac{3 A \cos (3 c) \sin (3 d x)}{7 d}+\frac{A \cos (4 c) \sin (4 d x)}{14 d}\right)}{\sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)^{5/2} (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 a^{5/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (32 A+49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{21 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(4*C*Cos[c + d*x]^3*(-Log[1 + Sec[c + d*x]] + Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(a*(1 + Sec[c + d*x]))^(5/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(1 - Cos[c + d*x]^2)*(A + 2*C + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(5/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)*(A + C*Sec[c + d*x]^2)*(((137*A + 196*C)*Cos[d*x]*Sin[c])/(21*d) + ((31*A + 14*C)*Cos[2*d*x]*Sin[2*c])/(21*d) + (3*A*Cos[3*d*x]*Sin[3*c])/(7*d) + (A*Cos[4*d*x]*Sin[4*c])/(14*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(32*A*Sin[(d*x)/2] + 49*C*Sin[(d*x)/2]))/(21*d) + ((137*A + 196*C)*Cos[c]*Sin[d*x])/(21*d) + ((31*A + 14*C)*Cos[2*c]*Sin[2*d*x])/(21*d) + (3*A*Cos[3*c]*Sin[3*d*x])/(7*d) + (A*Cos[4*c]*Sin[4*d*x])/(14*d) - (4*(32*A + 49*C)*Tan[c/2])/(21*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^(5/2))","B",0
274,1,105,216,1.6113042,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (4 (779 A+588 C) \cos (c+d x)+4 (254 A+63 C) \cos (2 (c+d x))+260 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+5653 A+7476 C)}{1260 d \sqrt{\sec (c+d x)}}","\frac{64 a^3 (13 A+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (13 A+21 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d \sqrt{\sec (c+d x)}}+\frac{2 a (13 A+21 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{10 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^2*(5653*A + 7476*C + 4*(779*A + 588*C)*Cos[c + d*x] + 4*(254*A + 63*C)*Cos[2*(c + d*x)] + 260*A*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d*Sqrt[Sec[c + d*x]])","A",1
275,1,127,266,2.0953341,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (6989 A+6666 C) \cos (c+d x)+16 (325 A+198 C) \cos (2 (c+d x))+1735 A \cos (3 (c+d x))+448 A \cos (4 (c+d x))+63 A \cos (5 (c+d x))+22928 A+396 C \cos (3 (c+d x))+27456 C)}{5544 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 (232 A+297 C) \sin (c+d x)}{693 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a^2*(22928*A + 27456*C + 2*(6989*A + 6666*C)*Cos[c + d*x] + 16*(325*A + 198*C)*Cos[2*(c + d*x)] + 1735*A*Cos[3*(c + d*x)] + 396*C*Cos[3*(c + d*x)] + 448*A*Cos[4*(c + d*x)] + 63*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(5544*d*Sqrt[Sec[c + d*x]])","A",1
276,1,148,313,2.4647072,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sec ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (8 (226573 A+222794 C) \cos (c+d x)+(746519 A+581152 C) \cos (2 (c+d x))+287060 A \cos (3 (c+d x))+94010 A \cos (4 (c+d x))+23940 A \cos (5 (c+d x))+3465 A \cos (6 (c+d x))+2798182 A+148720 C \cos (3 (c+d x))+20020 C \cos (4 (c+d x))+3233516 C)}{720720 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 (8368 A+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (2224 A+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+143 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{10 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d \sec ^{\frac{11}{2}}(c+d x)}",1,"(a^2*(2798182*A + 3233516*C + 8*(226573*A + 222794*C)*Cos[c + d*x] + (746519*A + 581152*C)*Cos[2*(c + d*x)] + 287060*A*Cos[3*(c + d*x)] + 148720*C*Cos[3*(c + d*x)] + 94010*A*Cos[4*(c + d*x)] + 20020*C*Cos[4*(c + d*x)] + 23940*A*Cos[5*(c + d*x)] + 3465*A*Cos[6*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(720720*d*Sqrt[Sec[c + d*x]])","A",1
277,1,368,226,5.3619171,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos ^2(c+d x) \sqrt{\sec (c+d x)+1} \left(A+C \sec ^2(c+d x)\right) \left(\frac{6 \tan (c+d x) \left((8 A+9 C) \log (\sec (c+d x)+1)-(8 A+9 C) \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)+2 \sqrt{2} (A+C) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{\sqrt{\tan ^2(c+d x)}}+\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} \sec ^{\frac{5}{2}}(c+d x) (3 (8 A+7 C) \cos (2 (c+d x))+24 A-4 C \cos (c+d x)+37 C)\right)}{24 d \sqrt{a (\sec (c+d x)+1)} (A \cos (2 (c+d x))+A+2 C)}","\frac{(8 A+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(8 A+9 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^2*Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((24*A + 37*C - 4*C*Cos[c + d*x] + 3*(8*A + 7*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2)*Sqrt[1 + Sec[c + d*x]]*Tan[(c + d*x)/2] + (6*((8*A + 9*C)*Log[1 + Sec[c + d*x]] - (8*A + 9*C)*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] + 2*Sqrt[2]*(A + C)*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]]))*Tan[c + d*x])/Sqrt[Tan[c + d*x]^2]))/(24*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])])","A",0
278,1,730,183,6.741999,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)+1} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(-\frac{3 C \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{C \sec (c) \sin (d x) \sec (c+d x)}{d}-\frac{C \sin \left(\frac{c}{2}\right) (\cos (c)-2)}{d \left(\cos \left(\frac{c}{2}\right)+\cos \left(\frac{3 c}{2}\right)\right)}\right)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}+\frac{\cos ^2(c+d x) \sqrt{\sec (c+d x)+1} \left(A+C \sec ^2(c+d x)\right) \left(-\frac{(-8 A-7 C) \sin (c+d x) \cos ^2(c+d x) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \left(\sqrt{2} \left(\log \left(-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)\right)+8 \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-8 \log (\sec (c+d x)+1)\right)}{4 d (\cos (c+d x)+1) \left(1-\cos ^2(c+d x)\right)}-\frac{C \sin (c+d x) \cos ^2(c+d x) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \left(\log \left(-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)-\log \left(-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)\right)}{2 d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)}}\right)}{4 \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}","-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(8 A+7 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}-\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(-((C*(-2 + Cos[c])*Sin[c/2])/(d*(Cos[c/2] + Cos[(3*c)/2]))) - (3*C*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/(2*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])]) + (Cos[c + d*x]^2*Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(-1/2*(C*Cos[c + d*x]^2*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])*Sqrt[2 - 2*Cos[c + d*x]^2]*Sqrt[1 - Cos[c + d*x]^2]) - ((-8*A - 7*C)*Cos[c + d*x]^2*(-8*Log[1 + Sec[c + d*x]] + 8*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Sqrt[2]*(-Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]]))*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(4*d*(1 + Cos[c + d*x])*(1 - Cos[c + d*x]^2))))/(4*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a*(1 + Sec[c + d*x])])","B",0
279,1,717,133,6.7652338,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{(2 A+C) \sin (c+d x) \cos ^4(c+d x) (\sec (c+d x)+1)^{3/2} \sqrt{\sec ^2(c+d x)-1} \left(\log \left(-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)-\log \left(-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)\right) \left(A+C \sec ^2(c+d x)\right)}{2 d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)} \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}-\frac{C \sin (c+d x) \cos ^4(c+d x) (\sec (c+d x)+1)^{3/2} \sqrt{\sec ^2(c+d x)-1} \left(\sqrt{2} \left(\log \left(-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)\right)+8 \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}+\sqrt{\sec (c+d x)}\right)-8 \log (\sec (c+d x)+1)\right) \left(A+C \sec ^2(c+d x)\right)}{4 d (\cos (c+d x)+1) \left(1-\cos ^2(c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}+\frac{\sqrt{\sec (c+d x)+1} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 C \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{2 C \tan \left(\frac{c}{2}\right)}{d}\right)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}-\frac{C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((2*A + C)*Cos[c + d*x]^4*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(1 + Sec[c + d*x])^(3/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(2*d*(1 + Cos[c + d*x])*Sqrt[2 - 2*Cos[c + d*x]^2]*Sqrt[1 - Cos[c + d*x]^2]*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a*(1 + Sec[c + d*x])]) - (C*Cos[c + d*x]^4*(-8*Log[1 + Sec[c + d*x]] + 8*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Sqrt[2]*(-Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]]))*(1 + Sec[c + d*x])^(3/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(4*d*(1 + Cos[c + d*x])*(1 - Cos[c + d*x]^2)*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a*(1 + Sec[c + d*x])]) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((2*C*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (2*C*Tan[c/2])/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","B",0
280,1,504,135,3.4423426,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{\tan (c+d x) \left(\frac{8 A}{\sqrt{\frac{1}{\cos (c+d x)+1}}}-8 A \sqrt{\sec (c+d x)} \sqrt{\sec (c+d x)+1}+\sqrt{2} A \sqrt{\tan ^2(c+d x)} \log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\sqrt{2} A \sqrt{\tan ^2(c+d x)} \log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-8 C \sqrt{\tan ^2(c+d x)} \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)+\sqrt{2} C \sqrt{\tan ^2(c+d x)} \log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\sqrt{2} C \sqrt{\tan ^2(c+d x)} \log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)+8 C \sqrt{\tan ^2(c+d x)} \log (\sec (c+d x)+1)\right)}{4 d (\sec (c+d x)-1) \sqrt{\sec (c+d x)+1} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"-1/4*(Tan[c + d*x]*((8*A)/Sqrt[(1 + Cos[c + d*x])^(-1)] - 8*A*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]] + 8*C*Log[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2] - 8*C*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]]*Sqrt[Tan[c + d*x]^2] + Sqrt[2]*A*Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]]*Sqrt[Tan[c + d*x]^2] + Sqrt[2]*C*Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]]*Sqrt[Tan[c + d*x]^2] - Sqrt[2]*A*Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]]*Sqrt[Tan[c + d*x]^2] - Sqrt[2]*C*Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]]*Sqrt[Tan[c + d*x]^2]))/(d*(-1 + Sec[c + d*x])*Sqrt[1 + Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","B",0
281,1,273,136,4.0249115,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)+1} \left(A+C \sec ^2(c+d x)\right) \left(3 \sqrt{2} (A+C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)-\frac{16 A \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1}}{\sec ^{\frac{5}{2}}(c+d x)}\right)}{6 d \sqrt{a (\sec (c+d x)+1)} (A \cos (2 (c+d x))+A+2 C)}","\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((-16*A*Sqrt[1 + Sec[c + d*x]]*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2])/Sec[c + d*x]^(5/2) + 3*Sqrt[2]*(A + C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(6*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])])","B",1
282,1,528,181,6.4444453,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sqrt{\sec (c+d x)+1} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{(71 A+60 C) \sin (c) \cos (d x)}{15 d}+\frac{(71 A+60 C) \cos (c) \sin (d x)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(17 A \sin \left(\frac{d x}{2}\right)+15 C \sin \left(\frac{d x}{2}\right)\right)}{15 d}-\frac{4 (17 A+15 C) \tan \left(\frac{c}{2}\right)}{15 d}-\frac{8 A \sin (2 c) \cos (2 d x)}{15 d}+\frac{A \sin (3 c) \cos (3 d x)}{5 d}-\frac{8 A \cos (2 c) \sin (2 d x)}{15 d}+\frac{A \cos (3 c) \sin (3 d x)}{5 d}\right)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}-\frac{(A+C) \sin (c+d x) \cos ^4(c+d x) (\sec (c+d x)+1)^{3/2} \sqrt{\sec ^2(c+d x)-1} \left(\log \left(-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)-\log \left(-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)\right) \left(A+C \sec ^2(c+d x)\right)}{d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)} \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 (13 A+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{2 A \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"-(((A + C)*Cos[c + d*x]^4*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(1 + Sec[c + d*x])^(3/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])*Sqrt[2 - 2*Cos[c + d*x]^2]*Sqrt[1 - Cos[c + d*x]^2]*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a*(1 + Sec[c + d*x])])) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(((71*A + 60*C)*Cos[d*x]*Sin[c])/(15*d) - (8*A*Cos[2*d*x]*Sin[2*c])/(15*d) + (A*Cos[3*d*x]*Sin[3*c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(17*A*Sin[(d*x)/2] + 15*C*Sin[(d*x)/2]))/(15*d) + ((71*A + 60*C)*Cos[c]*Sin[d*x])/(15*d) - (8*A*Cos[2*c]*Sin[2*d*x])/(15*d) + (A*Cos[3*c]*Sin[3*d*x])/(5*d) - (4*(17*A + 15*C)*Tan[c/2])/(15*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","B",0
283,1,573,224,6.5962227,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{(A+C) \sin (c+d x) \cos ^4(c+d x) (\sec (c+d x)+1)^{3/2} \sqrt{\sec ^2(c+d x)-1} \left(\log \left(-3 \sec ^2(c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)-\log \left(-3 \sec ^2(c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}-2 \sec (c+d x)+1\right)\right) \left(A+C \sec ^2(c+d x)\right)}{d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)} \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}+\frac{\sqrt{\sec (c+d x)+1} \sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(-\frac{2 (193 A+140 C) \sin (c) \cos (d x)}{105 d}+\frac{(113 A+70 C) \sin (2 c) \cos (2 d x)}{105 d}-\frac{2 (193 A+140 C) \cos (c) \sin (d x)}{105 d}+\frac{(113 A+70 C) \cos (2 c) \sin (2 d x)}{105 d}+\frac{8 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(46 A \sin \left(\frac{d x}{2}\right)+35 C \sin \left(\frac{d x}{2}\right)\right)}{105 d}+\frac{8 (46 A+35 C) \tan \left(\frac{c}{2}\right)}{105 d}-\frac{6 A \sin (3 c) \cos (3 d x)}{35 d}+\frac{A \sin (4 c) \cos (4 d x)}{14 d}-\frac{6 A \cos (3 c) \sin (3 d x)}{35 d}+\frac{A \cos (4 c) \sin (4 d x)}{14 d}\right)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 C)}","-\frac{2 (43 A+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (31 A+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 A \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"((A + C)*Cos[c + d*x]^4*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(1 + Sec[c + d*x])^(3/2)*Sqrt[-1 + Sec[c + d*x]^2]*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(1 + Cos[c + d*x])*Sqrt[2 - 2*Cos[c + d*x]^2]*Sqrt[1 - Cos[c + d*x]^2]*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a*(1 + Sec[c + d*x])]) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((-2*(193*A + 140*C)*Cos[d*x]*Sin[c])/(105*d) + ((113*A + 70*C)*Cos[2*d*x]*Sin[2*c])/(105*d) - (6*A*Cos[3*d*x]*Sin[3*c])/(35*d) + (A*Cos[4*d*x]*Sin[4*c])/(14*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(46*A*Sin[(d*x)/2] + 35*C*Sin[(d*x)/2]))/(105*d) - (2*(193*A + 140*C)*Cos[c]*Sin[d*x])/(105*d) + ((113*A + 70*C)*Cos[2*c]*Sin[2*d*x])/(105*d) - (6*A*Cos[3*c]*Sin[3*d*x])/(35*d) + (A*Cos[4*c]*Sin[4*d*x])/(14*d) + (8*(46*A + 35*C)*Tan[c/2])/(105*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","B",0
284,1,800,188,7.2686957,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{(\sec (c+d x)+1)^{3/2} \left(C \sec ^2(c+d x)+A\right) \left(\frac{(A+3 C) \cos ^2(c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)\right) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \sin (c+d x)}{2 d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)}}-\frac{3 C \cos ^2(c+d x) \left(-8 \log (\sec (c+d x)+1)+8 \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}\right)+\sqrt{2} \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)\right)\right) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \sin (c+d x)}{2 d (\cos (c+d x)+1) \left(1-\cos ^2(c+d x)\right)}\right) \cos ^2(c+d x)}{2 (\cos (2 c+2 d x) A+A+2 C) (a (\sec (c+d x)+1))^{3/2}}+\frac{\sqrt{(\cos (c+d x)+1) \sec (c+d x)} (\sec (c+d x)+1)^{3/2} \left(C \sec ^2(c+d x)+A\right) \left(\frac{\sec \left(\frac{c}{2}\right) \left(-A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\sec \left(\frac{c}{2}\right) \left(-A \sin \left(\frac{c}{2}\right)-C \sin \left(\frac{c}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{(A+3 C) \tan \left(\frac{c}{2}\right)}{d}\right)}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","\frac{(A+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(A+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^2*(1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(((A + 3*C)*Cos[c + d*x]^2*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(2*d*(1 + Cos[c + d*x])*Sqrt[2 - 2*Cos[c + d*x]^2]*Sqrt[1 - Cos[c + d*x]^2]) - (3*C*Cos[c + d*x]^2*(-8*Log[1 + Sec[c + d*x]] + 8*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Sqrt[2]*(-Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]]))*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(2*d*(1 + Cos[c + d*x])*(1 - Cos[c + d*x]^2))))/(2*(A + 2*C + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(3/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(-(A*Sin[c/2]) - C*Sin[c/2]))/(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) - C*Sin[(d*x)/2]))/(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/d + ((A + 3*C)*Tan[c/2])/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","B",0
285,1,795,145,7.4383571,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{(\sec (c+d x)+1)^{3/2} \left(C \sec ^2(c+d x)+A\right) \left(\frac{(3 A-C) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)\right) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \sin (c+d x) \cos ^2(c+d x)}{2 d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)}}+\frac{C \left(-8 \log (\sec (c+d x)+1)+8 \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}\right)+\sqrt{2} \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)\right)\right) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \sin (c+d x) \cos ^2(c+d x)}{d (\cos (c+d x)+1) \left(1-\cos ^2(c+d x)\right)}\right) \cos ^2(c+d x)}{2 (\cos (2 c+2 d x) A+A+2 C) (a (\sec (c+d x)+1))^{3/2}}+\frac{\sqrt{(\cos (c+d x)+1) \sec (c+d x)} (\sec (c+d x)+1)^{3/2} \left(C \sec ^2(c+d x)+A\right) \left(\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{c}{2}\right)+C \sin \left(\frac{c}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\sec \left(\frac{c}{2}\right) \left(-A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{(A+C) \tan \left(\frac{c}{2}\right)}{d}\right)}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","\frac{(3 A-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[c + d*x]^2*(1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(((3*A - C)*Cos[c + d*x]^2*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(2*d*(1 + Cos[c + d*x])*Sqrt[2 - 2*Cos[c + d*x]^2]*Sqrt[1 - Cos[c + d*x]^2]) + (C*Cos[c + d*x]^2*(-8*Log[1 + Sec[c + d*x]] + 8*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Sqrt[2]*(-Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]]))*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])*(1 - Cos[c + d*x]^2))))/(2*(A + 2*C + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(3/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(A*Sin[c/2] + C*Sin[c/2]))/(2*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) - C*Sin[(d*x)/2]))/d + (Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(2*d) - ((A + C)*Tan[c/2])/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","B",0
286,1,303,152,3.5665292,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{(\sec (c+d x)+1)^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 \left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} (4 A \cos (c+d x)+5 A+C)}{\sec ^{\frac{3}{2}}(c+d x)}-\sqrt{2} (7 A-C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{8 d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 C)}","-\frac{(7 A-C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"((1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((2*(5*A + C + 4*A*Cos[c + d*x])*Sec[(c + d*x)/2]^3*Sqrt[1 + Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/Sec[c + d*x]^(3/2) - Sqrt[2]*(7*A - C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(8*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
287,1,316,201,3.0332915,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{(\sec (c+d x)+1)^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\sec (c+d x)+1} \sec ^3\left(\frac{1}{2} (c+d x)\right) (12 A \cos (c+d x)-2 A \cos (2 (c+d x))+17 A+3 C)}{\sec ^{\frac{3}{2}}(c+d x)}+3 \sqrt{2} (11 A+3 C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{24 d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(11 A+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(19 A+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{(7 A+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"((1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((2*(17*A + 3*C + 12*A*Cos[c + d*x] - 2*A*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Sqrt[1 + Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/Sec[c + d*x]^(3/2) + 3*Sqrt[2]*(11*A + 3*C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(24*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
288,1,331,248,5.4778897,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{(\sec (c+d x)+1)^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 \left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} ((39 A+20 C) \cos (c+d x)-2 A \cos (2 (c+d x))+A \cos (3 (c+d x))+47 A+25 C)}{\sec ^{\frac{3}{2}}(c+d x)}-5 \sqrt{2} (15 A+7 C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{40 d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 C)}","-\frac{(15 A+7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A+5 C) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{(49 A+25 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{(13 A+5 C) \sin (c+d x)}{10 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"((1 + Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((2*(47*A + 25*C + (39*A + 20*C)*Cos[c + d*x] - 2*A*Cos[2*(c + d*x)] + A*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^3*Sqrt[1 + Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/Sec[c + d*x]^(3/2) - 5*Sqrt[2]*(15*A + 7*C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(40*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
289,1,903,237,7.4450459,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\cos ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{(3 A+35 C) \cos ^2(c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)\right) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \sin (c+d x)}{2 d (\cos (c+d x)+1) \sqrt{2-2 \cos ^2(c+d x)} \sqrt{1-\cos ^2(c+d x)}}-\frac{20 C \cos ^2(c+d x) \left(-8 \log (\sec (c+d x)+1)+8 \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1}\right)+\sqrt{2} \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\sec (c+d x)+1} \sqrt{\sec ^2(c+d x)-1} \sqrt{\sec (c+d x)}+1\right)\right)\right) (\sec (c+d x)+1) \sqrt{\sec ^2(c+d x)-1} \sin (c+d x)}{d (\cos (c+d x)+1) \left(1-\cos ^2(c+d x)\right)}\right) (\sec (c+d x)+1)^{5/2}}{16 (\cos (2 c+2 d x) A+A+2 C) (a (\sec (c+d x)+1))^{5/2}}+\frac{\sqrt{(\cos (c+d x)+1) \sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(\frac{\sec \left(\frac{c}{2}\right) \left(-A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d}+\frac{\sec \left(\frac{c}{2}\right) \left(-A \sin \left(\frac{c}{2}\right)-C \sin \left(\frac{c}{2}\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d}+\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-15 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{\sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{c}{2}\right)-15 C \sin \left(\frac{c}{2}\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{\sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+35 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d}+\frac{(3 A+35 C) \tan \left(\frac{c}{2}\right)}{8 d}\right) (\sec (c+d x)+1)^{5/2}}{(\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}","\frac{(3 A+115 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(A-15 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[c + d*x]^2*(1 + Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(((3*A + 35*C)*Cos[c + d*x]^2*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]])*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(2*d*(1 + Cos[c + d*x])*Sqrt[2 - 2*Cos[c + d*x]^2]*Sqrt[1 - Cos[c + d*x]^2]) - (20*C*Cos[c + d*x]^2*(-8*Log[1 + Sec[c + d*x]] + 8*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Sqrt[2]*(-Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]] + Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[-1 + Sec[c + d*x]^2]]))*(1 + Sec[c + d*x])*Sqrt[-1 + Sec[c + d*x]^2]*Sin[c + d*x])/(d*(1 + Cos[c + d*x])*(1 - Cos[c + d*x]^2))))/(16*(A + 2*C + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(5/2)) + (Sqrt[(1 + Cos[c + d*x])*Sec[c + d*x]]*(1 + Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((Sec[c/2]*Sec[c/2 + (d*x)/2]^2*(A*Sin[c/2] - 15*C*Sin[c/2]))/(16*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^4*(-(A*Sin[c/2]) - C*Sin[c/2]))/(8*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - 15*C*Sin[(d*x)/2]))/(16*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) - C*Sin[(d*x)/2]))/(8*d) + (Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] + 35*C*Sin[(d*x)/2]))/(8*d) + ((3*A + 35*C)*Tan[c/2])/(8*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(5/2))","B",0
290,1,445,192,7.2740714,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\left(A+C \sec ^2(c+d x)\right) \left(8 \tan \left(\frac{c}{2}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{\sec (c+d x)+1} ((5 A-11 C) \cos (c+d x)+A-15 C)+\sin (c+d x) \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)} \left(\sqrt{2} (5 A-43 C) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)-256 C \log (\sec (c+d x)+1)+256 C \log \left(\sec ^{\frac{3}{2}}(c+d x)+\sqrt{\sec (c+d x)}+\sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}\right)\right)+8 \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{\sec (c+d x)+1} ((5 A-11 C) \cos (c+d x)+A-15 C)\right)}{64 d (\sec (c+d x)-1) \left(\frac{\sec (c+d x)}{\sec (c+d x)+1}\right)^{3/2} (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(5 A-43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(5 A-11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((A + C*Sec[c + d*x]^2)*(8*(A - 15*C + (5*A - 11*C)*Cos[c + d*x])*Sec[c/2]*Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[1 + Sec[c + d*x]]*Sin[(d*x)/2]*Tan[(c + d*x)/2]^2 + 8*(A - 15*C + (5*A - 11*C)*Cos[c + d*x])*Sec[c + d*x]*Sqrt[1 + Sec[c + d*x]]*Tan[c/2]*Tan[(c + d*x)/2]^2 + (-256*C*Log[1 + Sec[c + d*x]] + 256*C*Log[Sqrt[Sec[c + d*x]] + Sec[c + d*x]^(3/2) + Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] + Sqrt[2]*(5*A - 43*C)*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]]))*Sqrt[Sec[c + d*x]]*Sin[c + d*x]*Sqrt[Tan[c + d*x]^2]))/(64*d*(A + 2*C + A*Cos[2*(c + d*x)])*(-1 + Sec[c + d*x])*(Sec[c + d*x]/(1 + Sec[c + d*x]))^(3/2)*(a*(1 + Sec[c + d*x]))^(5/2))","B",0
291,1,308,154,2.9293462,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{(\sec (c+d x)+1)^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\sec (c+d x)+1} \sec ^5\left(\frac{1}{2} (c+d x)\right) ((13 A-3 C) \cos (c+d x)+9 A-7 C)}{\sec ^{\frac{3}{2}}(c+d x)}+\sqrt{2} (19 A+3 C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{64 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(19 A+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"((1 + Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(((9*A - 7*C + (13*A - 3*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^5*Sqrt[1 + Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/Sec[c + d*x]^(3/2) + Sqrt[2]*(19*A + 3*C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(64*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
292,1,317,199,3.2037119,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{(\sec (c+d x)+1)^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} (5 (17 A+C) \cos (c+d x)+16 A \cos (2 (c+d x))+65 A+C)}{\sec ^{\frac{3}{2}}(c+d x)}-5 \sqrt{2} (15 A-C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{64 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","-\frac{5 (15 A-C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(13 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"((1 + Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(((65*A + C + 5*(17*A + C)*Cos[c + d*x] + 16*A*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^5*Sqrt[1 + Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/Sec[c + d*x]^(3/2) - 5*Sqrt[2]*(15*A - C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(64*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
293,1,331,246,3.4187666,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{(\sec (c+d x)+1)^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\left(\sin \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{3}{2} (c+d x)\right)\right) \sqrt{\sec (c+d x)+1} \sec ^5\left(\frac{1}{2} (c+d x)\right) ((479 A+39 C) \cos (c+d x)+80 A \cos (2 (c+d x))-8 A \cos (3 (c+d x))+379 A+27 C)}{\sec ^{\frac{3}{2}}(c+d x)}+3 \sqrt{2} (163 A+19 C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{192 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(163 A+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(299 A+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{5 (19 A+3 C) \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(17 A+C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"((1 + Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(((379*A + 27*C + (479*A + 39*C)*Cos[c + d*x] + 80*A*Cos[2*(c + d*x)] - 8*A*Cos[3*(c + d*x)])*Sec[(c + d*x)/2]^5*Sqrt[1 + Sec[c + d*x]]*(Sin[(c + d*x)/2] - Sin[(3*(c + d*x))/2]))/Sec[c + d*x]^(3/2) + 3*Sqrt[2]*(163*A + 19*C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(192*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
294,1,349,295,4.2456214,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{(\sec (c+d x)+1)^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} (5 (887 A+255 C) \cos (c+d x)+16 (52 A+15 C) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+12 A \cos (4 (c+d x))+3491 A+975 C)}{\sec ^{\frac{3}{2}}(c+d x)}-15 \sqrt{2} (283 A+75 C) \cos ^2(c+d x) \sqrt{\tan ^2(c+d x)} \cot (c+d x) \left(\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)-2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)-\log \left(-3 \sec ^2(c+d x)-2 \sec (c+d x)+2 \sqrt{2} \sqrt{\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1} \sqrt{\sec (c+d x)}+1\right)\right)\right)}{960 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","-\frac{(283 A+75 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(2671 A+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(787 A+195 C) \sin (c+d x)}{240 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(21 A+5 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"((1 + Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(((3491*A + 975*C + 5*(887*A + 255*C)*Cos[c + d*x] + 16*(52*A + 15*C)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 12*A*Cos[4*(c + d*x)])*Sec[(c + d*x)/2]^5*Sqrt[1 + Sec[c + d*x]]*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2]))/Sec[c + d*x]^(3/2) - 15*Sqrt[2]*(283*A + 75*C)*Cos[c + d*x]^2*Cot[c + d*x]*(Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 - 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]] - Log[1 - 2*Sec[c + d*x] - 3*Sec[c + d*x]^2 + 2*Sqrt[2]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]]*Sqrt[Tan[c + d*x]^2]])*Sqrt[Tan[c + d*x]^2]))/(960*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
295,0,0,434,4.5786188,"\int (a+a \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2),x]","\int (a+a \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","\frac{3 \sqrt{2} A \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 C \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}+\frac{3 C \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d (\sec (c+d x)+1)}-\frac{3^{3/4} C \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"Integrate[(a + a*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2), x]","F",-1
296,0,0,384,2.8036351,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(1/3),x]","\int \frac{A+C \sec ^2(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}+\frac{3 C \tan (c+d x)}{2 d \sqrt[3]{a \sec (c+d x)+a}}+\frac{3^{3/4} C \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(1/3), x]","F",-1
297,0,0,396,3.0926663,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{4/3}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(4/3),x]","\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{4/3}} \, dx","\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{a d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 (A+C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^{4/3}}+\frac{3^{3/4} (A-4 C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[3]{2} a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(4/3), x]","F",-1
298,0,0,457,4.0180416,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{7/3}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(7/3),x]","\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{7/3}} \, dx","-\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(-\frac{5}{6};\frac{1}{2},1;\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 a^2 d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 (4 A-7 C) \tan (c+d x)}{55 a^2 d (\sec (c+d x)+1) \sqrt[3]{a \sec (c+d x)+a}}+\frac{3^{3/4} (4 A-7 C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{55 \sqrt[3]{2} a^2 d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 (A+C) \tan (c+d x)}{11 d (a \sec (c+d x)+a)^{7/3}}",1,"Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(7/3), x]","F",-1
299,0,0,815,29.2705683,"\int (a+a \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\int (a+a \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","\frac{3 C \tan (c+d x) (\sec (c+d x) a+a)^{4/3}}{7 d}+\frac{3 \sqrt{2} a A F_1\left(\frac{11}{6};\frac{1}{2},1;\frac{17}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right) (\sec (c+d x)+1) \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{11 d \sqrt{1-\sec (c+d x)}}+\frac{15 \sqrt[3]{2} \sqrt[4]{3} a C E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{7 d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{5\ 3^{3/4} \left(1-\sqrt{3}\right) a C F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{7\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 a C \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{7 d}-\frac{15 \left(1+\sqrt{3}\right) a C \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{7 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}",1,"Integrate[(a + a*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2), x]","F",-1
300,0,0,774,15.8376993,"\int \sqrt[3]{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\int \sqrt[3]{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","\frac{3 \sqrt{2} A \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 d \sqrt{1-\sec (c+d x)}}+\frac{3 C \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{4 d}-\frac{3 \left(1+\sqrt{3}\right) C \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{4 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}+\frac{3^{3/4} \left(1-\sqrt{3}\right) C \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{4\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 \sqrt[4]{3} C \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"Integrate[(a + a*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x]","F",-1
301,0,0,791,18.9915132,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(2/3),x]","\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx","\frac{3 \sqrt{2} A \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 a d \sqrt{1-\sec (c+d x)}}-\frac{3 (A+C) \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}-\frac{3 \left(1+\sqrt{3}\right) (A+2 C) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{a d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}+\frac{3^{3/4} \left(1-\sqrt{3}\right) (A+2 C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 \sqrt[3]{2} \sqrt[4]{3} (A+2 C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}",1,"Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(2/3), x]","F",-1
302,0,0,841,11.0828224,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/3),x]","\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","\frac{3 \sqrt[3]{2} \sqrt[4]{3} (2 A-5 C) E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x)}{7 a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3^{3/4} \left(1-\sqrt{3}\right) (2 A-5 C) F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{3 \sqrt{2} A F_1\left(-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (\sec (c+d x) a+a)^{2/3}}-\frac{3 (2 A-5 C) \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3}}-\frac{3 \left(1+\sqrt{3}\right) (2 A-5 C) \sqrt[3]{\sec (c+d x)+1} \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}-\frac{3 (A+C) \tan (c+d x)}{7 d (\sec (c+d x) a+a)^{5/3}}",1,"Integrate[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/3), x]","F",-1
303,0,0,244,20.4029832,"\int \sec ^m(c+d x) (a+a \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\int \sec ^m(c+d x) (a+a \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","\frac{2^{n+\frac{1}{2}} (A (m+n+1)+C (m-n)) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};1-m,\frac{1}{2}-n;\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (m+n+1)}+\frac{C 2^{n+\frac{3}{2}} n \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};1-m,-n-\frac{1}{2};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (m+n+1)}+\frac{C \sin (c+d x) \sec ^{m+1}(c+d x) (a \sec (c+d x)+a)^n}{d (m+n+1)}",1,"Integrate[Sec[c + d*x]^m*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x]","F",-1
304,0,0,253,26.796087,"\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","-\frac{(-A n+C n+C) \sin (c+d x) \sec ^{1-n}(c+d x) \left(\frac{\sec (c+d x)+1}{1-\sec (c+d x)}\right)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n \, _2F_1\left(\frac{1}{2}-n,-n;1-n;-\frac{2 \sec (c+d x)}{1-\sec (c+d x)}\right)}{d n (n+1) (\sec (c+d x)+1)}+\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}+\frac{C 2^{n+\frac{3}{2}} \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};n+1,-n-\frac{1}{2};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d}",1,"Integrate[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2), x]","F",-1
305,1,38,38,0.2196197,"\int \left(\frac{\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a A n-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right)\right) \, dx","Integrate[((a + a*Sec[c + d*x])^n*(-(a*A*n) - a*C*(1 + n)*Sec[c + d*x]))/(a*(1 + n)*Sec[c + d*x]^n) + Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a (\sec (c+d x)+1))^n}{d (n+1)}","\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}",1,"(A*(a*(1 + Sec[c + d*x]))^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n)","A",1
306,1,337,106,0.6714537,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a \sec ^4(c+d x) \left(12 (4 B+3 C) \cos (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 (4 B+3 C) \cos (4 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-24 B \sin (c+d x)-64 B \sin (2 (c+d x))-24 B \sin (3 (c+d x))-16 B \sin (4 (c+d x))+36 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-36 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-66 C \sin (c+d x)-64 C \sin (2 (c+d x))-18 C \sin (3 (c+d x))-16 C \sin (4 (c+d x))+27 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-27 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{a (B+C) \tan ^3(c+d x)}{3 d}+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 B+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"-1/192*(a*Sec[c + d*x]^4*(36*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 27*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*(4*B + 3*C)*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*(4*B + 3*C)*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 36*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 27*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 24*B*Sin[c + d*x] - 66*C*Sin[c + d*x] - 64*B*Sin[2*(c + d*x)] - 64*C*Sin[2*(c + d*x)] - 24*B*Sin[3*(c + d*x)] - 18*C*Sin[3*(c + d*x)] - 16*B*Sin[4*(c + d*x)] - 16*C*Sin[4*(c + d*x)]))/d","B",1
307,1,181,86,0.5539992,"\int \sec (c+d x) (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a \sec ^3(c+d x) \left(-4 \sin (c+d x) (3 (B+C) \cos (c+d x)+(3 B+2 C) \cos (2 (c+d x))+3 B+4 C)+9 (B+C) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 (B+C) \cos (3 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{24 d}","\frac{a (3 B+2 C) \tan (c+d x)}{3 d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (B+C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"-1/24*(a*Sec[c + d*x]^3*(9*(B + C)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*(B + C)*Cos[3*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 4*(3*B + 4*C + 3*(B + C)*Cos[c + d*x] + (3*B + 2*C)*Cos[2*(c + d*x)])*Sin[c + d*x]))/d","B",1
308,1,75,56,0.0442531,"\int (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a (B+C) \tan (c+d x)}{d}+\frac{a (2 B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/(2*d) + (a*B*Tan[c + d*x])/d + (a*C*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
309,1,43,32,0.0146507,"\int \cos (c+d x) (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \tan (c+d x)}{d}",1,"a*B*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*C*Tan[c + d*x])/d","A",1
310,1,46,32,0.0264053,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a B \sin (c) \cos (d x)}{d}+\frac{a B \cos (c) \sin (d x)}{d}+a B x+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+a C x","\frac{a B \sin (c+d x)}{d}+a x (B+C)+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*B*x + a*C*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Cos[d*x]*Sin[c])/d + (a*B*Cos[c]*Sin[d*x])/d","A",1
311,1,44,47,0.1009379,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a (4 (B+C) \sin (c+d x)+B \sin (2 (c+d x))+2 B c+2 B d x+4 C d x)}{4 d}","\frac{a (B+C) \sin (c+d x)}{d}+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (B+2 C)",1,"(a*(2*B*c + 2*B*d*x + 4*C*d*x + 4*(B + C)*Sin[c + d*x] + B*Sin[2*(c + d*x)]))/(4*d)","A",1
312,1,65,77,0.2021295,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a (3 (3 B+4 C) \sin (c+d x)+3 (B+C) \sin (2 (c+d x))+B \sin (3 (c+d x))+6 B c+6 B d x+6 c C+6 C d x)}{12 d}","\frac{a (2 B+3 C) \sin (c+d x)}{3 d}+\frac{a (B+C) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{1}{2} a x (B+C)",1,"(a*(6*B*c + 6*c*C + 6*B*d*x + 6*C*d*x + 3*(3*B + 4*C)*Sin[c + d*x] + 3*(B + C)*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d)","A",1
313,1,75,97,0.2529504,"\int \cos ^5(c+d x) (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \left(-32 (B+C) \sin ^3(c+d x)+96 (B+C) \sin (c+d x)+24 (B+C) \sin (2 (c+d x))+3 B \sin (4 (c+d x))+36 B c+36 B d x+48 c C+48 C d x\right)}{96 d}","-\frac{a (B+C) \sin ^3(c+d x)}{3 d}+\frac{a (B+C) \sin (c+d x)}{d}+\frac{a (3 B+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a B \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 B+4 C)",1,"(a*(36*B*c + 48*c*C + 36*B*d*x + 48*C*d*x + 96*(B + C)*Sin[c + d*x] - 32*(B + C)*Sin[c + d*x]^3 + 24*(B + C)*Sin[2*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)","A",1
314,1,391,169,0.8334893,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^2 \sec ^5(c+d x) \left(150 (7 B+6 C) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+75 (7 B+6 C) \cos (3 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-640 B \sin (c+d x)-660 B \sin (2 (c+d x))-800 B \sin (3 (c+d x))-210 B \sin (4 (c+d x))-160 B \sin (5 (c+d x))+105 B \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-105 B \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-960 C \sin (c+d x)-840 C \sin (2 (c+d x))-720 C \sin (3 (c+d x))-180 C \sin (4 (c+d x))-144 C \sin (5 (c+d x))+90 C \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-90 C \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 d}","\frac{a^2 (10 B+9 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (10 B+9 C) \tan (c+d x)}{5 d}+\frac{a^2 (7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 B+6 C) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 (7 B+6 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{C \tan (c+d x) \sec ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d}",1,"-1/1920*(a^2*Sec[c + d*x]^5*(105*B*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 90*C*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 150*(7*B + 6*C)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 75*(7*B + 6*C)*Cos[3*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 105*B*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 90*C*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 640*B*Sin[c + d*x] - 960*C*Sin[c + d*x] - 660*B*Sin[2*(c + d*x)] - 840*C*Sin[2*(c + d*x)] - 800*B*Sin[3*(c + d*x)] - 720*C*Sin[3*(c + d*x)] - 210*B*Sin[4*(c + d*x)] - 180*C*Sin[4*(c + d*x)] - 160*B*Sin[5*(c + d*x)] - 144*C*Sin[5*(c + d*x)]))/d","B",1
315,1,339,138,0.775819,"\int \sec (c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^2 \sec ^4(c+d x) \left(12 (8 B+7 C) \cos (2 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 (8 B+7 C) \cos (4 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-48 B \sin (c+d x)-112 B \sin (2 (c+d x))-48 B \sin (3 (c+d x))-40 B \sin (4 (c+d x))+72 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-72 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-90 C \sin (c+d x)-128 C \sin (2 (c+d x))-42 C \sin (3 (c+d x))-32 C \sin (4 (c+d x))+63 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-63 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{a^2 (8 B+7 C) \tan (c+d x)}{6 d}+\frac{a^2 (8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (8 B+7 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(4 B-C) \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d}",1,"-1/192*(a^2*Sec[c + d*x]^4*(72*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 63*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*(8*B + 7*C)*Cos[2*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*(8*B + 7*C)*Cos[4*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 72*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 63*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 48*B*Sin[c + d*x] - 90*C*Sin[c + d*x] - 112*B*Sin[2*(c + d*x)] - 128*C*Sin[2*(c + d*x)] - 48*B*Sin[3*(c + d*x)] - 42*C*Sin[3*(c + d*x)] - 40*B*Sin[4*(c + d*x)] - 32*C*Sin[4*(c + d*x)]))/d","B",1
316,1,63,103,0.3724101,"\int (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \left((9 B+6 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (B+2 C) \sec (c+d x)+12 (B+C)+2 C \tan ^2(c+d x)\right)\right)}{6 d}","\frac{2 a^2 (3 B+2 C) \tan (c+d x)}{3 d}+\frac{a^2 (3 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (3 B+2 C) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^2*((9*B + 6*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(12*(B + C) + 3*(B + 2*C)*Sec[c + d*x] + 2*C*Tan[c + d*x]^2)))/(6*d)","A",1
317,1,277,82,1.3070034,"\int \cos (c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{1}{16} a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (B+2 C) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (B+2 C) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 (4 B+3 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (4 B+3 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+4 B x+\frac{C}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)","\frac{a^2 (2 B+3 C) \tan (c+d x)}{2 d}+\frac{a^2 (4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 B x+\frac{C \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}",1,"(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(4*B*x - (2*(4*B + 3*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(4*B + 3*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + C/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(B + 2*C)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - C/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(B + 2*C)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/16","B",1
318,1,143,73,0.3639502,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \left(B \sin (c+d x)-B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 B c+2 B d x+C \tan (c+d x)-2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+c C+C d x\right)}{d}","\frac{a^2 (B-C) \sin (c+d x)}{d}+\frac{a^2 (B+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 B+C)+\frac{C \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{d}",1,"(a^2*(2*B*c + c*C + 2*B*d*x + C*d*x - B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + B*Sin[c + d*x] + C*Tan[c + d*x]))/d","A",1
319,1,96,88,0.1689791,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \left(4 (2 B+C) \sin (c+d x)+B \sin (2 (c+d x))+6 B d x-4 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 C d x\right)}{4 d}","\frac{a^2 (3 B+2 C) \sin (c+d x)}{2 d}+\frac{B \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}+\frac{1}{2} a^2 x (3 B+4 C)+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a^2*(6*B*d*x + 8*C*d*x - 4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*(2*B + C)*Sin[c + d*x] + B*Sin[2*(c + d*x)]))/(4*d)","A",1
320,1,61,102,0.189788,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 (3 (7 B+8 C) \sin (c+d x)+3 (2 B+C) \sin (2 (c+d x))+B \sin (3 (c+d x))+12 B d x+18 C d x)}{12 d}","\frac{2 a^2 (2 B+3 C) \sin (c+d x)}{3 d}+\frac{a^2 (2 B+3 C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (2 B+3 C)+\frac{B \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^2*(12*B*d*x + 18*C*d*x + 3*(7*B + 8*C)*Sin[c + d*x] + 3*(2*B + C)*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d)","A",1
321,1,86,135,0.3802296,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 (24 (6 B+7 C) \sin (c+d x)+48 (B+C) \sin (2 (c+d x))+16 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))+84 B c+84 B d x+8 C \sin (3 (c+d x))+96 C d x)}{96 d}","\frac{a^2 (4 B+5 C) \sin (c+d x)}{3 d}+\frac{a^2 (5 B+4 C) \sin (c+d x) \cos ^2(c+d x)}{12 d}+\frac{a^2 (7 B+8 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{B \sin (c+d x) \cos ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{4 d}+\frac{1}{8} a^2 x (7 B+8 C)",1,"(a^2*(84*B*c + 84*B*d*x + 96*C*d*x + 24*(6*B + 7*C)*Sin[c + d*x] + 48*(B + C)*Sin[2*(c + d*x)] + 16*B*Sin[3*(c + d*x)] + 8*C*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)","A",1
322,1,108,160,0.3962957,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 (60 (11 B+12 C) \sin (c+d x)+240 (B+C) \sin (2 (c+d x))+90 B \sin (3 (c+d x))+30 B \sin (4 (c+d x))+6 B \sin (5 (c+d x))+360 B c+360 B d x+80 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+420 C d x)}{480 d}","-\frac{a^2 (9 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (9 B+10 C) \sin (c+d x)}{5 d}+\frac{a^2 (6 B+5 C) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (6 B+7 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{B \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d}+\frac{1}{8} a^2 x (6 B+7 C)",1,"(a^2*(360*B*c + 360*B*d*x + 420*C*d*x + 60*(11*B + 12*C)*Sin[c + d*x] + 240*(B + C)*Sin[2*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 80*C*Sin[3*(c + d*x)] + 30*B*Sin[4*(c + d*x)] + 15*C*Sin[4*(c + d*x)] + 6*B*Sin[5*(c + d*x)]))/(480*d)","A",1
323,1,391,163,1.0174849,"\int \sec (c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^3 \sec ^5(c+d x) \left(150 (15 B+13 C) \cos (c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+75 (15 B+13 C) \cos (3 (c+d x)) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-1200 B \sin (c+d x)-1140 B \sin (2 (c+d x))-1560 B \sin (3 (c+d x))-450 B \sin (4 (c+d x))-360 B \sin (5 (c+d x))+225 B \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-225 B \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-1600 C \sin (c+d x)-1500 C \sin (2 (c+d x))-1520 C \sin (3 (c+d x))-390 C \sin (4 (c+d x))-304 C \sin (5 (c+d x))+195 C \cos (5 (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-195 C \cos (5 (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{1920 d}","\frac{a^3 (15 B+13 C) \tan ^3(c+d x)}{60 d}+\frac{a^3 (15 B+13 C) \tan (c+d x)}{5 d}+\frac{a^3 (15 B+13 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (15 B+13 C) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{(5 B-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d}",1,"-1/1920*(a^3*Sec[c + d*x]^5*(225*B*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 195*C*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 150*(15*B + 13*C)*Cos[c + d*x]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 75*(15*B + 13*C)*Cos[3*(c + d*x)]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 225*B*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 195*C*Cos[5*(c + d*x)]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 1200*B*Sin[c + d*x] - 1600*C*Sin[c + d*x] - 1140*B*Sin[2*(c + d*x)] - 1500*C*Sin[2*(c + d*x)] - 1560*B*Sin[3*(c + d*x)] - 1520*C*Sin[3*(c + d*x)] - 450*B*Sin[4*(c + d*x)] - 390*C*Sin[4*(c + d*x)] - 360*B*Sin[5*(c + d*x)] - 304*C*Sin[5*(c + d*x)]))/d","B",1
324,1,81,125,0.5021625,"\int (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 \left(15 (4 B+3 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 (B+3 C) \tan ^2(c+d x)+9 (4 B+5 C) \sec (c+d x)+96 (B+C)+6 C \sec ^3(c+d x)\right)\right)}{24 d}","\frac{a^3 (4 B+3 C) \tan ^3(c+d x)}{12 d}+\frac{a^3 (4 B+3 C) \tan (c+d x)}{d}+\frac{5 a^3 (4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (4 B+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(15*(4*B + 3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(96*(B + C) + 9*(4*B + 5*C)*Sec[c + d*x] + 6*C*Sec[c + d*x]^3 + 8*(B + 3*C)*Tan[c + d*x]^2)))/(24*d)","A",1
325,1,772,111,6.4496389,"\int \cos (c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a^3 \left(\frac{(\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(9 B \sin \left(\frac{d x}{2}\right)+11 C \sin \left(\frac{d x}{2}\right)\right)}{24 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(9 B \sin \left(\frac{d x}{2}\right)+11 C \sin \left(\frac{d x}{2}\right)\right)}{24 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{(\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-3 B \sin \left(\frac{c}{2}\right)+3 B \cos \left(\frac{c}{2}\right)-8 C \sin \left(\frac{c}{2}\right)+10 C \cos \left(\frac{c}{2}\right)\right)}{96 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \left(-3 B \sin \left(\frac{c}{2}\right)-3 B \cos \left(\frac{c}{2}\right)-8 C \sin \left(\frac{c}{2}\right)-10 C \cos \left(\frac{c}{2}\right)\right)}{96 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{(-7 B-5 C) (\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{16 d}+\frac{(7 B+5 C) (\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{16 d}+\frac{1}{8} B x (\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{C \sin \left(\frac{d x}{2}\right) (\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{C \sin \left(\frac{d x}{2}\right) (\cos (c+d x)+1)^3 \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}\right)","\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{a^3 (7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(3 B+5 C) \tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{6 d}+a^3 B x+\frac{a C \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"a^3*((B*x*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6)/8 + ((-7*B - 5*C)*(1 + Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(16*d) + ((7*B + 5*C)*(1 + Cos[c + d*x])^3*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6)/(16*d) + (C*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(48*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + ((1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(3*B*Cos[c/2] + 10*C*Cos[c/2] - 3*B*Sin[c/2] - 8*C*Sin[c/2]))/(96*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + ((1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(9*B*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(24*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (C*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*Sin[(d*x)/2])/(48*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + ((1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-3*B*Cos[c/2] - 10*C*Cos[c/2] - 3*B*Sin[c/2] - 8*C*Sin[c/2]))/(96*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + ((1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(9*B*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(24*d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])))","B",1
326,1,208,108,1.8070256,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 \left(4 (B+3 C) \tan (c+d x)+4 B \sin (c+d x)-12 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 B c+12 B d x+\frac{C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-14 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+14 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 c C+4 C d x\right)}{4 d}","\frac{a^3 (6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(B+2 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{d}+a^3 x (3 B+C)-\frac{5 a^3 C \sin (c+d x)}{2 d}+\frac{a C \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}",1,"(a^3*(12*B*c + 4*c*C + 12*B*d*x + 4*C*d*x - 12*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 14*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 14*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + C/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - C/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + 4*B*Sin[c + d*x] + 4*(B + 3*C)*Tan[c + d*x]))/(4*d)","A",1
327,1,272,117,1.9272549,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{1}{32} a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (3 B+C) \sin (c) \cos (d x)}{d}+\frac{4 (3 B+C) \cos (c) \sin (d x)}{d}-\frac{4 (B+3 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 (B+3 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{B \sin (2 c) \cos (2 d x)}{d}+\frac{B \cos (2 c) \sin (2 d x)}{d}+2 x (7 B+6 C)+\frac{4 C \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 C \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)","\frac{a^3 (B+3 C) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(B-2 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{2 d}+\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{1}{2} a^3 x (7 B+6 C)+\frac{a B \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}",1,"(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(2*(7*B + 6*C)*x - (4*(B + 3*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (4*(B + 3*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(3*B + C)*Cos[d*x]*Sin[c])/d + (B*Cos[2*d*x]*Sin[2*c])/d + (4*(3*B + C)*Cos[c]*Sin[d*x])/d + (B*Cos[2*c]*Sin[2*d*x])/d + (4*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/32","B",1
328,1,113,125,0.2670665,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 \left(9 (5 B+4 C) \sin (c+d x)+3 (3 B+C) \sin (2 (c+d x))+B \sin (3 (c+d x))+30 B d x-12 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+42 C d x\right)}{12 d}","\frac{5 a^3 (B+C) \sin (c+d x)}{2 d}+\frac{(5 B+3 C) \sin (c+d x) \cos (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{6 d}+\frac{1}{2} a^3 x (5 B+7 C)+\frac{a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^3*(30*B*d*x + 42*C*d*x - 12*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*(5*B + 4*C)*Sin[c + d*x] + 3*(3*B + C)*Sin[2*(c + d*x)] + B*Sin[3*(c + d*x)]))/(12*d)","A",1
329,1,86,124,0.3125322,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 (24 (13 B+15 C) \sin (c+d x)+24 (4 B+3 C) \sin (2 (c+d x))+24 B \sin (3 (c+d x))+3 B \sin (4 (c+d x))+180 B d x+8 C \sin (3 (c+d x))+240 C d x)}{96 d}","-\frac{a^3 (3 B+4 C) \sin ^3(c+d x)}{12 d}+\frac{a^3 (3 B+4 C) \sin (c+d x)}{d}+\frac{3 a^3 (3 B+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (3 B+4 C)+\frac{B \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(180*B*d*x + 240*C*d*x + 24*(13*B + 15*C)*Sin[c + d*x] + 24*(4*B + 3*C)*Sin[2*(c + d*x)] + 24*B*Sin[3*(c + d*x)] + 8*C*Sin[3*(c + d*x)] + 3*B*Sin[4*(c + d*x)]))/(96*d)","A",1
330,1,108,176,0.4609118,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 (60 (23 B+26 C) \sin (c+d x)+480 (B+C) \sin (2 (c+d x))+170 B \sin (3 (c+d x))+45 B \sin (4 (c+d x))+6 B \sin (5 (c+d x))+780 B c+780 B d x+120 C \sin (3 (c+d x))+15 C \sin (4 (c+d x))+900 C d x)}{480 d}","\frac{a^3 (38 B+45 C) \sin (c+d x)}{15 d}+\frac{a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac{a^3 (13 B+15 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{20 d}+\frac{1}{8} a^3 x (13 B+15 C)+\frac{a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}",1,"(a^3*(780*B*c + 780*B*d*x + 900*C*d*x + 60*(23*B + 26*C)*Sin[c + d*x] + 480*(B + C)*Sin[2*(c + d*x)] + 170*B*Sin[3*(c + d*x)] + 120*C*Sin[3*(c + d*x)] + 45*B*Sin[4*(c + d*x)] + 15*C*Sin[4*(c + d*x)] + 6*B*Sin[5*(c + d*x)]))/(480*d)","A",1
331,1,134,201,0.5334125,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 (120 (21 B+23 C) \sin (c+d x)+15 (63 B+64 C) \sin (2 (c+d x))+380 B \sin (3 (c+d x))+135 B \sin (4 (c+d x))+36 B \sin (5 (c+d x))+5 B \sin (6 (c+d x))+1380 B c+1380 B d x+340 C \sin (3 (c+d x))+90 C \sin (4 (c+d x))+12 C \sin (5 (c+d x))+1560 C d x)}{960 d}","-\frac{a^3 (17 B+19 C) \sin ^3(c+d x)}{15 d}+\frac{a^3 (17 B+19 C) \sin (c+d x)}{5 d}+\frac{a^3 (21 B+22 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^3 (23 B+26 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{1}{16} a^3 x (23 B+26 C)+\frac{a B \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}",1,"(a^3*(1380*B*c + 1380*B*d*x + 1560*C*d*x + 120*(21*B + 23*C)*Sin[c + d*x] + 15*(63*B + 64*C)*Sin[2*(c + d*x)] + 380*B*Sin[3*(c + d*x)] + 340*C*Sin[3*(c + d*x)] + 135*B*Sin[4*(c + d*x)] + 90*C*Sin[4*(c + d*x)] + 36*B*Sin[5*(c + d*x)] + 12*C*Sin[5*(c + d*x)] + 5*B*Sin[6*(c + d*x)]))/(960*d)","A",1
332,1,550,131,1.4618356,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(27 (B-C) \cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+27 (B-C) \cos \left(\frac{3}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-12 B \sin \left(\frac{1}{2} (c+d x)\right)+18 B \sin \left(\frac{3}{2} (c+d x)\right)-6 B \sin \left(\frac{5}{2} (c+d x)\right)+12 B \sin \left(\frac{7}{2} (c+d x)\right)+9 B \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 B \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-9 B \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-9 B \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-30 C \sin \left(\frac{3}{2} (c+d x)\right)+2 C \sin \left(\frac{5}{2} (c+d x)\right)-16 C \sin \left(\frac{7}{2} (c+d x)\right)-9 C \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-9 C \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+9 C \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+9 C \cos \left(\frac{7}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 a d (\cos (c+d x)+1)}","-\frac{(3 B-4 C) \tan ^3(c+d x)}{3 a d}-\frac{(3 B-4 C) \tan (c+d x)}{a d}+\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(B-C) \tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"-1/24*(Cos[(c + d*x)/2]*Sec[c + d*x]^3*(9*B*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 9*C*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 9*B*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 9*C*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 27*(B - C)*Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 27*(B - C)*Cos[(3*(c + d*x))/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 9*B*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*C*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 9*B*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*C*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 12*B*Sin[(c + d*x)/2] + 18*B*Sin[(3*(c + d*x))/2] - 30*C*Sin[(3*(c + d*x))/2] - 6*B*Sin[(5*(c + d*x))/2] + 2*C*Sin[(5*(c + d*x))/2] + 12*B*Sin[(7*(c + d*x))/2] - 16*C*Sin[(7*(c + d*x))/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
333,1,383,108,0.7129516,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(2 (2 B-3 C) \cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+(2 B-3 C) \cos \left(\frac{3}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+4 B \sin \left(\frac{1}{2} (c+d x)\right)+4 B \sin \left(\frac{5}{2} (c+d x)\right)+2 B \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 B \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-2 C \sin \left(\frac{1}{2} (c+d x)\right)+2 C \sin \left(\frac{3}{2} (c+d x)\right)-4 C \sin \left(\frac{5}{2} (c+d x)\right)-3 C \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+3 C \cos \left(\frac{5}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 a d (\cos (c+d x)+1)}","\frac{2 (B-C) \tan (c+d x)}{a d}-\frac{(2 B-3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{(2 B-3 C) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^2*(2*B*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 3*C*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*B - 3*C)*Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (2*B - 3*C)*Cos[(3*(c + d*x))/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*B*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*C*Cos[(5*(c + d*x))/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*B*Sin[(c + d*x)/2] - 2*C*Sin[(c + d*x)/2] + 2*C*Sin[(3*(c + d*x))/2] + 4*B*Sin[(5*(c + d*x))/2] - 4*C*Sin[(5*(c + d*x))/2]))/(4*a*d*(1 + Cos[c + d*x]))","B",1
334,1,234,62,0.5348543,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(-2 \sin \left(\frac{1}{2} (c+d x)\right) (C-(B-2 C) \cos (c+d x))+(B-C) \cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+(B-C) \cos \left(\frac{3}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\cos (c+d x)+1) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{(B-C) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(B-C) \tan (c+d x)}{d (a \sec (c+d x)+a)}+\frac{C \tan (c+d x)}{a d}",1,"-((Cos[(c + d*x)/2]*((B - C)*Cos[(c + d*x)/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (B - C)*Cos[(3*(c + d*x))/2]*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*(C - (B - 2*C)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(a*d*(1 + Cos[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))","B",1
335,1,106,44,0.2179248,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((B-C) \sin \left(\frac{1}{2} (c+d x)\right)+C \cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(B-C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(2*Cos[(c + d*x)/2]*(C*Cos[(c + d*x)/2]*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (B - C)*Sin[(c + d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
336,1,72,35,0.1439988,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(B d x \cos \left(c+\frac{d x}{2}\right)+2 (C-B) \sin \left(\frac{d x}{2}\right)+B d x \cos \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{B x}{a}-\frac{(B-C) \tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(B*d*x*Cos[(d*x)/2] + B*d*x*Cos[c + (d*x)/2] + 2*(-B + C)*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
337,1,76,60,0.3803585,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) (B \sin (c+d x)+d x (C-B))+(B-C) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(2 B-C) \sin (c+d x)}{a d}-\frac{(B-C) \sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x (B-C)}{a}",1,"(2*Cos[(c + d*x)/2]*((B - C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((-B + C)*d*x + B*Sin[c + d*x])))/(a*d*(1 + Cos[c + d*x]))","A",1
338,1,197,98,0.4685492,"\int \frac{\cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(4 d x (3 B-2 C) \cos \left(c+\frac{d x}{2}\right)-4 B \sin \left(c+\frac{d x}{2}\right)-3 B \sin \left(c+\frac{3 d x}{2}\right)-3 B \sin \left(2 c+\frac{3 d x}{2}\right)+B \sin \left(2 c+\frac{5 d x}{2}\right)+B \sin \left(3 c+\frac{5 d x}{2}\right)+4 d x (3 B-2 C) \cos \left(\frac{d x}{2}\right)-20 B \sin \left(\frac{d x}{2}\right)+4 C \sin \left(c+\frac{d x}{2}\right)+4 C \sin \left(c+\frac{3 d x}{2}\right)+4 C \sin \left(2 c+\frac{3 d x}{2}\right)+20 C \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","-\frac{2 (B-C) \sin (c+d x)}{a d}+\frac{(3 B-2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(B-C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{x (3 B-2 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(4*(3*B - 2*C)*d*x*Cos[(d*x)/2] + 4*(3*B - 2*C)*d*x*Cos[c + (d*x)/2] - 20*B*Sin[(d*x)/2] + 20*C*Sin[(d*x)/2] - 4*B*Sin[c + (d*x)/2] + 4*C*Sin[c + (d*x)/2] - 3*B*Sin[c + (3*d*x)/2] + 4*C*Sin[c + (3*d*x)/2] - 3*B*Sin[2*c + (3*d*x)/2] + 4*C*Sin[2*c + (3*d*x)/2] + B*Sin[2*c + (5*d*x)/2] + B*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","B",1
339,1,249,122,0.747098,"\int \frac{\cos ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-36 d x (B-C) \cos \left(c+\frac{d x}{2}\right)+21 B \sin \left(c+\frac{d x}{2}\right)+18 B \sin \left(c+\frac{3 d x}{2}\right)+18 B \sin \left(2 c+\frac{3 d x}{2}\right)-2 B \sin \left(2 c+\frac{5 d x}{2}\right)-2 B \sin \left(3 c+\frac{5 d x}{2}\right)+B \sin \left(3 c+\frac{7 d x}{2}\right)+B \sin \left(4 c+\frac{7 d x}{2}\right)-36 d x (B-C) \cos \left(\frac{d x}{2}\right)+69 B \sin \left(\frac{d x}{2}\right)-12 C \sin \left(c+\frac{d x}{2}\right)-9 C \sin \left(c+\frac{3 d x}{2}\right)-9 C \sin \left(2 c+\frac{3 d x}{2}\right)+3 C \sin \left(2 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{5 d x}{2}\right)-60 C \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","-\frac{(4 B-3 C) \sin ^3(c+d x)}{3 a d}+\frac{(4 B-3 C) \sin (c+d x)}{a d}-\frac{3 (B-C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{3 x (B-C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-36*(B - C)*d*x*Cos[(d*x)/2] - 36*(B - C)*d*x*Cos[c + (d*x)/2] + 69*B*Sin[(d*x)/2] - 60*C*Sin[(d*x)/2] + 21*B*Sin[c + (d*x)/2] - 12*C*Sin[c + (d*x)/2] + 18*B*Sin[c + (3*d*x)/2] - 9*C*Sin[c + (3*d*x)/2] + 18*B*Sin[2*c + (3*d*x)/2] - 9*C*Sin[2*c + (3*d*x)/2] - 2*B*Sin[2*c + (5*d*x)/2] + 3*C*Sin[2*c + (5*d*x)/2] - 2*B*Sin[3*c + (5*d*x)/2] + 3*C*Sin[3*c + (5*d*x)/2] + B*Sin[3*c + (7*d*x)/2] + B*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","B",1
340,1,379,156,1.7233732,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-8 (5 B-8 C) \tan ^3\left(\frac{1}{2} (c+d x)\right)+(26 B-44 C) \tan \left(\frac{1}{2} (c+d x)\right)-64 (B-C) \sin ^8\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+8 (B+5 C) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+\tan ^5\left(\frac{1}{2} (c+d x)\right) \left(B \sec ^2\left(\frac{1}{2} (c+d x)\right)+14 B-20 C\right)+3 (4 B-7 C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 (4 B-7 C) \tan ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-6 (4 B-7 C) \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-128 C \sin ^{12}\left(\frac{1}{2} (c+d x)\right) \csc ^7(c+d x)\right)}{6 a^2 d}","\frac{2 (5 B-8 C) \tan (c+d x)}{3 a^2 d}-\frac{(4 B-7 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(5 B-8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{(4 B-7 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{(B-C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(3*(4*B - 7*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 8*(B + 5*C)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 - 64*(B - C)*Csc[c + d*x]^5*Sin[(c + d*x)/2]^8 - 128*C*Csc[c + d*x]^7*Sin[(c + d*x)/2]^12 + (26*B - 44*C)*Tan[(c + d*x)/2] - 6*(4*B - 7*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Tan[(c + d*x)/2]^2 - 8*(5*B - 8*C)*Tan[(c + d*x)/2]^3 + 3*(4*B - 7*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Tan[(c + d*x)/2]^4 + (14*B - 20*C + B*Sec[(c + d*x)/2]^2)*Tan[(c + d*x)/2]^5))/(6*a^2*d)","B",1
341,1,245,108,1.1381637,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left((4 B-7 C) \tan ^3\left(\frac{1}{2} (c+d x)\right)+(13 C-4 B) \tan \left(\frac{1}{2} (c+d x)\right)+16 (B-C) \sin ^8\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)-4 (B-C) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-3 (B-2 C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 (B-2 C) \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d}","-\frac{(B-4 C) \tan (c+d x)}{3 a^2 d}+\frac{(B-2 C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(B-2 C) \tan (c+d x)}{a^2 d (\sec (c+d x)+1)}+\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(-3*(B - 2*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 4*(B - C)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 16*(B - C)*Csc[c + d*x]^5*Sin[(c + d*x)/2]^8 + (-4*B + 13*C)*Tan[(c + d*x)/2] + 3*(B - 2*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Tan[(c + d*x)/2]^2 + (4*B - 7*C)*Tan[(c + d*x)/2]^3))/(3*a^2*d)","B",1
342,1,106,79,0.7937177,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{(B-4 C) \tan \left(\frac{1}{2} (c+d x)\right)+4 (B-C) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+3 C \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3 a^2 d}","\frac{(2 B-5 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(B-C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(3*C*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4*(B - C)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + (B - 4*C)*Tan[(c + d*x)/2])/(3*a^2*d)","A",1
343,1,46,62,0.3354538,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2,x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left((C-B) \sec ^2\left(\frac{1}{2} (c+d x)\right)+2 (2 B+C)\right)}{6 a^2 d}","\frac{(B+2 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(B-C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"((2*(2*B + C) + (-B + C)*Sec[(c + d*x)/2]^2)*Tan[(c + d*x)/2])/(6*a^2*d)","A",1
344,1,153,70,0.3897236,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(12 B \sin \left(c+\frac{d x}{2}\right)-10 B \sin \left(c+\frac{3 d x}{2}\right)+9 B d x \cos \left(c+\frac{d x}{2}\right)+3 B d x \cos \left(c+\frac{3 d x}{2}\right)+3 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 B \sin \left(\frac{d x}{2}\right)+9 B d x \cos \left(\frac{d x}{2}\right)-6 C \sin \left(c+\frac{d x}{2}\right)+4 C \sin \left(c+\frac{3 d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","-\frac{(4 B-C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{B x}{a^2}-\frac{(B-C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*B*d*x*Cos[(d*x)/2] + 9*B*d*x*Cos[c + (d*x)/2] + 3*B*d*x*Cos[c + (3*d*x)/2] + 3*B*d*x*Cos[2*c + (3*d*x)/2] - 18*B*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2] + 12*B*Sin[c + (d*x)/2] - 6*C*Sin[c + (d*x)/2] - 10*B*Sin[c + (3*d*x)/2] + 4*C*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
345,1,245,98,0.6468571,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-18 d x (2 B-C) \cos \left(c+\frac{d x}{2}\right)-30 B \sin \left(c+\frac{d x}{2}\right)+41 B \sin \left(c+\frac{3 d x}{2}\right)+9 B \sin \left(2 c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{5 d x}{2}\right)-12 B d x \cos \left(c+\frac{3 d x}{2}\right)-12 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 d x (2 B-C) \cos \left(\frac{d x}{2}\right)+66 B \sin \left(\frac{d x}{2}\right)+24 C \sin \left(c+\frac{d x}{2}\right)-20 C \sin \left(c+\frac{3 d x}{2}\right)+6 C d x \cos \left(c+\frac{3 d x}{2}\right)+6 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-36 C \sin \left(\frac{d x}{2}\right)\right)}{12 a^2 d (\cos (c+d x)+1)^2}","\frac{2 (5 B-2 C) \sin (c+d x)}{3 a^2 d}-\frac{(2 B-C) \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{x (2 B-C)}{a^2}-\frac{(B-C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-18*(2*B - C)*d*x*Cos[(d*x)/2] - 18*(2*B - C)*d*x*Cos[c + (d*x)/2] - 12*B*d*x*Cos[c + (3*d*x)/2] + 6*C*d*x*Cos[c + (3*d*x)/2] - 12*B*d*x*Cos[2*c + (3*d*x)/2] + 6*C*d*x*Cos[2*c + (3*d*x)/2] + 66*B*Sin[(d*x)/2] - 36*C*Sin[(d*x)/2] - 30*B*Sin[c + (d*x)/2] + 24*C*Sin[c + (d*x)/2] + 41*B*Sin[c + (3*d*x)/2] - 20*C*Sin[c + (3*d*x)/2] + 9*B*Sin[2*c + (3*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] + 3*B*Sin[3*c + (5*d*x)/2]))/(12*a^2*d*(1 + Cos[c + d*x])^2)","B",1
346,1,315,143,0.8375195,"\int \frac{\cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(36 d x (7 B-4 C) \cos \left(c+\frac{d x}{2}\right)+147 B \sin \left(c+\frac{d x}{2}\right)-239 B \sin \left(c+\frac{3 d x}{2}\right)-63 B \sin \left(2 c+\frac{3 d x}{2}\right)-15 B \sin \left(2 c+\frac{5 d x}{2}\right)-15 B \sin \left(3 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{7 d x}{2}\right)+3 B \sin \left(4 c+\frac{7 d x}{2}\right)+84 B d x \cos \left(c+\frac{3 d x}{2}\right)+84 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+36 d x (7 B-4 C) \cos \left(\frac{d x}{2}\right)-381 B \sin \left(\frac{d x}{2}\right)-120 C \sin \left(c+\frac{d x}{2}\right)+164 C \sin \left(c+\frac{3 d x}{2}\right)+36 C \sin \left(2 c+\frac{3 d x}{2}\right)+12 C \sin \left(2 c+\frac{5 d x}{2}\right)+12 C \sin \left(3 c+\frac{5 d x}{2}\right)-48 C d x \cos \left(c+\frac{3 d x}{2}\right)-48 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+264 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{2 (8 B-5 C) \sin (c+d x)}{3 a^2 d}+\frac{(7 B-4 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(8 B-5 C) \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{x (7 B-4 C)}{2 a^2}-\frac{(B-C) \sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(36*(7*B - 4*C)*d*x*Cos[(d*x)/2] + 36*(7*B - 4*C)*d*x*Cos[c + (d*x)/2] + 84*B*d*x*Cos[c + (3*d*x)/2] - 48*C*d*x*Cos[c + (3*d*x)/2] + 84*B*d*x*Cos[2*c + (3*d*x)/2] - 48*C*d*x*Cos[2*c + (3*d*x)/2] - 381*B*Sin[(d*x)/2] + 264*C*Sin[(d*x)/2] + 147*B*Sin[c + (d*x)/2] - 120*C*Sin[c + (d*x)/2] - 239*B*Sin[c + (3*d*x)/2] + 164*C*Sin[c + (3*d*x)/2] - 63*B*Sin[2*c + (3*d*x)/2] + 36*C*Sin[2*c + (3*d*x)/2] - 15*B*Sin[2*c + (5*d*x)/2] + 12*C*Sin[2*c + (5*d*x)/2] - 15*B*Sin[3*c + (5*d*x)/2] + 12*C*Sin[3*c + (5*d*x)/2] + 3*B*Sin[3*c + (7*d*x)/2] + 3*B*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
347,1,369,170,0.7819512,"\int \frac{\cos ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-36 d x (10 B-7 C) \cos \left(c+\frac{d x}{2}\right)-156 B \sin \left(c+\frac{d x}{2}\right)+342 B \sin \left(c+\frac{3 d x}{2}\right)+118 B \sin \left(2 c+\frac{3 d x}{2}\right)+30 B \sin \left(2 c+\frac{5 d x}{2}\right)+30 B \sin \left(3 c+\frac{5 d x}{2}\right)-3 B \sin \left(3 c+\frac{7 d x}{2}\right)-3 B \sin \left(4 c+\frac{7 d x}{2}\right)+B \sin \left(4 c+\frac{9 d x}{2}\right)+B \sin \left(5 c+\frac{9 d x}{2}\right)-120 B d x \cos \left(c+\frac{3 d x}{2}\right)-120 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-36 d x (10 B-7 C) \cos \left(\frac{d x}{2}\right)+516 B \sin \left(\frac{d x}{2}\right)+147 C \sin \left(c+\frac{d x}{2}\right)-239 C \sin \left(c+\frac{3 d x}{2}\right)-63 C \sin \left(2 c+\frac{3 d x}{2}\right)-15 C \sin \left(2 c+\frac{5 d x}{2}\right)-15 C \sin \left(3 c+\frac{5 d x}{2}\right)+3 C \sin \left(3 c+\frac{7 d x}{2}\right)+3 C \sin \left(4 c+\frac{7 d x}{2}\right)+84 C d x \cos \left(c+\frac{3 d x}{2}\right)+84 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 C \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{4 (3 B-2 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{4 (3 B-2 C) \sin (c+d x)}{a^2 d}-\frac{(10 B-7 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(10 B-7 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{x (10 B-7 C)}{2 a^2}-\frac{(B-C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-36*(10*B - 7*C)*d*x*Cos[(d*x)/2] - 36*(10*B - 7*C)*d*x*Cos[c + (d*x)/2] - 120*B*d*x*Cos[c + (3*d*x)/2] + 84*C*d*x*Cos[c + (3*d*x)/2] - 120*B*d*x*Cos[2*c + (3*d*x)/2] + 84*C*d*x*Cos[2*c + (3*d*x)/2] + 516*B*Sin[(d*x)/2] - 381*C*Sin[(d*x)/2] - 156*B*Sin[c + (d*x)/2] + 147*C*Sin[c + (d*x)/2] + 342*B*Sin[c + (3*d*x)/2] - 239*C*Sin[c + (3*d*x)/2] + 118*B*Sin[2*c + (3*d*x)/2] - 63*C*Sin[2*c + (3*d*x)/2] + 30*B*Sin[2*c + (5*d*x)/2] - 15*C*Sin[2*c + (5*d*x)/2] + 30*B*Sin[3*c + (5*d*x)/2] - 15*C*Sin[3*c + (5*d*x)/2] - 3*B*Sin[3*c + (7*d*x)/2] + 3*C*Sin[3*c + (7*d*x)/2] - 3*B*Sin[4*c + (7*d*x)/2] + 3*C*Sin[4*c + (7*d*x)/2] + B*Sin[4*c + (9*d*x)/2] + B*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
348,1,428,202,2.1158424,"\int \frac{\sec ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-64 (9 B-19 C) \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 (87 B-197 C) \tan \left(\frac{1}{2} (c+d x)\right)+16 (12 B+13 C) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+\tan ^5\left(\frac{1}{2} (c+d x)\right) \left(3 (B-C) \sec ^4\left(\frac{1}{2} (c+d x)\right)+228 B-428 C\right)+30 (6 B-13 C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\tan ^3\left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) ((12 B-17 C) \cos (c+d x)-6 B+11 C)+\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) ((24 B-34 C) \cos (c+d x)-21 B+31 C)+30 (6 B-13 C) \tan ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-60 (6 B-13 C) \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{60 a^3 d}","\frac{8 (9 B-19 C) \tan (c+d x)}{15 a^3 d}-\frac{(6 B-13 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{4 (9 B-19 C) \tan (c+d x) \sec ^2(c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(6 B-13 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}+\frac{(B-C) \tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(6 B-11 C) \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*Sec[c + d*x]^2*(30*(6*B - 13*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 16*(12*B + 13*C)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 4*(87*B - 197*C)*Tan[(c + d*x)/2] + (-21*B + 31*C + (24*B - 34*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] - 60*(6*B - 13*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Tan[(c + d*x)/2]^2 - 64*(9*B - 19*C)*Tan[(c + d*x)/2]^3 - (-6*B + 11*C + (12*B - 17*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^3 + 30*(6*B - 13*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Tan[(c + d*x)/2]^4 + (228*B - 428*C + 3*(B - C)*Sec[(c + d*x)/2]^4)*Tan[(c + d*x)/2]^5))/(60*a^3*d)","B",1
349,1,294,156,1.8491447,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left((22 B-57 C) \tan ^3\left(\frac{1}{2} (c+d x)\right)+(87 C-22 B) \tan \left(\frac{1}{2} (c+d x)\right)+96 (B-C) \sin ^{10}\left(\frac{1}{2} (c+d x)\right) \csc ^7(c+d x)-4 (7 B-12 C) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)-15 (B-3 C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\frac{1}{4} \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) ((7 B-12 C) \cos (c+d x)-4 B+9 C)+15 (B-3 C) \tan ^2\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{15 a^3 d}","-\frac{(7 B-27 C) \tan (c+d x)}{15 a^3 d}+\frac{(B-3 C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(B-3 C) \tan (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(B-C) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(4 B-9 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^2*Sec[c + d*x]*(-15*(B - 3*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 4*(7*B - 12*C)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 96*(B - C)*Csc[c + d*x]^7*Sin[(c + d*x)/2]^10 + (-22*B + 87*C)*Tan[(c + d*x)/2] - ((-4*B + 9*C + (7*B - 12*C)*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/4 + 15*(B - 3*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Tan[(c + d*x)/2]^2 + (22*B - 57*C)*Tan[(c + d*x)/2]^3))/(15*a^3*d)","A",1
350,1,136,125,0.8093139,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{2 (B-11 C) \tan \left(\frac{1}{2} (c+d x)\right)+24 (B-C) \sin ^6\left(\frac{1}{2} (c+d x)\right) \csc ^5(c+d x)+4 (2 B-7 C) \sin ^4\left(\frac{1}{2} (c+d x)\right) \csc ^3(c+d x)+15 C \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15 a^3 d}","\frac{(4 B-29 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(2 B-7 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(15*C*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4*(2*B - 7*C)*Csc[c + d*x]^3*Sin[(c + d*x)/2]^4 + 24*(B - C)*Csc[c + d*x]^5*Sin[(c + d*x)/2]^6 + 2*(B - 11*C)*Tan[(c + d*x)/2])/(15*a^3*d)","A",1
351,1,70,102,0.1822538,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (6 (3 B+2 C) \cos (c+d x)+(3 B+2 C) \cos (2 (c+d x))+9 B+16 C)}{120 a^3 d}","\frac{(3 B+7 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 B-8 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"((9*B + 16*C + 6*(3*B + 2*C)*Cos[c + d*x] + (3*B + 2*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(120*a^3*d)","A",1
352,1,70,102,0.3775346,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3,x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (6 (2 B+3 C) \cos (c+d x)+(7 B+3 C) \cos (2 (c+d x))+11 B+9 C)}{120 a^3 d}","\frac{(2 B+3 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(2 B+3 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{(B-C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"((11*B + 9*C + 6*(2*B + 3*C)*Cos[c + d*x] + (7*B + 3*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])/(120*a^3*d)","A",1
353,1,241,108,0.5931066,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(270 B \sin \left(c+\frac{d x}{2}\right)-230 B \sin \left(c+\frac{3 d x}{2}\right)+90 B \sin \left(2 c+\frac{3 d x}{2}\right)-64 B \sin \left(2 c+\frac{5 d x}{2}\right)+150 B d x \cos \left(c+\frac{d x}{2}\right)+75 B d x \cos \left(c+\frac{3 d x}{2}\right)+75 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 B \sin \left(\frac{d x}{2}\right)+150 B d x \cos \left(\frac{d x}{2}\right)-60 C \sin \left(c+\frac{d x}{2}\right)+40 C \sin \left(c+\frac{3 d x}{2}\right)-30 C \sin \left(2 c+\frac{3 d x}{2}\right)+14 C \sin \left(2 c+\frac{5 d x}{2}\right)+80 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","-\frac{2 (11 B-C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{B x}{a^3}-\frac{(7 B-2 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(B-C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(150*B*d*x*Cos[(d*x)/2] + 150*B*d*x*Cos[c + (d*x)/2] + 75*B*d*x*Cos[c + (3*d*x)/2] + 75*B*d*x*Cos[2*c + (3*d*x)/2] + 15*B*d*x*Cos[2*c + (5*d*x)/2] + 15*B*d*x*Cos[3*c + (5*d*x)/2] - 370*B*Sin[(d*x)/2] + 80*C*Sin[(d*x)/2] + 270*B*Sin[c + (d*x)/2] - 60*C*Sin[c + (d*x)/2] - 230*B*Sin[c + (3*d*x)/2] + 40*C*Sin[c + (3*d*x)/2] + 90*B*Sin[2*c + (3*d*x)/2] - 30*C*Sin[2*c + (3*d*x)/2] - 64*B*Sin[2*c + (5*d*x)/2] + 14*C*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","B",1
354,1,365,136,1.0450583,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-300 d x (3 B-C) \cos \left(c+\frac{d x}{2}\right)-1125 B \sin \left(c+\frac{d x}{2}\right)+1215 B \sin \left(c+\frac{3 d x}{2}\right)-225 B \sin \left(2 c+\frac{3 d x}{2}\right)+363 B \sin \left(2 c+\frac{5 d x}{2}\right)+75 B \sin \left(3 c+\frac{5 d x}{2}\right)+15 B \sin \left(3 c+\frac{7 d x}{2}\right)+15 B \sin \left(4 c+\frac{7 d x}{2}\right)-450 B d x \cos \left(c+\frac{3 d x}{2}\right)-450 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-90 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-90 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-300 d x (3 B-C) \cos \left(\frac{d x}{2}\right)+1755 B \sin \left(\frac{d x}{2}\right)+540 C \sin \left(c+\frac{d x}{2}\right)-460 C \sin \left(c+\frac{3 d x}{2}\right)+180 C \sin \left(2 c+\frac{3 d x}{2}\right)-128 C \sin \left(2 c+\frac{5 d x}{2}\right)+150 C d x \cos \left(c+\frac{3 d x}{2}\right)+150 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+30 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+30 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-740 C \sin \left(\frac{d x}{2}\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","\frac{2 (36 B-11 C) \sin (c+d x)}{15 a^3 d}-\frac{(3 B-C) \sin (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{x (3 B-C)}{a^3}-\frac{(9 B-4 C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-300*(3*B - C)*d*x*Cos[(d*x)/2] - 300*(3*B - C)*d*x*Cos[c + (d*x)/2] - 450*B*d*x*Cos[c + (3*d*x)/2] + 150*C*d*x*Cos[c + (3*d*x)/2] - 450*B*d*x*Cos[2*c + (3*d*x)/2] + 150*C*d*x*Cos[2*c + (3*d*x)/2] - 90*B*d*x*Cos[2*c + (5*d*x)/2] + 30*C*d*x*Cos[2*c + (5*d*x)/2] - 90*B*d*x*Cos[3*c + (5*d*x)/2] + 30*C*d*x*Cos[3*c + (5*d*x)/2] + 1755*B*Sin[(d*x)/2] - 740*C*Sin[(d*x)/2] - 1125*B*Sin[c + (d*x)/2] + 540*C*Sin[c + (d*x)/2] + 1215*B*Sin[c + (3*d*x)/2] - 460*C*Sin[c + (3*d*x)/2] - 225*B*Sin[2*c + (3*d*x)/2] + 180*C*Sin[2*c + (3*d*x)/2] + 363*B*Sin[2*c + (5*d*x)/2] - 128*C*Sin[2*c + (5*d*x)/2] + 75*B*Sin[3*c + (5*d*x)/2] + 15*B*Sin[3*c + (7*d*x)/2] + 15*B*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
355,1,435,187,0.7481946,"\int \frac{\cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(600 d x (13 B-6 C) \cos \left(c+\frac{d x}{2}\right)+7560 B \sin \left(c+\frac{d x}{2}\right)-9230 B \sin \left(c+\frac{3 d x}{2}\right)+930 B \sin \left(2 c+\frac{3 d x}{2}\right)-2782 B \sin \left(2 c+\frac{5 d x}{2}\right)-750 B \sin \left(3 c+\frac{5 d x}{2}\right)-105 B \sin \left(3 c+\frac{7 d x}{2}\right)-105 B \sin \left(4 c+\frac{7 d x}{2}\right)+15 B \sin \left(4 c+\frac{9 d x}{2}\right)+15 B \sin \left(5 c+\frac{9 d x}{2}\right)+3900 B d x \cos \left(c+\frac{3 d x}{2}\right)+3900 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+600 d x (13 B-6 C) \cos \left(\frac{d x}{2}\right)-12760 B \sin \left(\frac{d x}{2}\right)-4500 C \sin \left(c+\frac{d x}{2}\right)+4860 C \sin \left(c+\frac{3 d x}{2}\right)-900 C \sin \left(2 c+\frac{3 d x}{2}\right)+1452 C \sin \left(2 c+\frac{5 d x}{2}\right)+300 C \sin \left(3 c+\frac{5 d x}{2}\right)+60 C \sin \left(3 c+\frac{7 d x}{2}\right)+60 C \sin \left(4 c+\frac{7 d x}{2}\right)-1800 C d x \cos \left(c+\frac{3 d x}{2}\right)-1800 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-360 C d x \cos \left(2 c+\frac{5 d x}{2}\right)-360 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+7020 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{8 (19 B-9 C) \sin (c+d x)}{15 a^3 d}+\frac{(13 B-6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{4 (19 B-9 C) \sin (c+d x) \cos (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{x (13 B-6 C)}{2 a^3}-\frac{(11 B-6 C) \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(B-C) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(600*(13*B - 6*C)*d*x*Cos[(d*x)/2] + 600*(13*B - 6*C)*d*x*Cos[c + (d*x)/2] + 3900*B*d*x*Cos[c + (3*d*x)/2] - 1800*C*d*x*Cos[c + (3*d*x)/2] + 3900*B*d*x*Cos[2*c + (3*d*x)/2] - 1800*C*d*x*Cos[2*c + (3*d*x)/2] + 780*B*d*x*Cos[2*c + (5*d*x)/2] - 360*C*d*x*Cos[2*c + (5*d*x)/2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] - 360*C*d*x*Cos[3*c + (5*d*x)/2] - 12760*B*Sin[(d*x)/2] + 7020*C*Sin[(d*x)/2] + 7560*B*Sin[c + (d*x)/2] - 4500*C*Sin[c + (d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] + 4860*C*Sin[c + (3*d*x)/2] + 930*B*Sin[2*c + (3*d*x)/2] - 900*C*Sin[2*c + (3*d*x)/2] - 2782*B*Sin[2*c + (5*d*x)/2] + 1452*C*Sin[2*c + (5*d*x)/2] - 750*B*Sin[3*c + (5*d*x)/2] + 300*C*Sin[3*c + (5*d*x)/2] - 105*B*Sin[3*c + (7*d*x)/2] + 60*C*Sin[3*c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] + 60*C*Sin[4*c + (7*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2] + 15*B*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
356,1,283,230,6.0784547,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 B \tan (c+d x) \left(35 (1-\sec (c+d x))^{9/2}-180 (1-\sec (c+d x))^{7/2}+378 (1-\sec (c+d x))^{5/2}-420 (1-\sec (c+d x))^{3/2}+315 \sqrt{1-\sec (c+d x)}\right) \sqrt{a (\sec (c+d x)+1)}}{315 d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1)}+\frac{2 C \tan (c+d x) \left(-63 (1-\sec (c+d x))^{11/2}+385 (1-\sec (c+d x))^{9/2}-990 (1-\sec (c+d x))^{7/2}+1386 (1-\sec (c+d x))^{5/2}-1155 (1-\sec (c+d x))^{3/2}+693 \sqrt{1-\sec (c+d x)}\right) \sqrt{a (\sec (c+d x)+1)}}{693 d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1)}","\frac{2 a (11 B+10 C) \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a (11 B+10 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{32 (11 B+10 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac{64 (11 B+10 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{32 a (11 B+10 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sec ^5(c+d x)}{11 d \sqrt{a \sec (c+d x)+a}}",1,"(2*B*(315*Sqrt[1 - Sec[c + d*x]] - 420*(1 - Sec[c + d*x])^(3/2) + 378*(1 - Sec[c + d*x])^(5/2) - 180*(1 - Sec[c + d*x])^(7/2) + 35*(1 - Sec[c + d*x])^(9/2))*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(315*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])) + (2*C*(693*Sqrt[1 - Sec[c + d*x]] - 1155*(1 - Sec[c + d*x])^(3/2) + 1386*(1 - Sec[c + d*x])^(5/2) - 990*(1 - Sec[c + d*x])^(7/2) + 385*(1 - Sec[c + d*x])^(9/2) - 63*(1 - Sec[c + d*x])^(11/2))*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(693*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x]))","A",1
357,1,98,187,0.5682211,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a \tan (c+d x) \left(5 (9 B+8 C) \sec ^3(c+d x)+6 (9 B+8 C) \sec ^2(c+d x)+8 (9 B+8 C) \sec (c+d x)+16 (9 B+8 C)+35 C \sec ^4(c+d x)\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a (9 B+8 C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (9 B+8 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac{8 (9 B+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{4 a (9 B+8 C) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(16*(9*B + 8*C) + 8*(9*B + 8*C)*Sec[c + d*x] + 6*(9*B + 8*C)*Sec[c + d*x]^2 + 5*(9*B + 8*C)*Sec[c + d*x]^3 + 35*C*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
358,1,81,144,0.2817022,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a \tan (c+d x) \left(3 (7 B+6 C) \sec ^2(c+d x)+4 (7 B+6 C) \sec (c+d x)+8 (7 B+6 C)+15 C \sec ^3(c+d x)\right)}{105 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (7 B+6 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 B+6 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (7 B+6 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(8*(7*B + 6*C) + 4*(7*B + 6*C)*Sec[c + d*x] + 3*(7*B + 6*C)*Sec[c + d*x]^2 + 15*C*Sec[c + d*x]^3)*Tan[c + d*x])/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
359,1,80,101,0.3271467,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} ((5 B+4 C) \cos (c+d x)+(5 B+4 C) \cos (2 (c+d x))+5 B+7 C)}{15 d (\cos (c+d x)+1)}","\frac{2 (5 B-2 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a (5 B+7 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}",1,"(2*(5*B + 7*C + (5*B + 4*C)*Cos[c + d*x] + (5*B + 4*C)*Cos[2*(c + d*x)])*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(15*d*(1 + Cos[c + d*x]))","A",1
360,1,43,62,0.1517198,"\int \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a \tan (c+d x) (3 B+C \sec (c+d x)+2 C)}{3 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a (3 B+C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a*(3*B + 2*C + C*Sec[c + d*x])*Tan[c + d*x])/(3*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
361,1,76,66,0.2492309,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}","\frac{2 \sqrt{a} B \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a C \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*C*Sin[(c + d*x)/2]))/d","A",1
362,1,93,68,0.2464795,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (B+2 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{2 d}","\frac{\sqrt{a} (B+2 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a B \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(B + 2*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*B*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
363,1,117,117,0.4518652,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 B \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)+C \left(\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{a (3 B+4 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (3 B+4 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"((C*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]]) + 2*B*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","C",1
364,1,70,160,0.2055103,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(B \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+C \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)\right)}{d}","\frac{a (5 B+6 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (5 B+6 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a (5 B+6 C) \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(2*(C*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]] + B*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
365,1,487,234,6.2317575,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 B \tan (c+d x) \sec ^4(c+d x) (a (\sec (c+d x)+1))^{3/2}}{9 d (\sec (c+d x)+1)^2}+\frac{34 B \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{3/2}}{63 d (\sec (c+d x)+1)^2}+\frac{68 B \tan (c+d x) \sec ^2(c+d x) (a (\sec (c+d x)+1))^{3/2}}{105 d (\sec (c+d x)+1)^2}+\frac{272 B \tan (c+d x) \sec (c+d x) (a (\sec (c+d x)+1))^{3/2}}{315 d (\sec (c+d x)+1)^2}+\frac{544 B \tan (c+d x) (a (\sec (c+d x)+1))^{3/2}}{315 d (\sec (c+d x)+1)^2}+\frac{2 C \tan (c+d x) \sec ^5(c+d x) (a (\sec (c+d x)+1))^{3/2}}{11 d (\sec (c+d x)+1)^2}+\frac{14 C \tan (c+d x) \sec ^4(c+d x) (a (\sec (c+d x)+1))^{3/2}}{33 d (\sec (c+d x)+1)^2}+\frac{16 C \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{3/2}}{33 d (\sec (c+d x)+1)^2}+\frac{32 C \tan (c+d x) \sec ^2(c+d x) (a (\sec (c+d x)+1))^{3/2}}{55 d (\sec (c+d x)+1)^2}+\frac{128 C \tan (c+d x) \sec (c+d x) (a (\sec (c+d x)+1))^{3/2}}{165 d (\sec (c+d x)+1)^2}+\frac{256 C \tan (c+d x) (a (\sec (c+d x)+1))^{3/2}}{165 d (\sec (c+d x)+1)^2}","\frac{2 a^2 (11 B+12 C) \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (187 B+168 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (187 B+168 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (187 B+168 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}-\frac{8 a (187 B+168 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}",1,"(544*B*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(315*d*(1 + Sec[c + d*x])^2) + (256*C*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(165*d*(1 + Sec[c + d*x])^2) + (272*B*Sec[c + d*x]*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(315*d*(1 + Sec[c + d*x])^2) + (128*C*Sec[c + d*x]*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(165*d*(1 + Sec[c + d*x])^2) + (68*B*Sec[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(105*d*(1 + Sec[c + d*x])^2) + (32*C*Sec[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(55*d*(1 + Sec[c + d*x])^2) + (34*B*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(63*d*(1 + Sec[c + d*x])^2) + (16*C*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(33*d*(1 + Sec[c + d*x])^2) + (2*B*Sec[c + d*x]^4*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(9*d*(1 + Sec[c + d*x])^2) + (14*C*Sec[c + d*x]^4*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(33*d*(1 + Sec[c + d*x])^2) + (2*C*Sec[c + d*x]^5*(a*(1 + Sec[c + d*x]))^(3/2)*Tan[c + d*x])/(11*d*(1 + Sec[c + d*x])^2)","B",1
366,1,100,189,0.7363437,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a^2 \tan (c+d x) \left(5 (9 B+17 C) \sec ^3(c+d x)+3 (39 B+34 C) \sec ^2(c+d x)+4 (39 B+34 C) \sec (c+d x)+8 (39 B+34 C)+35 C \sec ^4(c+d x)\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^2 (9 B+10 C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (39 B+34 C) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (39 B+34 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}-\frac{4 a (39 B+34 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}",1,"(2*a^2*(8*(39*B + 34*C) + 4*(39*B + 34*C)*Sec[c + d*x] + 3*(39*B + 34*C)*Sec[c + d*x]^2 + 5*(9*B + 17*C)*Sec[c + d*x]^3 + 35*C*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
367,1,82,138,0.417954,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a^2 \tan (c+d x) \left(3 (7 B+13 C) \sec ^2(c+d x)+(63 B+52 C) \sec (c+d x)+2 (63 B+52 C)+15 C \sec ^3(c+d x)\right)}{105 d \sqrt{a (\sec (c+d x)+1)}}","\frac{8 a^2 (21 B+19 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 B+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}",1,"(2*a^2*(2*(63*B + 52*C) + (63*B + 52*C)*Sec[c + d*x] + 3*(7*B + 13*C)*Sec[c + d*x]^2 + 15*C*Sec[c + d*x]^3)*Tan[c + d*x])/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
368,1,62,101,0.2943609,"\int (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a^2 \tan (c+d x) \left((5 B+9 C) \sec (c+d x)+25 B+3 C \sec ^2(c+d x)+18 C\right)}{15 d \sqrt{a (\sec (c+d x)+1)}}","\frac{8 a^2 (5 B+3 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (5 B+3 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^2*(25*B + 18*C + (5*B + 9*C)*Sec[c + d*x] + 3*C*Sec[c + d*x]^2)*Tan[c + d*x])/(15*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
369,1,102,105,0.5229103,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((3 B+5 C) \cos (c+d x)+C)+3 \sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{3 d}","\frac{2 a^{3/2} B \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (3 B+4 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(C + (3*B + 5*C)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
370,1,97,103,0.5460943,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (3 B+2 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (B \cos (c+d x)+2 C)\right)}{2 d}","\frac{a^{3/2} (3 B+2 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 (B-2 C) \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(3*B + 2*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*C + B*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
371,1,101,119,0.7608297,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\cos (c+d x) \sqrt{\sec (c+d x)-1} (2 B \cos (c+d x)+7 B+4 C)+(7 B+12 C) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{4 d \sqrt{\sec (c+d x)-1}}","\frac{a^{3/2} (7 B+12 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(a*((7*B + 12*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]] + Cos[c + d*x]*(7*B + 4*C + 2*B*Cos[c + d*x])*Sqrt[-1 + Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(4*d*Sqrt[-1 + Sec[c + d*x]])","A",1
372,1,740,164,11.9017045,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a \left(\frac{B (\cos (c+d x)+1) \tan (c+d x) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a (\sec (c+d x)+1)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)}{d (\sec (c+d x)+1)}+\frac{B (\cos (c+d x)+1) \tan (c+d x) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a (\sec (c+d x)+1)} \left(\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{4 d \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}}+\frac{B (\cos (c+d x)+1) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a (\sec (c+d x)+1)} \left(\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} \left(7 \sin \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+2 \sin \left(\frac{7}{2} (c+d x)\right)-3 \sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)+\frac{12 \tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{\sqrt{-\tan ^2(c+d x)}}\right)}{96 d \sqrt{\sec (c+d x)+1}}-\frac{C (\cos (c+d x)+1) \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{5}{2} (c+d x)\right)+\sqrt{2} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}\right)}{16 d}+\frac{C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a (\sec (c+d x)+1)}}{2 \sqrt{2} d}+\frac{C (\cos (c+d x)+1) \tan (c+d x) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \sqrt{a (\sec (c+d x)+1)} \left(\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{2 d \sqrt{-\tan ^2(c+d x)} \sqrt{\sec (c+d x)+1}}\right)","\frac{a^{3/2} (11 B+14 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (11 B+14 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (7 B+6 C) \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"a*((C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])])/(2*Sqrt[2]*d) - (C*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + Sin[(c + d*x)/2] - 2*Sin[(3*(c + d*x))/2] - Sin[(5*(c + d*x))/2]))/(16*d) + (B*(1 + Cos[c + d*x])*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sec[c/2 + (d*x)/2]^2*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(d*(1 + Sec[c + d*x])) + (B*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(4*d*Sqrt[1 + Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2]) + (C*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(2*d*Sqrt[1 + Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2]) + (B*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sqrt[a*(1 + Sec[c + d*x])]*(Sec[(c + d*x)/2]*Sqrt[1 + Sec[c + d*x]]*(-3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 7*Sin[(c + d*x)/2] - 2*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2] + 2*Sin[(7*(c + d*x))/2]) + (12*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/Sqrt[-Tan[c + d*x]^2]))/(96*d*Sqrt[1 + Sec[c + d*x]]))","C",1
373,1,1031,209,12.7395773,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a \left(-\frac{C (\cos (c+d x)+1) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} \sqrt{\cos (c+d x)} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{5}{2} (c+d x)\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d}+\frac{C (\cos (c+d x)+1) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(-3 \sqrt{2} \sqrt{\cos (c+d x)} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+7 \sin \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+2 \sin \left(\frac{7}{2} (c+d x)\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d}-\frac{B (\cos (c+d x)+1) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} \sqrt{\cos (c+d x)} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+5 \sin \left(\frac{1}{2} (c+d x)\right)-16 \sin \left(\frac{3}{2} (c+d x)\right)-9 \sin \left(\frac{5}{2} (c+d x)\right)-8 \sin \left(\frac{7}{2} (c+d x)\right)-6 \sin \left(\frac{9}{2} (c+d x)\right)\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{768 d}+\frac{3 B (\cos (c+d x)+1) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \tan (c+d x) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\sec (c+d x)+1)}+\frac{B (\cos (c+d x)+1) \sqrt{a (\sec (c+d x)+1)} \left(\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)+1} \left(-3 \sqrt{2} \sqrt{\cos (c+d x)} \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+7 \sin \left(\frac{1}{2} (c+d x)\right)-2 \sin \left(\frac{3}{2} (c+d x)\right)+3 \sin \left(\frac{5}{2} (c+d x)\right)+2 \sin \left(\frac{7}{2} (c+d x)\right)\right)+\frac{12 \left(\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+\cos (c+d x) \sqrt{1-\sec (c+d x)}\right) \tan (c+d x)}{\sqrt{-\tan ^2(c+d x)}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d \sqrt{\sec (c+d x)+1}}+\frac{C \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 \sqrt{2} d}+\frac{B (\cos (c+d x)+1) \left(\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+\cos (c+d x) \sqrt{1-\sec (c+d x)}\right) \sqrt{a (\sec (c+d x)+1)} \tan (c+d x) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d \sqrt{\sec (c+d x)+1} \sqrt{-\tan ^2(c+d x)}}+\frac{3 C (\cos (c+d x)+1) \left(\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+\cos (c+d x) \sqrt{1-\sec (c+d x)}\right) \sqrt{a (\sec (c+d x)+1)} \tan (c+d x) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d \sqrt{\sec (c+d x)+1} \sqrt{-\tan ^2(c+d x)}}\right)","\frac{a^{3/2} (75 B+88 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (75 B+88 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (9 B+8 C) \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (75 B+88 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"a*((C*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])])/(2*Sqrt[2]*d) - (C*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + Sin[(c + d*x)/2] - 2*Sin[(3*(c + d*x))/2] - Sin[(5*(c + d*x))/2]))/(16*d) + (C*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(-3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 7*Sin[(c + d*x)/2] - 2*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2] + 2*Sin[(7*(c + d*x))/2]))/(96*d) - (B*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 5*Sin[(c + d*x)/2] - 16*Sin[(3*(c + d*x))/2] - 9*Sin[(5*(c + d*x))/2] - 8*Sin[(7*(c + d*x))/2] - 6*Sin[(9*(c + d*x))/2]))/(768*d) + (3*B*(1 + Cos[c + d*x])*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sec[c/2 + (d*x)/2]^2*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(4*d*(1 + Sec[c + d*x])) + (B*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(4*d*Sqrt[1 + Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2]) + (3*C*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(8*d*Sqrt[1 + Sec[c + d*x]]*Sqrt[-Tan[c + d*x]^2]) + (B*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*Sqrt[a*(1 + Sec[c + d*x])]*(Sec[(c + d*x)/2]*Sqrt[1 + Sec[c + d*x]]*(-3*Sqrt[2]*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 7*Sin[(c + d*x)/2] - 2*Sin[(3*(c + d*x))/2] + 3*Sin[(5*(c + d*x))/2] + 2*Sin[(7*(c + d*x))/2]) + (12*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/Sqrt[-Tan[c + d*x]^2]))/(96*d*Sqrt[1 + Sec[c + d*x]]))","C",1
374,1,131,282,0.5850196,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a^3 \tan (c+d x) \left(315 (13 B+38 C) \sec ^5(c+d x)+35 (416 B+523 C) \sec ^4(c+d x)+5 (4615 B+4184 C) \sec ^3(c+d x)+6 (4615 B+4184 C) \sec ^2(c+d x)+8 (4615 B+4184 C) \sec (c+d x)+73840 B+3465 C \sec ^6(c+d x)+66944 C\right)}{45045 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^3 (299 B+280 C) \tan (c+d x) \sec ^4(c+d x)}{1287 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (4615 B+4184 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (13 B+16 C) \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{143 d}-\frac{8 a^2 (4615 B+4184 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{45045 d}+\frac{4 a (4615 B+4184 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac{2 a C \tan (c+d x) \sec ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{13 d}",1,"(2*a^3*(73840*B + 66944*C + 8*(4615*B + 4184*C)*Sec[c + d*x] + 6*(4615*B + 4184*C)*Sec[c + d*x]^2 + 5*(4615*B + 4184*C)*Sec[c + d*x]^3 + 35*(416*B + 523*C)*Sec[c + d*x]^4 + 315*(13*B + 38*C)*Sec[c + d*x]^5 + 3465*C*Sec[c + d*x]^6)*Tan[c + d*x])/(45045*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
375,1,487,237,6.1711695,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 B \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{5/2}}{9 d (\sec (c+d x)+1)^2}+\frac{38 B \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{5/2}}{63 d (\sec (c+d x)+1)^3}+\frac{146 B \tan (c+d x) \sec ^2(c+d x) (a (\sec (c+d x)+1))^{5/2}}{105 d (\sec (c+d x)+1)^3}+\frac{584 B \tan (c+d x) \sec (c+d x) (a (\sec (c+d x)+1))^{5/2}}{315 d (\sec (c+d x)+1)^3}+\frac{1168 B \tan (c+d x) (a (\sec (c+d x)+1))^{5/2}}{315 d (\sec (c+d x)+1)^3}+\frac{2 C \tan (c+d x) \sec ^4(c+d x) (a (\sec (c+d x)+1))^{5/2}}{11 d (\sec (c+d x)+1)^2}+\frac{46 C \tan (c+d x) \sec ^4(c+d x) (a (\sec (c+d x)+1))^{5/2}}{99 d (\sec (c+d x)+1)^3}+\frac{710 C \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{5/2}}{693 d (\sec (c+d x)+1)^3}+\frac{284 C \tan (c+d x) \sec ^2(c+d x) (a (\sec (c+d x)+1))^{5/2}}{231 d (\sec (c+d x)+1)^3}+\frac{1136 C \tan (c+d x) \sec (c+d x) (a (\sec (c+d x)+1))^{5/2}}{693 d (\sec (c+d x)+1)^3}+\frac{2272 C \tan (c+d x) (a (\sec (c+d x)+1))^{5/2}}{693 d (\sec (c+d x)+1)^3}","\frac{2 a^3 (209 B+194 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (803 B+710 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (11 B+14 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}-\frac{4 a^2 (803 B+710 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{2 a (803 B+710 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}",1,"(1168*B*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(315*d*(1 + Sec[c + d*x])^3) + (2272*C*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(693*d*(1 + Sec[c + d*x])^3) + (584*B*Sec[c + d*x]*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(315*d*(1 + Sec[c + d*x])^3) + (1136*C*Sec[c + d*x]*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(693*d*(1 + Sec[c + d*x])^3) + (146*B*Sec[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(105*d*(1 + Sec[c + d*x])^3) + (284*C*Sec[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(231*d*(1 + Sec[c + d*x])^3) + (38*B*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(63*d*(1 + Sec[c + d*x])^3) + (710*C*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(693*d*(1 + Sec[c + d*x])^3) + (46*C*Sec[c + d*x]^4*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(99*d*(1 + Sec[c + d*x])^3) + (2*B*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(9*d*(1 + Sec[c + d*x])^2) + (2*C*Sec[c + d*x]^4*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(11*d*(1 + Sec[c + d*x])^2)","B",1
376,1,96,175,0.6469035,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a^3 \tan (c+d x) \left(5 (9 B+26 C) \sec ^3(c+d x)+3 (60 B+73 C) \sec ^2(c+d x)+(345 B+292 C) \sec (c+d x)+690 B+35 C \sec ^4(c+d x)+584 C\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{64 a^3 (15 B+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 (9 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 B+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}",1,"(2*a^3*(690*B + 584*C + (345*B + 292*C)*Sec[c + d*x] + 3*(60*B + 73*C)*Sec[c + d*x]^2 + 5*(9*B + 26*C)*Sec[c + d*x]^3 + 35*C*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
377,1,79,138,0.3650095,"\int (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a^3 \tan (c+d x) \left(3 (7 B+20 C) \sec ^2(c+d x)+(98 B+115 C) \sec (c+d x)+301 B+15 C \sec ^3(c+d x)+230 C\right)}{105 d \sqrt{a (\sec (c+d x)+1)}}","\frac{64 a^3 (7 B+5 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (7 B+5 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (7 B+5 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(2*a^3*(301*B + 230*C + (98*B + 115*C)*Sec[c + d*x] + 3*(7*B + 20*C)*Sec[c + d*x]^2 + 15*C*Sec[c + d*x]^3)*Tan[c + d*x])/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
378,1,128,142,0.8340565,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 (5 B+14 C) \cos (c+d x)+(40 B+43 C) \cos (2 (c+d x))+40 B+49 C)+30 \sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{30 d}","\frac{2 a^{5/2} B \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (35 B+32 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (5 B+8 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*(30*Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(40*B + 49*C + 2*(5*B + 14*C)*Cos[c + d*x] + (40*B + 43*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(30*d)","A",1
379,1,126,143,0.9513922,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (5 B+2 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+\sin \left(\frac{1}{2} (c+d x)\right) (4 (3 B+8 C) \cos (c+d x)+3 B \cos (2 (c+d x))+3 B+4 C)\right)}{6 d}","\frac{a^{5/2} (5 B+2 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a^3 (3 B+14 C) \sin (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (B+2 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(5*B + 2*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (3*B + 4*C + 4*(3*B + 8*C)*Cos[c + d*x] + 3*B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
380,1,116,154,0.8419353,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (19 B+20 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) ((11 B+4 C) \cos (c+d x)+B \cos (2 (c+d x))+B+8 C)\right)}{8 d}","\frac{a^{5/2} (19 B+20 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (9 B-4 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (B-4 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{a B \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(19*B + 20*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(B + 8*C + (11*B + 4*C)*Cos[c + d*x] + B*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
381,1,121,164,1.3430617,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\cos (c+d x) \sqrt{\sec (c+d x)-1} (2 (17 B+6 C) \cos (c+d x)+4 B \cos (2 (c+d x))+79 B+66 C)+3 (25 B+38 C) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{24 d \sqrt{\sec (c+d x)-1}}","\frac{a^{5/2} (25 B+38 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (49 B+54 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (3 B+2 C) \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a B \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^2*(3*(25*B + 38*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]] + Cos[c + d*x]*(79*B + 66*C + 2*(17*B + 6*C)*Cos[c + d*x] + 4*B*Cos[2*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(24*d*Sqrt[-1 + Sec[c + d*x]])","A",1
382,1,366,209,1.7883137,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4608 B \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+2079 B \sqrt{1-\sec (c+d x)}+7641 B \cos (c+d x) \sqrt{1-\sec (c+d x)}+2097 B \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}+522 B \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+18 B \cos (4 (c+d x)) \sqrt{1-\sec (c+d x)}+6075 B \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+7680 C \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+1240 C \sqrt{1-\sec (c+d x)}+6360 C \cos (c+d x) \sqrt{1-\sec (c+d x)}+1240 C \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}-80 C \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+6600 C \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{2880 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{a^{5/2} (163 B+200 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (95 B+104 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (11 B+8 C) \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a^2*(6075*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 6600*C*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 2079*B*Sqrt[1 - Sec[c + d*x]] + 1240*C*Sqrt[1 - Sec[c + d*x]] + 7641*B*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 6360*C*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 2097*B*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 1240*C*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 522*B*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] - 80*C*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 18*B*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 7680*C*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 4608*B*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(2880*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
383,1,416,254,2.0340798,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(15360 B \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},6;\frac{3}{2};1-\sec (c+d x)\right)+11651 B \sqrt{1-\sec (c+d x)}+37029 B \cos (c+d x) \sqrt{1-\sec (c+d x)}+12653 B \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}+3818 B \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+1002 B \cos (4 (c+d x)) \sqrt{1-\sec (c+d x)}+72 B \cos (5 (c+d x)) \sqrt{1-\sec (c+d x)}+25935 B \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+21504 C \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+9702 C \sqrt{1-\sec (c+d x)}+35658 C \cos (c+d x) \sqrt{1-\sec (c+d x)}+9786 C \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}+2436 C \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+84 C \cos (4 (c+d x)) \sqrt{1-\sec (c+d x)}+28350 C \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{13440 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{a^{5/2} (283 B+326 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (283 B+326 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (157 B+170 C) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (283 B+326 C) \sin (c+d x) \cos (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (13 B+10 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*(25935*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 28350*C*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 11651*B*Sqrt[1 - Sec[c + d*x]] + 9702*C*Sqrt[1 - Sec[c + d*x]] + 37029*B*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 35658*C*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 12653*B*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 9786*C*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 3818*B*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 2436*C*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 1002*B*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 84*C*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 72*B*Cos[5*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 21504*C*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 15360*B*Hypergeometric2F1[1/2, 6, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(13440*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
384,1,183,243,1.3076471,"\int \frac{\sec ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\frac{1}{4} \sqrt{1-\sec (c+d x)} \sec ^4(c+d x) ((918 B-214 C) \cos (c+d x)-8 (69 B-157 C) \cos (2 (c+d x))+186 B \cos (3 (c+d x))-129 B \cos (4 (c+d x))-423 B-58 C \cos (3 (c+d x))+257 C \cos (4 (c+d x))+1279 C)+315 \sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{315 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (9 B-C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}-\frac{2 (3 B-19 C) \tan (c+d x) \sec ^2(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (93 B-29 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 a d}-\frac{4 (111 B-143 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}",1,"((315*Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + ((-423*B + 1279*C + (918*B - 214*C)*Cos[c + d*x] - 8*(69*B - 157*C)*Cos[2*(c + d*x)] + 186*B*Cos[3*(c + d*x)] - 58*C*Cos[3*(c + d*x)] - 129*B*Cos[4*(c + d*x)] + 257*C*Cos[4*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^4)/4)*Tan[c + d*x])/(315*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
385,1,140,202,0.6610774,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(3 (7 B-C) \sec ^2(c+d x)+(31 C-7 B) \sec (c+d x)+91 B+15 C \sec ^3(c+d x)-43 C\right)-105 \sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{105 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (7 B-C) \tan (c+d x) \sec ^2(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (7 B-31 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 a d}+\frac{4 (49 B-37 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"((-105*Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*Sqrt[1 - Sec[c + d*x]]*(91*B - 43*C + (-7*B + 31*C)*Sec[c + d*x] + 3*(7*B - C)*Sec[c + d*x]^2 + 15*C*Sec[c + d*x]^3))*Tan[c + d*x])/(105*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
386,1,123,159,0.478529,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left((5 B-C) \sec (c+d x)-5 B+3 C \sec ^2(c+d x)+13 C\right)+15 \sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{15 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (5 B-C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 a d}-\frac{4 (5 B-7 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^2(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"((15*Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*Sqrt[1 - Sec[c + d*x]]*(-5*B + 13*C + (5*B - C)*Sec[c + d*x] + 3*C*Sec[c + d*x]^2))*Tan[c + d*x])/(15*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
387,1,106,118,0.4069423,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} (3 B+C \sec (c+d x)-C)-3 \sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{3 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (3 B-2 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 a d}",1,"((-3*Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*Sqrt[1 - Sec[c + d*x]]*(3*B - C + C*Sec[c + d*x]))*Tan[c + d*x])/(3*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
388,1,88,78,0.1797366,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)+2 C \sqrt{1-\sec (c+d x)}\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"((Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*C*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
389,1,92,91,0.2952573,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((C-B) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+\sqrt{2} B \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 B \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*(Sqrt[2]*B*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + (-B + C)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]])*Cos[(c + d*x)/2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
390,1,11162,119,26.868719,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{(B-2 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{B \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"Result too large to show","C",0
391,1,135,165,0.5072691,"\int \frac{\cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} (2 B \cos (c+d x)-B+4 C)+(7 B-4 C) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-4 \sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{(B-4 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{(7 B-4 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(((7*B - 4*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-B + 4*C + 2*B*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
392,1,150,206,0.7606668,"\int \frac{\cos ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} \left(-2 (B-6 C) \cos (c+d x)+8 B \cos ^2(c+d x)+21 B-6 C\right)+(42 C-27 B) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+24 \sqrt{2} (B-C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{24 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(7 B-2 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{(9 B-14 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(B-6 C) \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{B \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(((-27*B + 42*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 24*Sqrt[2]*(B - C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(21*B - 6*C - 2*(B - 6*C)*Cos[c + d*x] + 8*B*Cos[c + d*x]^2)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(24*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
393,1,204,261,1.430043,"\int \frac{\sec ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(\frac{1}{4} \sqrt{1-\sec (c+d x)} \sec ^4(c+d x) (24 (217 B-213 C) \cos (c+d x)+60 (63 B-67 C) \cos (2 (c+d x))+1512 B \cos (3 (c+d x))+1029 B \cos (4 (c+d x))+2751 B-1608 C \cos (3 (c+d x))-1201 C \cos (4 (c+d x))-2339 C)-210 \sqrt{2} (15 B-19 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{420 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{(15 B-19 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(273 B-397 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{210 a^2 d}+\frac{(B-C) \tan (c+d x) \sec ^4(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(7 B-11 C) \tan (c+d x) \sec ^3(c+d x)}{14 a d \sqrt{a \sec (c+d x)+a}}+\frac{(63 B-67 C) \tan (c+d x) \sec ^2(c+d x)}{70 a d \sqrt{a \sec (c+d x)+a}}+\frac{(651 B-799 C) \tan (c+d x)}{105 a d \sqrt{a \sec (c+d x)+a}}",1,"((-210*Sqrt[2]*(15*B - 19*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + ((2751*B - 2339*C + 24*(217*B - 213*C)*Cos[c + d*x] + 60*(63*B - 67*C)*Cos[2*(c + d*x)] + 1512*B*Cos[3*(c + d*x)] - 1608*C*Cos[3*(c + d*x)] + 1029*B*Cos[4*(c + d*x)] - 1201*C*Cos[4*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^4)/4)*Tan[c + d*x])/(420*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
394,1,160,216,2.3539632,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(4 (5 B-3 C) \sec ^2(c+d x)-12 (5 B-9 C) \sec (c+d x)-95 B+12 C \sec ^3(c+d x)+147 C\right)+15 \sqrt{2} (11 B-15 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{30 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{(11 B-15 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(35 B-39 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{30 a^2 d}+\frac{(B-C) \tan (c+d x) \sec ^3(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(5 B-9 C) \tan (c+d x) \sec ^2(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{(65 B-93 C) \tan (c+d x)}{15 a d \sqrt{a \sec (c+d x)+a}}",1,"((15*Sqrt[2]*(11*B - 15*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(-95*B + 147*C - 12*(5*B - 9*C)*Sec[c + d*x] + 4*(5*B - 3*C)*Sec[c + d*x]^2 + 12*C*Sec[c + d*x]^3))*Tan[c + d*x])/(30*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
395,1,141,171,1.4033013,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(12 (B-C) \sec (c+d x)+15 B+4 C \sec ^2(c+d x)-19 C\right)-3 \sqrt{2} (7 B-11 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{6 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{(7 B-11 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(3 B-7 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{6 a^2 d}+\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(9 B-13 C) \tan (c+d x)}{3 a d \sqrt{a \sec (c+d x)+a}}",1,"((-3*Sqrt[2]*(7*B - 11*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(15*B - 19*C + 12*(B - C)*Sec[c + d*x] + 4*C*Sec[c + d*x]^2))*Tan[c + d*x])/(6*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
396,1,125,118,0.9027275,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} (-B+4 C \sec (c+d x)+5 C)+\sqrt{2} (3 B-7 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{2 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{(3 B-7 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(B-C) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 C \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}",1,"((Sqrt[2]*(3*B - 7*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(-B + 5*C + 4*C*Sec[c + d*x]))*Tan[c + d*x])/(2*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
397,1,127,87,0.7970701,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 (B-C) \sin (c+d x) \sqrt{1-\sec (c+d x)}+2 \sqrt{2} (B+3 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tan (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)}{4 a d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(B-C) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*(B - C)*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 2*Sqrt[2]*(B + 3*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Tan[c + d*x])/(4*a*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
398,1,147,127,1.6583512,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\csc (c+d x) \left(2 (C-B) \sin ^2\left(\frac{1}{2} (c+d x)\right)-\sqrt{2} (5 B-C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)+8 B \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{2 a d \sqrt{a (\sec (c+d x)+1)}}","-\frac{(5 B-C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 B \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(B-C) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Csc[c + d*x]*(8*B*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] - Sqrt[2]*(5*B - C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + 2*(-B + C)*Sin[(c + d*x)/2]^2))/(2*a*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
399,1,11954,170,27.0358083,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(3 B-2 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(9 B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(3 B-C) \sin (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(B-C) \sin (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
400,1,395,221,2.4424957,"\int \frac{\cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sec (c+d x) \left((91 B-48 C) (\sin (c+d x)+\tan (c+d x)) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-40 B \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right) (\sin (c+d x)+\tan (c+d x))-13 B \sin (c+d x) \sqrt{1-\sec (c+d x)}+\frac{13}{2} B \sin (2 (c+d x)) \sqrt{1-\sec (c+d x)}+18 B \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec (c+d x)}-52 \sqrt{2} B \sin (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)-52 \sqrt{2} B \tan (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)+24 C \sin (c+d x) \sqrt{1-\sec (c+d x)}+8 C \sin (2 (c+d x)) \sqrt{1-\sec (c+d x)}+36 \sqrt{2} C \sin (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)+36 \sqrt{2} C \tan (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{(19 B-12 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 B-9 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 B-6 C) \sin (c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}+\frac{(2 B-C) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(B-C) \sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Sec[c + d*x]*(-52*Sqrt[2]*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Sin[c + d*x] + 36*Sqrt[2]*C*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Sin[c + d*x] - 13*B*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 24*C*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 18*B*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + (13*B*Sqrt[1 - Sec[c + d*x]]*Sin[2*(c + d*x)])/2 + 8*C*Sqrt[1 - Sec[c + d*x]]*Sin[2*(c + d*x)] - 52*Sqrt[2]*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Tan[c + d*x] + 36*Sqrt[2]*C*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Tan[c + d*x] + (91*B - 48*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*(Sin[c + d*x] + Tan[c + d*x]) - 40*B*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]*(Sin[c + d*x] + Tan[c + d*x])))/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
401,1,177,261,2.874553,"\int \frac{\sec ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(160 (B-C) \sec ^3(c+d x)-32 (25 B-49 C) \sec ^2(c+d x)-5 (503 B-911 C) \sec (c+d x)-1495 B+96 C \sec ^4(c+d x)+2671 C\right)+30 \sqrt{2} (163 B-283 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{240 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{(163 B-283 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(475 B-787 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{240 a^3 d}-\frac{(85 B-157 C) \tan (c+d x) \sec ^2(c+d x)}{80 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(985 B-1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(B-C) \tan (c+d x) \sec ^4(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(13 B-21 C) \tan (c+d x) \sec ^3(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((30*Sqrt[2]*(163*B - 283*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(-1495*B + 2671*C - 5*(503*B - 911*C)*Sec[c + d*x] - 32*(25*B - 49*C)*Sec[c + d*x]^2 + 160*(B - C)*Sec[c + d*x]^3 + 96*C*Sec[c + d*x]^4))*Tan[c + d*x])/(240*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
402,1,161,216,2.7211614,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(32 (3 B-5 C) \sec ^2(c+d x)+(255 B-503 C) \sec (c+d x)+147 B+32 C \sec ^3(c+d x)-299 C\right)-6 \sqrt{2} (75 B-163 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{48 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{(75 B-163 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(39 B-95 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}+\frac{(93 B-197 C) \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(B-C) \tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(9 B-17 C) \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((-6*Sqrt[2]*(75*B - 163*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(147*B - 299*C + (255*B - 503*C)*Sec[c + d*x] + 32*(3*B - 5*C)*Sec[c + d*x]^2 + 32*C*Sec[c + d*x]^3))*Tan[c + d*x])/(48*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
403,1,144,169,1.5463419,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left((85 C-13 B) \sec (c+d x)-9 B+32 C \sec ^2(c+d x)+49 C\right)+2 \sqrt{2} (19 B-75 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{(19 B-75 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(B-9 C) \tan (c+d x)}{4 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(B-C) \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(5 B-13 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((2*Sqrt[2]*(19*B - 75*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(-9*B + 49*C + (-13*B + 85*C)*Sec[c + d*x] + 32*C*Sec[c + d*x]^2))*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
404,1,131,126,1.4671721,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} ((5 B-13 C) \sec (c+d x)+B-9 C)+2 \sqrt{2} (5 B+19 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{(5 B+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(5 B-13 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(B-C) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"((2*Sqrt[2]*(5*B + 19*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(B - 9*C + (5*B - 13*C)*Sec[c + d*x]))*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
405,1,206,126,1.7702312,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2),x]","\frac{64 B \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\sec (c+d x)} \sec (c+d x) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+C (10 \sin (c+d x)+\sin (2 (c+d x))) \sqrt{1-\sec (c+d x)}+40 \sqrt{2} C \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)}{32 a^2 d (\cos (c+d x)+1)^2 \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(3 B+5 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(3 B+5 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{(B-C) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(40*Sqrt[2]*C*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^5*Sec[c + d*x]*Sin[(c + d*x)/2] + 64*B*Cos[(c + d*x)/2]^5*Hypergeometric2F1[1/2, 3, 3/2, (1 - Sec[c + d*x])/2]*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]*Sin[(c + d*x)/2] + C*Sqrt[1 - Sec[c + d*x]]*(10*Sin[c + d*x] + Sin[2*(c + d*x)]))/(32*a^2*d*(1 + Cos[c + d*x])^2*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","C",1
406,1,11199,164,27.5250949,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{(43 B-3 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 B \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 B-3 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(B-C) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
407,1,12012,207,27.8271047,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{(5 B-2 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(115 B-43 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(35 B-11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(15 B-7 C) \sin (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(B-C) \sin (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
408,1,101,152,1.2389694,"\int \sec ^3(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \left(15 (4 A+3 (B+C)) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 (A+B+2 C) \tan ^2(c+d x)+15 (A+B+C)+3 C \tan ^4(c+d x)\right)+15 (4 A+3 (B+C)) \sec (c+d x)+30 (B+C) \sec ^3(c+d x)\right)\right)}{120 d}","\frac{a (5 A+5 B+4 C) \tan ^3(c+d x)}{15 d}+\frac{a (5 A+5 B+4 C) \tan (c+d x)}{5 d}+\frac{a (4 A+3 (B+C)) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 (B+C)) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a (B+C) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a C \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(a*(15*(4*A + 3*(B + C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(4*A + 3*(B + C))*Sec[c + d*x] + 30*(B + C)*Sec[c + d*x]^3 + 8*(15*(A + B + C) + 5*(A + B + 2*C)*Tan[c + d*x]^2 + 3*C*Tan[c + d*x]^4))))/(120*d)","A",1
409,1,84,127,0.7769944,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \left(3 (4 A+4 B+3 C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (4 A+4 B+3 C) \sec (c+d x)+24 (A+B+C)+8 (B+C) \tan ^2(c+d x)+6 C \sec ^3(c+d x)\right)\right)}{24 d}","\frac{a (3 A+2 (B+C)) \tan (c+d x)}{3 d}+\frac{a (4 A+4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+4 B+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a (B+C) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*(4*A + 4*B + 3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(A + B + C) + 3*(4*A + 4*B + 3*C)*Sec[c + d*x] + 6*C*Sec[c + d*x]^3 + 8*(B + C)*Tan[c + d*x]^2)))/(24*d)","A",1
410,1,485,92,12.2282622,"\int \sec (c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 (3 A+3 B+2 C) \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (3 A+3 B+2 C) \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-6 (2 A+B+C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (2 A+B+C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{(3 B+4 C) \cos \left(\frac{c}{2}\right)-(3 B+2 C) \sin \left(\frac{c}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{(3 B+2 C) \sin \left(\frac{c}{2}\right)+(3 B+4 C) \cos \left(\frac{c}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 C \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 C \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)}{6 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a (3 A+3 B+2 C) \tan (c+d x)}{3 d}+\frac{a (2 A+B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (B+C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-6*(2*A + B + C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(2*A + B + C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*C*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((3*B + 4*C)*Cos[c/2] - (3*B + 2*C)*Sin[c/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(3*A + 3*B + 2*C)*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*C*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - ((3*B + 4*C)*Cos[c/2] + (3*B + 2*C)*Sin[c/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(3*A + 3*B + 2*C)*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(6*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
411,1,305,63,1.8571327,"\int (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{2 (2 A+2 B+C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (2 A+2 B+C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+4 A x+\frac{4 (B+C) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (B+C) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a (2 A+2 B+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a A x+\frac{a (B+C) \tan (c+d x)}{d}+\frac{a C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(4*A*x - (2*(2*A + 2*B + C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(2*A + 2*B + C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + C/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(B + C)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - C/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(B + C)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
412,1,71,46,0.0341099,"\int \cos (c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \tan (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}","a x (A+B)+\frac{a A \sin (c+d x)}{d}+\frac{a (B+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tan (c+d x)}{d}",1,"a*A*x + a*B*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Cos[d*x]*Sin[c])/d + (a*A*Cos[c]*Sin[d*x])/d + (a*C*Tan[c + d*x])/d","A",1
413,1,59,62,0.146975,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \left(4 (A+B) \sin (c+d x)+A \sin (2 (c+d x))+2 A c+2 A d x+4 B d x+4 C \tanh ^{-1}(\sin (c+d x))+4 C d x\right)}{4 d}","\frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (A+2 (B+C))+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a*(2*A*c + 2*A*d*x + 4*B*d*x + 4*C*d*x + 4*C*ArcTanh[Sin[c + d*x]] + 4*(A + B)*Sin[c + d*x] + A*Sin[2*(c + d*x)]))/(4*d)","A",1
414,1,64,82,0.2393965,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a (3 (3 A+4 (B+C)) \sin (c+d x)+3 (A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+6 A d x+6 B d x+12 C d x)}{12 d}","\frac{a (2 A+3 (B+C)) \sin (c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+B+2 C)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(a*(6*A*d*x + 6*B*d*x + 12*C*d*x + 3*(3*A + 4*(B + C))*Sin[c + d*x] + 3*(A + B)*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)","A",1
415,1,97,102,0.412024,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a (24 (3 A+3 B+4 C) \sin (c+d x)+24 (A+B+C) \sin (2 (c+d x))+8 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+24 A c+36 A d x+8 B \sin (3 (c+d x))+48 B c+48 B d x+48 C d x)}{96 d}","\frac{a (A+B+C) \sin (c+d x)}{d}+\frac{a (3 A+4 (B+C)) \sin (c+d x) \cos (c+d x)}{8 d}-\frac{a (A+B) \sin ^3(c+d x)}{3 d}+\frac{1}{8} a x (3 A+4 (B+C))+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(24*A*c + 48*B*c + 36*A*d*x + 48*B*d*x + 48*C*d*x + 24*(3*A + 3*B + 4*C)*Sin[c + d*x] + 24*(A + B + C)*Sin[2*(c + d*x)] + 8*A*Sin[3*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
416,1,94,141,0.4518505,"\int \cos ^5(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \left(-160 (2 A+B+C) \sin ^3(c+d x)+480 (A+B+C) \sin (c+d x)+15 (4 (3 A+3 B+4 C) (c+d x)+8 (A+B+C) \sin (2 (c+d x))+(A+B) \sin (4 (c+d x)))+96 A \sin ^5(c+d x)\right)}{480 d}","-\frac{a (4 A+5 (B+C)) \sin ^3(c+d x)}{15 d}+\frac{a (4 A+5 (B+C)) \sin (c+d x)}{5 d}+\frac{a (3 (A+B)+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a (A+B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 (A+B)+4 C)+\frac{a A \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(a*(480*(A + B + C)*Sin[c + d*x] - 160*(2*A + B + C)*Sin[c + d*x]^3 + 96*A*Sin[c + d*x]^5 + 15*(4*(3*A + 3*B + 4*C)*(c + d*x) + 8*(A + B + C)*Sin[2*(c + d*x)] + (A + B)*Sin[4*(c + d*x)])))/(480*d)","A",1
417,1,359,222,3.6006575,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(15 (14 A+12 B+11 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) \cos ^5(c+d x) (15 \sin (c) (14 A+12 B+11 C)+32 (10 A+9 B+8 C) \sin (d x))-\sec (c) \cos ^4(c+d x) (16 \sin (c) (10 A+9 B+8 C)+15 (14 A+12 B+11 C) \sin (d x))-2 \sec (c) \cos ^3(c+d x) (5 \sin (c) (6 A+12 B+11 C)+8 (10 A+9 B+8 C) \sin (d x))-2 \sec (c) \cos ^2(c+d x) (5 (6 A+12 B+11 C) \sin (d x)+24 (B+2 C) \sin (c))-8 \sec (c) \cos (c+d x) (6 (B+2 C) \sin (d x)+5 C \sin (c))-40 C \sec (c) \sin (d x)\right)}{480 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^2 (10 A+9 B+8 C) \tan ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B+8 C) \tan (c+d x)}{5 d}+\frac{a^2 (14 A+12 B+11 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^2 (10 A+12 B+9 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{a^2 (14 A+12 B+11 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(3 B+C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{15 d}+\frac{C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^2}{6 d}",1,"-1/480*(a^2*(1 + Cos[c + d*x])^2*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^4*Sec[c + d*x]^6*(15*(14*A + 12*B + 11*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 40*C*Sec[c]*Sin[d*x] - 8*Cos[c + d*x]*Sec[c]*(5*C*Sin[c] + 6*(B + 2*C)*Sin[d*x]) - 2*Cos[c + d*x]^3*Sec[c]*(5*(6*A + 12*B + 11*C)*Sin[c] + 8*(10*A + 9*B + 8*C)*Sin[d*x]) - Cos[c + d*x]^5*Sec[c]*(15*(14*A + 12*B + 11*C)*Sin[c] + 32*(10*A + 9*B + 8*C)*Sin[d*x]) - 2*Cos[c + d*x]^2*Sec[c]*(24*(B + 2*C)*Sin[c] + 5*(6*A + 12*B + 11*C)*Sin[d*x]) - Cos[c + d*x]^4*Sec[c]*(16*(10*A + 9*B + 8*C)*Sin[c] + 15*(14*A + 12*B + 11*C)*Sin[d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
418,1,417,190,3.313218,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(240 (8 A+7 B+6 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (3 A+2 B+C) \sin (2 c+d x)+80 (16 A+14 B+15 C) \sin (d x)+240 A \sin (c+2 d x)+240 A \sin (3 c+2 d x)+880 A \sin (2 c+3 d x)-120 A \sin (4 c+3 d x)+120 A \sin (3 c+4 d x)+120 A \sin (5 c+4 d x)+200 A \sin (4 c+5 d x)+330 B \sin (c+2 d x)+330 B \sin (3 c+2 d x)+800 B \sin (2 c+3 d x)+105 B \sin (3 c+4 d x)+105 B \sin (5 c+4 d x)+160 B \sin (4 c+5 d x)+420 C \sin (c+2 d x)+420 C \sin (3 c+2 d x)+720 C \sin (2 c+3 d x)+90 C \sin (3 c+4 d x)+90 C \sin (5 c+4 d x)+144 C \sin (4 c+5 d x))\right)}{3840 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^2 (8 A+7 B+6 C) \tan (c+d x)}{6 d}+\frac{a^2 (8 A+7 B+6 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (8 A+7 B+6 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(20 A-5 B+6 C) \tan (c+d x) (a \sec (c+d x)+a)^2}{60 d}+\frac{(5 B+2 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 a d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^2}{5 d}",1,"-1/3840*(a^2*(1 + Cos[c + d*x])^2*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^4*Sec[c + d*x]^5*(240*(8*A + 7*B + 6*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(16*A + 14*B + 15*C)*Sin[d*x] - 240*(3*A + 2*B + C)*Sin[2*c + d*x] + 240*A*Sin[c + 2*d*x] + 330*B*Sin[c + 2*d*x] + 420*C*Sin[c + 2*d*x] + 240*A*Sin[3*c + 2*d*x] + 330*B*Sin[3*c + 2*d*x] + 420*C*Sin[3*c + 2*d*x] + 880*A*Sin[2*c + 3*d*x] + 800*B*Sin[2*c + 3*d*x] + 720*C*Sin[2*c + 3*d*x] - 120*A*Sin[4*c + 3*d*x] + 120*A*Sin[3*c + 4*d*x] + 105*B*Sin[3*c + 4*d*x] + 90*C*Sin[3*c + 4*d*x] + 120*A*Sin[5*c + 4*d*x] + 105*B*Sin[5*c + 4*d*x] + 90*C*Sin[5*c + 4*d*x] + 200*A*Sin[4*c + 5*d*x] + 160*B*Sin[4*c + 5*d*x] + 144*C*Sin[4*c + 5*d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
419,1,386,147,2.7790303,"\int \sec (c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(24 (12 A+8 B+7 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-24 \sin (c) (6 A+5 B+4 C)+3 (4 A+8 B+15 C) \sin (d x)+12 A \sin (2 c+d x)+144 A \sin (c+2 d x)-48 A \sin (3 c+2 d x)+12 A \sin (2 c+3 d x)+12 A \sin (4 c+3 d x)+48 A \sin (3 c+4 d x)+24 B \sin (2 c+d x)+136 B \sin (c+2 d x)-24 B \sin (3 c+2 d x)+24 B \sin (2 c+3 d x)+24 B \sin (4 c+3 d x)+40 B \sin (3 c+4 d x)+45 C \sin (2 c+d x)+128 C \sin (c+2 d x)+21 C \sin (2 c+3 d x)+21 C \sin (4 c+3 d x)+32 C \sin (3 c+4 d x))\right)}{384 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^2 (12 A+8 B+7 C) \tan (c+d x)}{6 d}+\frac{a^2 (12 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (12 A+8 B+7 C) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(4 B-C) \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d}",1,"-1/384*(a^2*(1 + Cos[c + d*x])^2*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^4*Sec[c + d*x]^4*(24*(12*A + 8*B + 7*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-24*(6*A + 5*B + 4*C)*Sin[c] + 3*(4*A + 8*B + 15*C)*Sin[d*x] + 12*A*Sin[2*c + d*x] + 24*B*Sin[2*c + d*x] + 45*C*Sin[2*c + d*x] + 144*A*Sin[c + 2*d*x] + 136*B*Sin[c + 2*d*x] + 128*C*Sin[c + 2*d*x] - 48*A*Sin[3*c + 2*d*x] - 24*B*Sin[3*c + 2*d*x] + 12*A*Sin[2*c + 3*d*x] + 24*B*Sin[2*c + 3*d*x] + 21*C*Sin[2*c + 3*d*x] + 12*A*Sin[4*c + 3*d*x] + 24*B*Sin[4*c + 3*d*x] + 21*C*Sin[4*c + 3*d*x] + 48*A*Sin[3*c + 4*d*x] + 40*B*Sin[3*c + 4*d*x] + 32*C*Sin[3*c + 4*d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
420,1,542,120,5.4441717,"\int (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \cos ^4(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 (3 A+6 B+5 C) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (3 A+6 B+5 C) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{6 (4 A+3 B+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{6 (4 A+3 B+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+12 A x+\frac{(3 B+7 C) \cos \left(\frac{c}{2}\right)-(3 B+5 C) \sin \left(\frac{c}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{(3 B+5 C) \sin \left(\frac{c}{2}\right)+(3 B+7 C) \cos \left(\frac{c}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 C \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 C \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)}{24 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^2 (2 A+3 B+2 C) \tan (c+d x)}{2 d}+\frac{a^2 (4 A+3 B+2 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac{(3 B+2 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{6 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^2*Cos[c + d*x]^4*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(12*A*x - (6*(4*A + 3*B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (6*(4*A + 3*B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (2*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((3*B + 7*C)*Cos[c/2] - (3*B + 5*C)*Sin[c/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(3*A + 6*B + 5*C)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - ((3*B + 7*C)*Cos[c/2] + (3*B + 5*C)*Sin[c/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(3*A + 6*B + 5*C)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(24*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
421,1,365,121,3.5723189,"\int \cos (c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \cos ^4(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{2 (2 A+4 B+3 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (2 A+4 B+3 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+4 x (2 A+B)+\frac{4 A \sin (c) \cos (d x)}{d}+\frac{4 A \cos (c) \sin (d x)}{d}+\frac{4 (B+2 C) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (B+2 C) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{8 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{a^2 (2 A-2 B-3 C) \tan (c+d x)}{2 d}+\frac{a^2 (2 A+4 B+3 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 x (2 A+B)-\frac{(2 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^2}{d}",1,"(a^2*Cos[c + d*x]^4*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(4*(2*A + B)*x - (2*(2*A + 4*B + 3*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(2*A + 4*B + 3*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*A*Cos[d*x]*Sin[c])/d + (4*A*Cos[c]*Sin[d*x])/d + C/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(B + 2*C)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - C/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(B + 2*C)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(8*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
422,1,329,128,3.7056417,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \cos ^2(c+d x) (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 (2 A+B) \sin (c) \cos (d x)}{d}+\frac{4 (2 A+B) \cos (c) \sin (d x)}{d}+2 x (3 A+4 B+2 C)+\frac{A \sin (2 c) \cos (2 d x)}{d}+\frac{A \cos (2 c) \sin (2 d x)}{d}-\frac{4 (B+2 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 (B+2 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 C \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 C \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{8 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^2 (3 A+2 B-2 C) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (3 A+4 B+2 C)-\frac{(A-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}+\frac{a^2 (B+2 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}",1,"(a^2*Cos[c + d*x]^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(3*A + 4*B + 2*C)*x - (4*(B + 2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (4*(B + 2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(2*A + B)*Cos[d*x]*Sin[c])/d + (A*Cos[2*d*x]*Sin[2*c])/d + (4*(2*A + B)*Cos[c]*Sin[d*x])/d + (A*Cos[2*c]*Sin[2*d*x])/d + (4*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(8*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
423,1,121,134,0.2972774,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \left(3 (7 A+8 B+4 C) \sin (c+d x)+3 (2 A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+12 A d x+18 B d x-12 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+24 C d x\right)}{12 d}","\frac{a^2 (2 A+3 B+2 C) \sin (c+d x)}{2 d}+\frac{(2 A+3 B) \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{6 d}+\frac{1}{2} a^2 x (2 A+3 B+4 C)+\frac{a^2 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^2*(12*A*d*x + 18*B*d*x + 24*C*d*x - 12*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(7*A + 8*B + 4*C)*Sin[c + d*x] + 3*(2*A + B)*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)","A",1
424,1,95,149,0.3395897,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 (24 (6 A+7 B+8 C) \sin (c+d x)+24 (2 A+2 B+C) \sin (2 (c+d x))+16 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+84 A d x+8 B \sin (3 (c+d x))+96 B d x+144 C d x)}{96 d}","\frac{a^2 (7 A+8 B+12 C) \sin (c+d x)}{6 d}+\frac{a^2 (7 A+8 B+12 C) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} a^2 x (7 A+8 B+12 C)+\frac{(A+2 B) \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{6 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^2}{4 d}",1,"(a^2*(84*A*d*x + 96*B*d*x + 144*C*d*x + 24*(6*A + 7*B + 8*C)*Sin[c + d*x] + 24*(2*A + 2*B + C)*Sin[2*(c + d*x)] + 16*A*Sin[3*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
425,1,132,187,0.6072665,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 (60 (11 A+12 B+14 C) \sin (c+d x)+240 (A+B+C) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+240 A c+360 A d x+80 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+420 B c+420 B d x+40 C \sin (3 (c+d x))+480 C d x)}{480 d}","\frac{a^2 (18 A+20 B+25 C) \sin (c+d x)}{15 d}+\frac{a^2 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac{a^2 (6 A+7 B+8 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{20 d}+\frac{1}{8} a^2 x (6 A+7 B+8 C)+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}",1,"(a^2*(240*A*c + 420*B*c + 360*A*d*x + 420*B*d*x + 480*C*d*x + 60*(11*A + 12*B + 14*C)*Sin[c + d*x] + 240*(A + B + C)*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 80*B*Sin[3*(c + d*x)] + 40*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)","A",1
426,1,170,213,0.9522724,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 (120 (10 A+11 B+12 C) \sin (c+d x)+15 (31 A+32 (B+C)) \sin (2 (c+d x))+200 A \sin (3 (c+d x))+75 A \sin (4 (c+d x))+24 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+240 A c+660 A d x+180 B \sin (3 (c+d x))+60 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+720 B c+720 B d x+160 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+840 C d x)}{960 d}","-\frac{a^2 (8 A+9 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{a^2 (8 A+9 B+10 C) \sin (c+d x)}{5 d}+\frac{a^2 (9 A+12 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^2 (11 A+12 B+14 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(A+3 B) \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{15 d}+\frac{1}{16} a^2 x (11 A+12 B+14 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}",1,"(a^2*(240*A*c + 720*B*c + 660*A*d*x + 720*B*d*x + 840*C*d*x + 120*(10*A + 11*B + 12*C)*Sin[c + d*x] + 15*(31*A + 32*(B + C))*Sin[2*(c + d*x)] + 200*A*Sin[3*(c + d*x)] + 180*B*Sin[3*(c + d*x)] + 160*C*Sin[3*(c + d*x)] + 75*A*Sin[4*(c + d*x)] + 60*B*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 24*A*Sin[5*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
427,1,402,274,6.2543899,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^7(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(105 (26 A+23 B+21 C) \cos ^7(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) \cos ^6(c+d x) (105 \sin (c) (26 A+23 B+21 C)+32 (133 A+119 B+108 C) \sin (d x))-\sec (c) \cos ^5(c+d x) (16 \sin (c) (133 A+119 B+108 C)+105 (26 A+23 B+21 C) \sin (d x))-2 \sec (c) \cos ^4(c+d x) (35 \sin (c) (18 A+23 B+21 C)+8 (133 A+119 B+108 C) \sin (d x))-2 \sec (c) \cos ^3(c+d x) (24 \sin (c) (7 A+21 B+27 C)+35 (18 A+23 B+21 C) \sin (d x))-8 \sec (c) \cos ^2(c+d x) (6 (7 A+21 B+27 C) \sin (d x)+35 (B+3 C) \sin (c))-40 \sec (c) \cos (c+d x) (7 (B+3 C) \sin (d x)+6 C \sin (c))-240 C \sec (c) \sin (d x)\right)}{6720 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^3 (133 A+119 B+108 C) \tan ^3(c+d x)}{105 d}+\frac{a^3 (133 A+119 B+108 C) \tan (c+d x)}{35 d}+\frac{a^3 (26 A+23 B+21 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (154 A+147 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac{(3 A+4 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{a^3 (26 A+23 B+21 C) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{(7 B+3 C) \tan (c+d x) \sec ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{42 a d}+\frac{C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^3}{7 d}",1,"-1/6720*(a^3*(1 + Cos[c + d*x])^3*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^6*Sec[c + d*x]^7*(105*(26*A + 23*B + 21*C)*Cos[c + d*x]^7*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 240*C*Sec[c]*Sin[d*x] - 40*Cos[c + d*x]*Sec[c]*(6*C*Sin[c] + 7*(B + 3*C)*Sin[d*x]) - 2*Cos[c + d*x]^3*Sec[c]*(24*(7*A + 21*B + 27*C)*Sin[c] + 35*(18*A + 23*B + 21*C)*Sin[d*x]) - Cos[c + d*x]^5*Sec[c]*(16*(133*A + 119*B + 108*C)*Sin[c] + 105*(26*A + 23*B + 21*C)*Sin[d*x]) - 8*Cos[c + d*x]^2*Sec[c]*(35*(B + 3*C)*Sin[c] + 6*(7*A + 21*B + 27*C)*Sin[d*x]) - 2*Cos[c + d*x]^4*Sec[c]*(35*(18*A + 23*B + 21*C)*Sin[c] + 8*(133*A + 119*B + 108*C)*Sin[d*x]) - Cos[c + d*x]^6*Sec[c]*(105*(26*A + 23*B + 21*C)*Sin[c] + 32*(133*A + 119*B + 108*C)*Sin[d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
428,1,359,216,4.300243,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(15 (30 A+26 B+23 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) \cos ^5(c+d x) (15 \sin (c) (30 A+26 B+23 C)+16 (45 A+38 B+34 C) \sin (d x))-\sec (c) \cos ^4(c+d x) (16 \sin (c) (15 A+19 B+17 C)+15 (30 A+26 B+23 C) \sin (d x))-2 \sec (c) \cos ^3(c+d x) (5 \sin (c) (6 A+18 B+23 C)+8 (15 A+19 B+17 C) \sin (d x))-2 \sec (c) \cos ^2(c+d x) (5 (6 A+18 B+23 C) \sin (d x)+24 (B+3 C) \sin (c))-8 \sec (c) \cos (c+d x) (6 (B+3 C) \sin (d x)+5 C \sin (c))-40 C \sec (c) \sin (d x)\right)}{960 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^3 (30 A+26 B+23 C) \tan ^3(c+d x)}{120 d}+\frac{a^3 (30 A+26 B+23 C) \tan (c+d x)}{10 d}+\frac{a^3 (30 A+26 B+23 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{3 a^3 (30 A+26 B+23 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{(30 A-6 B+7 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{120 d}+\frac{(2 B+C) \tan (c+d x) (a \sec (c+d x)+a)^4}{10 a d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^3}{6 d}",1,"-1/960*(a^3*(1 + Cos[c + d*x])^3*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^6*Sec[c + d*x]^6*(15*(30*A + 26*B + 23*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 40*C*Sec[c]*Sin[d*x] - 8*Cos[c + d*x]*Sec[c]*(5*C*Sin[c] + 6*(B + 3*C)*Sin[d*x]) - 2*Cos[c + d*x]^3*Sec[c]*(5*(6*A + 18*B + 23*C)*Sin[c] + 8*(15*A + 19*B + 17*C)*Sin[d*x]) - 2*Cos[c + d*x]^2*Sec[c]*(24*(B + 3*C)*Sin[c] + 5*(6*A + 18*B + 23*C)*Sin[d*x]) - Cos[c + d*x]^4*Sec[c]*(16*(15*A + 19*B + 17*C)*Sin[c] + 15*(30*A + 26*B + 23*C)*Sin[d*x]) - Cos[c + d*x]^5*Sec[c]*(15*(30*A + 26*B + 23*C)*Sin[c] + 16*(45*A + 38*B + 34*C)*Sin[d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
429,1,431,175,4.0118936,"\int \sec (c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(240 (20 A+15 B+13 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (7 A+5 B+3 C) \sin (2 c+d x)+80 (34 A+30 B+29 C) \sin (d x)+360 A \sin (c+2 d x)+360 A \sin (3 c+2 d x)+1840 A \sin (2 c+3 d x)-360 A \sin (4 c+3 d x)+180 A \sin (3 c+4 d x)+180 A \sin (5 c+4 d x)+440 A \sin (4 c+5 d x)+570 B \sin (c+2 d x)+570 B \sin (3 c+2 d x)+1680 B \sin (2 c+3 d x)-120 B \sin (4 c+3 d x)+225 B \sin (3 c+4 d x)+225 B \sin (5 c+4 d x)+360 B \sin (4 c+5 d x)+750 C \sin (c+2 d x)+750 C \sin (3 c+2 d x)+1520 C \sin (2 c+3 d x)+195 C \sin (3 c+4 d x)+195 C \sin (5 c+4 d x)+304 C \sin (4 c+5 d x))\right)}{7680 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^3 (20 A+15 B+13 C) \tan ^3(c+d x)}{60 d}+\frac{a^3 (20 A+15 B+13 C) \tan (c+d x)}{5 d}+\frac{a^3 (20 A+15 B+13 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (20 A+15 B+13 C) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{(5 B-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d}",1,"-1/7680*(a^3*(1 + Cos[c + d*x])^3*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^6*Sec[c + d*x]^5*(240*(20*A + 15*B + 13*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(34*A + 30*B + 29*C)*Sin[d*x] - 240*(7*A + 5*B + 3*C)*Sin[2*c + d*x] + 360*A*Sin[c + 2*d*x] + 570*B*Sin[c + 2*d*x] + 750*C*Sin[c + 2*d*x] + 360*A*Sin[3*c + 2*d*x] + 570*B*Sin[3*c + 2*d*x] + 750*C*Sin[3*c + 2*d*x] + 1840*A*Sin[2*c + 3*d*x] + 1680*B*Sin[2*c + 3*d*x] + 1520*C*Sin[2*c + 3*d*x] - 360*A*Sin[4*c + 3*d*x] - 120*B*Sin[4*c + 3*d*x] + 180*A*Sin[3*c + 4*d*x] + 225*B*Sin[3*c + 4*d*x] + 195*C*Sin[3*c + 4*d*x] + 180*A*Sin[5*c + 4*d*x] + 225*B*Sin[5*c + 4*d*x] + 195*C*Sin[5*c + 4*d*x] + 440*A*Sin[4*c + 5*d*x] + 360*B*Sin[4*c + 5*d*x] + 304*C*Sin[4*c + 5*d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
430,1,464,162,3.204707,"\int (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(\sec (c) (12 A \sin (2 c+d x)+216 A \sin (c+2 d x)-72 A \sin (3 c+2 d x)+12 A \sin (2 c+3 d x)+12 A \sin (4 c+3 d x)+72 A \sin (3 c+4 d x)+72 A d x \cos (c)+48 A d x \cos (c+2 d x)+48 A d x \cos (3 c+2 d x)+12 A d x \cos (3 c+4 d x)+12 A d x \cos (5 c+4 d x)-216 A \sin (c)+12 A \sin (d x)+36 B \sin (2 c+d x)+280 B \sin (c+2 d x)-72 B \sin (3 c+2 d x)+36 B \sin (2 c+3 d x)+36 B \sin (4 c+3 d x)+88 B \sin (3 c+4 d x)-264 B \sin (c)+36 B \sin (d x)+69 C \sin (2 c+d x)+264 C \sin (c+2 d x)-24 C \sin (3 c+2 d x)+45 C \sin (2 c+3 d x)+45 C \sin (4 c+3 d x)+72 C \sin (3 c+4 d x)-216 C \sin (c)+69 C \sin (d x))-24 (28 A+20 B+15 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{768 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{5 a^3 (4 A+4 B+3 C) \tan (c+d x)}{8 d}+\frac{a^3 (28 A+20 B+15 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(12 A+20 B+15 C) \tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{24 d}+a^3 A x+\frac{(4 B+3 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 a d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(1 + Cos[c + d*x])^3*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^6*Sec[c + d*x]^4*(-24*(28*A + 20*B + 15*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(72*A*d*x*Cos[c] + 48*A*d*x*Cos[c + 2*d*x] + 48*A*d*x*Cos[3*c + 2*d*x] + 12*A*d*x*Cos[3*c + 4*d*x] + 12*A*d*x*Cos[5*c + 4*d*x] - 216*A*Sin[c] - 264*B*Sin[c] - 216*C*Sin[c] + 12*A*Sin[d*x] + 36*B*Sin[d*x] + 69*C*Sin[d*x] + 12*A*Sin[2*c + d*x] + 36*B*Sin[2*c + d*x] + 69*C*Sin[2*c + d*x] + 216*A*Sin[c + 2*d*x] + 280*B*Sin[c + 2*d*x] + 264*C*Sin[c + 2*d*x] - 72*A*Sin[3*c + 2*d*x] - 72*B*Sin[3*c + 2*d*x] - 24*C*Sin[3*c + 2*d*x] + 12*A*Sin[2*c + 3*d*x] + 36*B*Sin[2*c + 3*d*x] + 45*C*Sin[2*c + 3*d*x] + 12*A*Sin[4*c + 3*d*x] + 36*B*Sin[4*c + 3*d*x] + 45*C*Sin[4*c + 3*d*x] + 72*A*Sin[3*c + 4*d*x] + 88*B*Sin[3*c + 4*d*x] + 72*C*Sin[3*c + 4*d*x])))/(768*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
431,1,1503,156,6.4679051,"\int \cos (c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(-6 A-7 B-5 C) \cos ^5(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(6 A+7 B+5 C) \cos ^5(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(3 A+B) x \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{A \cos (d x) \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{A \cos (c) \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+9 B \sin \left(\frac{d x}{2}\right)+11 C \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)+9 B \sin \left(\frac{d x}{2}\right)+11 C \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(3 B \cos \left(\frac{c}{2}\right)+10 C \cos \left(\frac{c}{2}\right)-3 B \sin \left(\frac{c}{2}\right)-8 C \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-3 B \cos \left(\frac{c}{2}\right)-10 C \cos \left(\frac{c}{2}\right)-3 B \sin \left(\frac{c}{2}\right)-8 C \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{C \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{C \cos ^5(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(6 A-3 B-5 C) \tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{6 d}+a^3 x (3 A+B)+\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}-\frac{(3 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 a d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}",1,"((3*A + B)*x*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((-6*A - 7*B - 5*C)*Cos[c + d*x]^5*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((6*A + 7*B + 5*C)*Cos[c + d*x]^5*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Cos[d*x]*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Cos[c]*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[d*x])/(4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(24*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(3*B*Cos[c/2] + 10*C*Cos[c/2] - 3*B*Sin[c/2] - 8*C*Sin[c/2]))/(48*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(12*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (C*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(24*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-3*B*Cos[c/2] - 10*C*Cos[c/2] - 3*B*Sin[c/2] - 8*C*Sin[c/2]))/(48*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(12*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
432,1,406,171,5.8327375,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 \cos ^5(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{2 (2 A+6 B+7 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (2 A+6 B+7 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 (3 A+B) \sin (c) \cos (d x)}{d}+\frac{4 (3 A+B) \cos (c) \sin (d x)}{d}+2 x (7 A+6 B+2 C)+\frac{A \sin (2 c) \cos (2 d x)}{d}+\frac{A \cos (2 c) \sin (2 d x)}{d}+\frac{4 (B+3 C) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (B+3 C) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{16 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^3 (2 A+6 B+7 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(A-2 B-4 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{2 d}+\frac{1}{2} a^3 x (7 A+6 B+2 C)+\frac{5 a^3 (A-C) \sin (c+d x)}{2 d}-\frac{(A-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{2 d}",1,"(a^3*Cos[c + d*x]^5*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(7*A + 6*B + 2*C)*x - (2*(2*A + 6*B + 7*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(2*A + 6*B + 7*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(3*A + B)*Cos[d*x]*Sin[c])/d + (A*Cos[2*d*x]*Sin[2*c])/d + (4*(3*A + B)*Cos[c]*Sin[d*x])/d + (A*Cos[2*c]*Sin[2*d*x])/d + C/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(B + 3*C)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - C/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(B + 3*C)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(16*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
433,1,379,169,2.2725993,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 \cos ^2(c+d x) (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{3 \sin (c) (15 A+4 (3 B+C)) \cos (d x)}{d}+\frac{3 \cos (c) (15 A+4 (3 B+C)) \sin (d x)}{d}+\frac{3 (3 A+B) \sin (2 c) \cos (2 d x)}{d}+\frac{3 (3 A+B) \cos (2 c) \sin (2 d x)}{d}+6 x (5 A+7 B+6 C)+\frac{A \sin (3 c) \cos (3 d x)}{d}+\frac{A \cos (3 c) \sin (3 d x)}{d}-\frac{12 (B+3 C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{12 (B+3 C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{12 C \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{12 C \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{48 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(5 A+3 B-6 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{6 d}+\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{1}{2} a^3 x (5 A+7 B+6 C)+\frac{a^3 (B+3 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(A+B) \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"(a^3*Cos[c + d*x]^2*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*(5*A + 7*B + 6*C)*x - (12*(B + 3*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (12*(B + 3*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (3*(15*A + 4*(3*B + C))*Cos[d*x]*Sin[c])/d + (3*(3*A + B)*Cos[2*d*x]*Sin[2*c])/d + (A*Cos[3*d*x]*Sin[3*c])/d + (3*(15*A + 4*(3*B + C))*Cos[c]*Sin[d*x])/d + (3*(3*A + B)*Cos[2*c]*Sin[2*d*x])/d + (A*Cos[3*c]*Sin[3*d*x])/d + (12*C*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (12*C*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(48*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
434,1,147,183,0.442611,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 \left(24 (13 A+15 B+12 C) \sin (c+d x)+24 (4 A+3 B+C) \sin (2 (c+d x))+24 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+180 A d x+8 B \sin (3 (c+d x))+240 B d x-96 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+336 C d x\right)}{96 d}","\frac{5 a^3 (3 A+4 (B+C)) \sin (c+d x)}{8 d}+\frac{(15 A+20 B+12 C) \sin (c+d x) \cos (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{24 d}+\frac{1}{8} a^3 x (15 A+20 B+28 C)+\frac{a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(3 A+4 B) \sin (c+d x) \cos ^2(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 a d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(180*A*d*x + 240*B*d*x + 336*C*d*x - 96*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(13*A + 15*B + 12*C)*Sin[c + d*x] + 24*(4*A + 3*B + C)*Sin[2*(c + d*x)] + 24*A*Sin[3*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
435,1,130,179,0.5000934,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 (60 (23 A+26 B+30 C) \sin (c+d x)+120 (4 A+4 B+3 C) \sin (2 (c+d x))+170 A \sin (3 (c+d x))+45 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+780 A d x+120 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+900 B d x+40 C \sin (3 (c+d x))+1200 C d x)}{480 d}","-\frac{a^3 (13 A+15 B+20 C) \sin ^3(c+d x)}{60 d}+\frac{a^3 (13 A+15 B+20 C) \sin (c+d x)}{5 d}+\frac{3 a^3 (13 A+15 B+20 C) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} a^3 x (13 A+15 B+20 C)+\frac{(3 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^3}{5 d}",1,"(a^3*(780*A*d*x + 900*B*d*x + 1200*C*d*x + 60*(23*A + 26*B + 30*C)*Sin[c + d*x] + 120*(4*A + 4*B + 3*C)*Sin[2*(c + d*x)] + 170*A*Sin[3*(c + d*x)] + 120*B*Sin[3*(c + d*x)] + 40*C*Sin[3*(c + d*x)] + 45*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)","A",1
436,1,170,235,0.8889053,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 (120 (21 A+23 B+26 C) \sin (c+d x)+15 (63 A+64 (B+C)) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+135 A \sin (4 (c+d x))+36 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+900 A c+1380 A d x+340 B \sin (3 (c+d x))+90 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+1560 B c+1560 B d x+240 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+1800 C d x)}{960 d}","\frac{a^3 (34 A+38 B+45 C) \sin (c+d x)}{15 d}+\frac{a^3 (73 A+86 B+90 C) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac{a^3 (23 A+26 B+30 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(31 A+42 B+30 C) \sin (c+d x) \cos ^3(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{120 d}+\frac{1}{16} a^3 x (23 A+26 B+30 C)+\frac{(A+2 B) \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{10 a d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}",1,"(a^3*(900*A*c + 1560*B*c + 1380*A*d*x + 1560*B*d*x + 1800*C*d*x + 120*(21*A + 23*B + 26*C)*Sin[c + d*x] + 15*(63*A + 64*(B + C))*Sin[2*(c + d*x)] + 380*A*Sin[3*(c + d*x)] + 340*B*Sin[3*(c + d*x)] + 240*C*Sin[3*(c + d*x)] + 135*A*Sin[4*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 36*A*Sin[5*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
437,1,204,265,1.3087156,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 (105 (155 A+168 B+184 C) \sin (c+d x)+105 (61 A+63 B+64 C) \sin (2 (c+d x))+2835 A \sin (3 (c+d x))+1155 A \sin (4 (c+d x))+399 A \sin (5 (c+d x))+105 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+3360 A c+8820 A d x+2660 B \sin (3 (c+d x))+945 B \sin (4 (c+d x))+252 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+9660 B c+9660 B d x+2380 C \sin (3 (c+d x))+630 C \sin (4 (c+d x))+84 C \sin (5 (c+d x))+10920 C d x)}{6720 d}","-\frac{a^3 (108 A+119 B+133 C) \sin ^3(c+d x)}{105 d}+\frac{a^3 (108 A+119 B+133 C) \sin (c+d x)}{35 d}+\frac{a^3 (129 A+147 B+154 C) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{a^3 (21 A+23 B+26 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(3 A+4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{1}{16} a^3 x (21 A+23 B+26 C)+\frac{(3 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{42 a d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}",1,"(a^3*(3360*A*c + 9660*B*c + 8820*A*d*x + 9660*B*d*x + 10920*C*d*x + 105*(155*A + 168*B + 184*C)*Sin[c + d*x] + 105*(61*A + 63*B + 64*C)*Sin[2*(c + d*x)] + 2835*A*Sin[3*(c + d*x)] + 2660*B*Sin[3*(c + d*x)] + 2380*C*Sin[3*(c + d*x)] + 1155*A*Sin[4*(c + d*x)] + 945*B*Sin[4*(c + d*x)] + 630*C*Sin[4*(c + d*x)] + 399*A*Sin[5*(c + d*x)] + 252*B*Sin[5*(c + d*x)] + 84*C*Sin[5*(c + d*x)] + 105*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 15*A*Sin[7*(c + d*x)]))/(6720*d)","A",1
438,1,1087,252,6.4688015,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(-56 A-49 B-44 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(56 A+49 B+44 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{128 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{C \sec (c) \sec (c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{56 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\sec (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (6 C \sin (c)+7 B \sin (d x)+28 C \sin (d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{336 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\cos (c+d x) \sec (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (35 B \sin (c)+140 C \sin (c)+42 A \sin (d x)+168 B \sin (d x)+288 C \sin (d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{1680 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\cos ^2(c+d x) \sec (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (168 A \sin (c)+672 B \sin (c)+1152 C \sin (c)+840 A \sin (d x)+1435 B \sin (d x)+1540 C \sin (d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{6720 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\cos ^3(c+d x) \sec (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (840 A \sin (c)+1435 B \sin (c)+1540 C \sin (c)+1904 A \sin (d x)+2016 B \sin (d x)+1816 C \sin (d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{6720 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\cos ^4(c+d x) \sec (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (3808 A \sin (c)+4032 B \sin (c)+3632 C \sin (c)+5880 A \sin (d x)+5145 B \sin (d x)+4620 C \sin (d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{13440 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\cos ^5(c+d x) \sec (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (5880 A \sin (c)+5145 B \sin (c)+4620 C \sin (c)+9296 A \sin (d x)+8064 B \sin (d x)+7264 C \sin (d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{13440 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}","\frac{2 a^4 (56 A+49 B+44 C) \tan ^3(c+d x)}{105 d}+\frac{4 a^4 (56 A+49 B+44 C) \tan (c+d x)}{35 d}+\frac{a^4 (56 A+49 B+44 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (56 A+49 B+44 C) \tan (c+d x) \sec ^3(c+d x)}{280 d}+\frac{27 a^4 (56 A+49 B+44 C) \tan (c+d x) \sec (c+d x)}{560 d}+\frac{(42 A-7 B+8 C) \tan (c+d x) (a \sec (c+d x)+a)^4}{210 d}+\frac{(7 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^5}{42 a d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^4}{7 d}",1,"((-56*A - 49*B - 44*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(128*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((56*A + 49*B + 44*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(128*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (C*Sec[c]*Sec[c/2 + (d*x)/2]^8*Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[d*x])/(56*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Sec[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*C*Sin[c] + 7*B*Sin[d*x] + 28*C*Sin[d*x]))/(336*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]*Sec[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(35*B*Sin[c] + 140*C*Sin[c] + 42*A*Sin[d*x] + 168*B*Sin[d*x] + 288*C*Sin[d*x]))/(1680*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sec[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(168*A*Sin[c] + 672*B*Sin[c] + 1152*C*Sin[c] + 840*A*Sin[d*x] + 1435*B*Sin[d*x] + 1540*C*Sin[d*x]))/(6720*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^3*Sec[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(840*A*Sin[c] + 1435*B*Sin[c] + 1540*C*Sin[c] + 1904*A*Sin[d*x] + 2016*B*Sin[d*x] + 1816*C*Sin[d*x]))/(6720*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^4*Sec[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(3808*A*Sin[c] + 4032*B*Sin[c] + 3632*C*Sin[c] + 5880*A*Sin[d*x] + 5145*B*Sin[d*x] + 4620*C*Sin[d*x]))/(13440*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^5*Sec[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(5880*A*Sin[c] + 5145*B*Sin[c] + 4620*C*Sin[c] + 9296*A*Sin[d*x] + 8064*B*Sin[d*x] + 7264*C*Sin[d*x]))/(13440*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","B",1
439,1,359,209,6.3323417,"\int \sec (c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(105 (10 A+8 B+7 C) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) \cos ^5(c+d x) (15 \sin (c) (54 A+56 B+49 C)+16 (100 A+83 B+72 C) \sin (d x))-\sec (c) \cos ^4(c+d x) (32 \sin (c) (10 A+17 B+18 C)+15 (54 A+56 B+49 C) \sin (d x))-2 \sec (c) \cos ^3(c+d x) (5 \sin (c) (6 A+24 B+41 C)+16 (10 A+17 B+18 C) \sin (d x))-2 \sec (c) \cos ^2(c+d x) (5 (6 A+24 B+41 C) \sin (d x)+24 (B+4 C) \sin (c))-8 \sec (c) \cos (c+d x) (6 (B+4 C) \sin (d x)+5 C \sin (c))-40 C \sec (c) \sin (d x)\right)}{1920 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 a^4 (10 A+8 B+7 C) \tan ^3(c+d x)}{15 d}+\frac{4 a^4 (10 A+8 B+7 C) \tan (c+d x)}{5 d}+\frac{7 a^4 (10 A+8 B+7 C) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (10 A+8 B+7 C) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{27 a^4 (10 A+8 B+7 C) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{(6 B-C) \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}",1,"-1/1920*(a^4*(1 + Cos[c + d*x])^4*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^6*(105*(10*A + 8*B + 7*C)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 40*C*Sec[c]*Sin[d*x] - 8*Cos[c + d*x]*Sec[c]*(5*C*Sin[c] + 6*(B + 4*C)*Sin[d*x]) - 2*Cos[c + d*x]^3*Sec[c]*(5*(6*A + 24*B + 41*C)*Sin[c] + 16*(10*A + 17*B + 18*C)*Sin[d*x]) - 2*Cos[c + d*x]^2*Sec[c]*(24*(B + 4*C)*Sin[c] + 5*(6*A + 24*B + 41*C)*Sin[d*x]) - Cos[c + d*x]^4*Sec[c]*(32*(10*A + 17*B + 18*C)*Sin[c] + 15*(54*A + 56*B + 49*C)*Sin[d*x]) - Cos[c + d*x]^5*Sec[c]*(15*(54*A + 56*B + 49*C)*Sin[c] + 16*(100*A + 83*B + 72*C)*Sin[d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
440,1,538,195,5.5219896,"\int (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(\sec (c) (-3120 A \sin (2 c+d x)+480 A \sin (c+2 d x)+480 A \sin (3 c+2 d x)+3280 A \sin (2 c+3 d x)-720 A \sin (4 c+3 d x)+240 A \sin (3 c+4 d x)+240 A \sin (5 c+4 d x)+800 A \sin (4 c+5 d x)+600 A d x \cos (2 c+d x)+300 A d x \cos (2 c+3 d x)+300 A d x \cos (4 c+3 d x)+60 A d x \cos (4 c+5 d x)+60 A d x \cos (6 c+5 d x)+4880 A \sin (d x)+600 A d x \cos (d x)-2880 B \sin (2 c+d x)+930 B \sin (c+2 d x)+930 B \sin (3 c+2 d x)+3520 B \sin (2 c+3 d x)-480 B \sin (4 c+3 d x)+405 B \sin (3 c+4 d x)+405 B \sin (5 c+4 d x)+800 B \sin (4 c+5 d x)+5120 B \sin (d x)-1920 C \sin (2 c+d x)+1320 C \sin (c+2 d x)+1320 C \sin (3 c+2 d x)+3200 C \sin (2 c+3 d x)-120 C \sin (4 c+3 d x)+420 C \sin (3 c+4 d x)+420 C \sin (5 c+4 d x)+664 C \sin (4 c+5 d x)+4720 C \sin (d x))-240 (48 A+35 B+28 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{15360 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^4 (40 A+35 B+28 C) \tan (c+d x)}{8 d}+\frac{a^4 (48 A+35 B+28 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(32 A+35 B+28 C) \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{24 d}+a^4 A x+\frac{(20 A+35 B+28 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{60 d}+\frac{a (5 B+4 C) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{C \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}",1,"(a^4*(1 + Cos[c + d*x])^4*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^5*(-240*(48*A + 35*B + 28*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(600*A*d*x*Cos[d*x] + 600*A*d*x*Cos[2*c + d*x] + 300*A*d*x*Cos[2*c + 3*d*x] + 300*A*d*x*Cos[4*c + 3*d*x] + 60*A*d*x*Cos[4*c + 5*d*x] + 60*A*d*x*Cos[6*c + 5*d*x] + 4880*A*Sin[d*x] + 5120*B*Sin[d*x] + 4720*C*Sin[d*x] - 3120*A*Sin[2*c + d*x] - 2880*B*Sin[2*c + d*x] - 1920*C*Sin[2*c + d*x] + 480*A*Sin[c + 2*d*x] + 930*B*Sin[c + 2*d*x] + 1320*C*Sin[c + 2*d*x] + 480*A*Sin[3*c + 2*d*x] + 930*B*Sin[3*c + 2*d*x] + 1320*C*Sin[3*c + 2*d*x] + 3280*A*Sin[2*c + 3*d*x] + 3520*B*Sin[2*c + 3*d*x] + 3200*C*Sin[2*c + 3*d*x] - 720*A*Sin[4*c + 3*d*x] - 480*B*Sin[4*c + 3*d*x] - 120*C*Sin[4*c + 3*d*x] + 240*A*Sin[3*c + 4*d*x] + 405*B*Sin[3*c + 4*d*x] + 420*C*Sin[3*c + 4*d*x] + 240*A*Sin[5*c + 4*d*x] + 405*B*Sin[5*c + 4*d*x] + 420*C*Sin[5*c + 4*d*x] + 800*A*Sin[4*c + 5*d*x] + 800*B*Sin[4*c + 5*d*x] + 664*C*Sin[4*c + 5*d*x])))/(15360*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
441,1,530,196,4.9799576,"\int \cos (c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(\sec (c) (72 d x (4 A+B) \cos (c)+48 d x (4 A+B) \cos (c+2 d x)+24 A \sin (2 c+d x)+288 A \sin (c+2 d x)-96 A \sin (3 c+2 d x)+30 A \sin (2 c+3 d x)+30 A \sin (4 c+3 d x)+96 A \sin (3 c+4 d x)+6 A \sin (4 c+5 d x)+6 A \sin (6 c+5 d x)+192 A d x \cos (3 c+2 d x)+48 A d x \cos (3 c+4 d x)+48 A d x \cos (5 c+4 d x)-288 A \sin (c)+24 A \sin (d x)+48 B \sin (2 c+d x)+496 B \sin (c+2 d x)-144 B \sin (3 c+2 d x)+48 B \sin (2 c+3 d x)+48 B \sin (4 c+3 d x)+160 B \sin (3 c+4 d x)+48 B d x \cos (3 c+2 d x)+12 B d x \cos (3 c+4 d x)+12 B d x \cos (5 c+4 d x)-480 B \sin (c)+48 B \sin (d x)+105 C \sin (2 c+d x)+544 C \sin (c+2 d x)-96 C \sin (3 c+2 d x)+81 C \sin (2 c+3 d x)+81 C \sin (4 c+3 d x)+160 C \sin (3 c+4 d x)-480 C \sin (c)+105 C \sin (d x))-24 (52 A+48 B+35 C) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1536 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{5 a^4 (4 A+8 B+7 C) \tan (c+d x)}{8 d}+\frac{a^4 (52 A+48 B+35 C) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{(12 A-32 B-35 C) \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{24 d}+a^4 x (4 A+B)-\frac{(12 A-4 B-7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 d}-\frac{a (4 A-C) \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^4}{d}",1,"(a^4*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(-24*(52*A + 48*B + 35*C)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(72*(4*A + B)*d*x*Cos[c] + 48*(4*A + B)*d*x*Cos[c + 2*d*x] + 192*A*d*x*Cos[3*c + 2*d*x] + 48*B*d*x*Cos[3*c + 2*d*x] + 48*A*d*x*Cos[3*c + 4*d*x] + 12*B*d*x*Cos[3*c + 4*d*x] + 48*A*d*x*Cos[5*c + 4*d*x] + 12*B*d*x*Cos[5*c + 4*d*x] - 288*A*Sin[c] - 480*B*Sin[c] - 480*C*Sin[c] + 24*A*Sin[d*x] + 48*B*Sin[d*x] + 105*C*Sin[d*x] + 24*A*Sin[2*c + d*x] + 48*B*Sin[2*c + d*x] + 105*C*Sin[2*c + d*x] + 288*A*Sin[c + 2*d*x] + 496*B*Sin[c + 2*d*x] + 544*C*Sin[c + 2*d*x] - 96*A*Sin[3*c + 2*d*x] - 144*B*Sin[3*c + 2*d*x] - 96*C*Sin[3*c + 2*d*x] + 30*A*Sin[2*c + 3*d*x] + 48*B*Sin[2*c + 3*d*x] + 81*C*Sin[2*c + 3*d*x] + 30*A*Sin[4*c + 3*d*x] + 48*B*Sin[4*c + 3*d*x] + 81*C*Sin[4*c + 3*d*x] + 96*A*Sin[3*c + 4*d*x] + 160*B*Sin[3*c + 4*d*x] + 160*C*Sin[3*c + 4*d*x] + 6*A*Sin[4*c + 5*d*x] + 6*A*Sin[6*c + 5*d*x])))/(1536*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
442,1,524,209,4.6356408,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(\sec (c) (36 d x (13 A+8 B+2 C) \cos (2 c+d x)+36 d x (13 A+8 B+2 C) \cos (d x)-42 A \sin (2 c+d x)+96 A \sin (c+2 d x)+96 A \sin (3 c+2 d x)+57 A \sin (2 c+3 d x)+9 A \sin (4 c+3 d x)+48 A \sin (3 c+4 d x)+48 A \sin (5 c+4 d x)+3 A \sin (4 c+5 d x)+3 A \sin (6 c+5 d x)+156 A d x \cos (2 c+3 d x)+156 A d x \cos (4 c+3 d x)+102 A \sin (d x)-192 B \sin (2 c+d x)+48 B \sin (c+2 d x)+48 B \sin (3 c+2 d x)+192 B \sin (2 c+3 d x)+12 B \sin (3 c+4 d x)+12 B \sin (5 c+4 d x)+96 B d x \cos (2 c+3 d x)+96 B d x \cos (4 c+3 d x)+384 B \sin (d x)-288 C \sin (2 c+d x)+96 C \sin (c+2 d x)+96 C \sin (3 c+2 d x)+320 C \sin (2 c+3 d x)+24 C d x \cos (2 c+3 d x)+24 C d x \cos (4 c+3 d x)+672 C \sin (d x))-96 (8 A+13 B+12 C) \cos ^3(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1536 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{5 a^4 (A-B-2 C) \sin (c+d x)}{2 d}+\frac{a^4 (8 A+13 B+12 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(3 A+18 B+22 C) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (13 A+8 B+2 C)-\frac{(A-B-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 d}-\frac{a (3 A-2 C) \sin (c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^4}{2 d}",1,"(a^4*(1 + Cos[c + d*x])^4*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]^8*Sec[c + d*x]^3*(-96*(8*A + 13*B + 12*C)*Cos[c + d*x]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(36*(13*A + 8*B + 2*C)*d*x*Cos[d*x] + 36*(13*A + 8*B + 2*C)*d*x*Cos[2*c + d*x] + 156*A*d*x*Cos[2*c + 3*d*x] + 96*B*d*x*Cos[2*c + 3*d*x] + 24*C*d*x*Cos[2*c + 3*d*x] + 156*A*d*x*Cos[4*c + 3*d*x] + 96*B*d*x*Cos[4*c + 3*d*x] + 24*C*d*x*Cos[4*c + 3*d*x] + 102*A*Sin[d*x] + 384*B*Sin[d*x] + 672*C*Sin[d*x] - 42*A*Sin[2*c + d*x] - 192*B*Sin[2*c + d*x] - 288*C*Sin[2*c + d*x] + 96*A*Sin[c + 2*d*x] + 48*B*Sin[c + 2*d*x] + 96*C*Sin[c + 2*d*x] + 96*A*Sin[3*c + 2*d*x] + 48*B*Sin[3*c + 2*d*x] + 96*C*Sin[3*c + 2*d*x] + 57*A*Sin[2*c + 3*d*x] + 192*B*Sin[2*c + 3*d*x] + 320*C*Sin[2*c + 3*d*x] + 9*A*Sin[4*c + 3*d*x] + 48*A*Sin[3*c + 4*d*x] + 12*B*Sin[3*c + 4*d*x] + 48*A*Sin[5*c + 4*d*x] + 12*B*Sin[5*c + 4*d*x] + 3*A*Sin[4*c + 5*d*x] + 3*A*Sin[6*c + 5*d*x])))/(1536*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
443,1,1518,217,6.3066644,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(-2 A-8 B-13 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(2 A+8 B+13 C) \cos ^6(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(12 A+13 B+8 C) x \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(27 A+16 B+4 C) \cos (d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(4 A+B) \cos (2 d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (2 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{A \cos (3 d x) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (3 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(27 A+16 B+4 C) \cos (c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(4 A+B) \cos (2 c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (2 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{A \cos (3 c) \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (3 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(B \sin \left(\frac{d x}{2}\right)+4 C \sin \left(\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(B \sin \left(\frac{d x}{2}\right)+4 C \sin \left(\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}-\frac{C \cos ^6(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}","\frac{5 a^4 (2 A+B-C) \sin (c+d x)}{2 d}+\frac{a^4 (2 A+8 B+13 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(8 A-3 B-18 C) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{6 d}+\frac{1}{2} a^4 x (12 A+13 B+8 C)-\frac{(2 A+B-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 d}+\frac{a (4 A+3 B) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{6 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^4}{3 d}",1,"((12*A + 13*B + 8*C)*x*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(16*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((-2*A - 8*B - 13*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(16*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((2*A + 8*B + 13*C)*Cos[c + d*x]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(16*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((27*A + 16*B + 4*C)*Cos[d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((4*A + B)*Cos[2*d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[2*c])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Cos[3*d*x]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[3*c])/(96*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((27*A + 16*B + 4*C)*Cos[c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[d*x])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((4*A + B)*Cos[2*c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[2*d*x])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Cos[3*c]*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[3*d*x])/(96*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(B*Sin[(d*x)/2] + 4*C*Sin[(d*x)/2]))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) - (C*Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^6*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(B*Sin[(d*x)/2] + 4*C*Sin[(d*x)/2]))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
444,1,1436,217,6.2293922,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a^4 \left(\frac{(-B-4 C) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(B+4 C) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(35 A+48 B+52 C) x \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{64 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(28 A+27 B+16 C) \cos (d x) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(7 A+4 B+C) \cos (2 d x) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (2 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(4 A+B) \cos (3 d x) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (3 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{A \cos (4 d x) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (4 c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(28 A+27 B+16 C) \cos (c) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(7 A+4 B+C) \cos (2 c) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (2 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(4 A+B) \cos (3 c) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (3 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{A \cos (4 c) \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (4 d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{256 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{C \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{C \cos ^2(c+d x) (\cos (c+d x)+1)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}\right)","\frac{5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}-\frac{(35 A+32 B-12 C) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{24 d}+\frac{1}{8} a^4 x (35 A+48 B+52 C)+\frac{a^4 (B+4 C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(7 A+8 B+4 C) \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{8 d}+\frac{a (A+B) \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^4}{4 d}",1,"a^4*(((35*A + 48*B + 52*C)*x*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(64*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((-B - 4*C)*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((B + 4*C)*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((28*A + 27*B + 16*C)*Cos[d*x]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((7*A + 4*B + C)*Cos[2*d*x]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[2*c])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((4*A + B)*Cos[3*d*x]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[3*c])/(96*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Cos[4*d*x]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[4*c])/(256*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((28*A + 27*B + 16*C)*Cos[c]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[d*x])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((7*A + 4*B + C)*Cos[2*c]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[2*d*x])/(32*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((4*A + B)*Cos[3*c]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[3*d*x])/(96*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Cos[4*c]*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[4*d*x])/(256*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (C*Cos[c + d*x]^2*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[(d*x)/2])/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])))","B",1
445,1,182,225,0.6301817,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 \left(60 (49 A+56 B+54 C) \sin (c+d x)+120 (8 A+7 B+4 C) \sin (2 (c+d x))+290 A \sin (3 (c+d x))+60 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+1680 A d x+160 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+2100 B d x+40 C \sin (3 (c+d x))-480 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2880 C d x\right)}{480 d}","\frac{a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 d}+\frac{(28 A+35 B+32 C) \sin (c+d x) \cos (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{24 d}+\frac{1}{8} a^4 x (28 A+35 B+48 C)+\frac{a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(28 A+35 B+20 C) \sin (c+d x) \cos ^2(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{60 d}+\frac{a (4 A+5 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}",1,"(a^4*(1680*A*d*x + 2100*B*d*x + 2880*C*d*x - 480*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*(49*A + 56*B + 54*C)*Sin[c + d*x] + 120*(8*A + 7*B + 4*C)*Sin[2*(c + d*x)] + 290*A*Sin[3*(c + d*x)] + 160*B*Sin[3*(c + d*x)] + 40*C*Sin[3*(c + d*x)] + 60*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)","A",1
446,1,163,213,0.5012623,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 (120 (44 A+49 B+56 C) \sin (c+d x)+15 (127 A+128 B+112 C) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+225 A \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+2940 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x+320 C \sin (3 (c+d x))+30 C \sin (4 (c+d x))+4200 C d x)}{960 d}","-\frac{2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac{4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac{a^4 (7 A+8 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{27 a^4 (7 A+8 B+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac{7}{16} a^4 x (7 A+8 B+10 C)+\frac{(2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d}",1,"(a^4*(2940*A*d*x + 3360*B*d*x + 4200*C*d*x + 120*(44*A + 49*B + 56*C)*Sin[c + d*x] + 15*(127*A + 128*B + 112*C)*Sin[2*(c + d*x)] + 720*A*Sin[3*(c + d*x)] + 580*B*Sin[3*(c + d*x)] + 320*C*Sin[3*(c + d*x)] + 225*A*Sin[4*(c + d*x)] + 120*B*Sin[4*(c + d*x)] + 30*C*Sin[4*(c + d*x)] + 48*A*Sin[5*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
447,1,204,278,1.0163631,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 (105 (323 A+352 B+392 C) \sin (c+d x)+105 (124 A+127 B+128 C) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+651 A \sin (5 (c+d x))+140 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+11760 A c+18480 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B c+20580 B d x+4060 C \sin (3 (c+d x))+840 C \sin (4 (c+d x))+84 C \sin (5 (c+d x))+23520 C d x)}{6720 d}","\frac{a^4 (454 A+504 B+581 C) \sin (c+d x)}{105 d}+\frac{a^4 (988 A+1113 B+1232 C) \sin (c+d x) \cos ^2(c+d x)}{840 d}+\frac{a^4 (44 A+49 B+56 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(436 A+511 B+504 C) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{840 d}+\frac{1}{16} a^4 x (44 A+49 B+56 C)+\frac{(16 A+21 B+14 C) \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{70 d}+\frac{a (4 A+7 B) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^4}{7 d}",1,"(a^4*(11760*A*c + 20580*B*c + 18480*A*d*x + 20580*B*d*x + 23520*C*d*x + 105*(323*A + 352*B + 392*C)*Sin[c + d*x] + 105*(124*A + 127*B + 128*C)*Sin[2*(c + d*x)] + 5495*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 4060*C*Sin[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)] + 840*C*Sin[4*(c + d*x)] + 651*A*Sin[5*(c + d*x)] + 336*B*Sin[5*(c + d*x)] + 84*C*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 15*A*Sin[7*(c + d*x)]))/(6720*d)","A",1
448,1,237,303,1.9358409,"\int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 (1680 (300 A+323 B+352 C) \sin (c+d x)+1680 (120 A+124 B+127 C) \sin (2 (c+d x))+91840 A \sin (3 (c+d x))+39480 A \sin (4 (c+d x))+14784 A \sin (5 (c+d x))+4480 A \sin (6 (c+d x))+960 A \sin (7 (c+d x))+105 A \sin (8 (c+d x))+106680 A c+271320 A d x+87920 B \sin (3 (c+d x))+33600 B \sin (4 (c+d x))+10416 B \sin (5 (c+d x))+2240 B \sin (6 (c+d x))+240 B \sin (7 (c+d x))+295680 B c+295680 B d x+80640 C \sin (3 (c+d x))+25200 C \sin (4 (c+d x))+5376 C \sin (5 (c+d x))+560 C \sin (6 (c+d x))+329280 C d x)}{107520 d}","-\frac{a^4 (208 A+227 B+252 C) \sin ^3(c+d x)}{105 d}+\frac{a^4 (208 A+227 B+252 C) \sin (c+d x)}{35 d}+\frac{a^4 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{2240 d}+\frac{a^4 (323 A+352 B+392 C) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{7 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{120 d}+\frac{1}{128} a^4 x (323 A+352 B+392 C)+\frac{(61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{336 d}+\frac{a (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{14 d}+\frac{A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}",1,"(a^4*(106680*A*c + 295680*B*c + 271320*A*d*x + 295680*B*d*x + 329280*C*d*x + 1680*(300*A + 323*B + 352*C)*Sin[c + d*x] + 1680*(120*A + 124*B + 127*C)*Sin[2*(c + d*x)] + 91840*A*Sin[3*(c + d*x)] + 87920*B*Sin[3*(c + d*x)] + 80640*C*Sin[3*(c + d*x)] + 39480*A*Sin[4*(c + d*x)] + 33600*B*Sin[4*(c + d*x)] + 25200*C*Sin[4*(c + d*x)] + 14784*A*Sin[5*(c + d*x)] + 10416*B*Sin[5*(c + d*x)] + 5376*C*Sin[5*(c + d*x)] + 4480*A*Sin[6*(c + d*x)] + 2240*B*Sin[6*(c + d*x)] + 560*C*Sin[6*(c + d*x)] + 960*A*Sin[7*(c + d*x)] + 240*B*Sin[7*(c + d*x)] + 105*A*Sin[8*(c + d*x)]))/(107520*d)","A",1
449,1,1099,183,6.4220364,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-60 A \sin \left(\frac{d x}{2}\right)+108 B \sin \left(\frac{d x}{2}\right)-75 C \sin \left(\frac{d x}{2}\right)-60 A \sin \left(\frac{3 d x}{2}\right)+124 B \sin \left(\frac{3 d x}{2}\right)-91 C \sin \left(\frac{3 d x}{2}\right)+204 A \sin \left(c-\frac{d x}{2}\right)-252 B \sin \left(c-\frac{d x}{2}\right)+219 C \sin \left(c-\frac{d x}{2}\right)-60 A \sin \left(c+\frac{d x}{2}\right)+12 B \sin \left(c+\frac{d x}{2}\right)+21 C \sin \left(c+\frac{d x}{2}\right)+84 A \sin \left(2 c+\frac{d x}{2}\right)-132 B \sin \left(2 c+\frac{d x}{2}\right)+165 C \sin \left(2 c+\frac{d x}{2}\right)+36 A \sin \left(c+\frac{3 d x}{2}\right)+28 B \sin \left(c+\frac{3 d x}{2}\right)+5 C \sin \left(c+\frac{3 d x}{2}\right)+36 A \sin \left(2 c+\frac{3 d x}{2}\right)-36 B \sin \left(2 c+\frac{3 d x}{2}\right)+69 C \sin \left(2 c+\frac{3 d x}{2}\right)+132 A \sin \left(3 c+\frac{3 d x}{2}\right)-132 B \sin \left(3 c+\frac{3 d x}{2}\right)+165 C \sin \left(3 c+\frac{3 d x}{2}\right)-156 A \sin \left(c+\frac{5 d x}{2}\right)+220 B \sin \left(c+\frac{5 d x}{2}\right)-211 C \sin \left(c+\frac{5 d x}{2}\right)-60 A \sin \left(2 c+\frac{5 d x}{2}\right)+124 B \sin \left(2 c+\frac{5 d x}{2}\right)-115 C \sin \left(2 c+\frac{5 d x}{2}\right)-60 A \sin \left(3 c+\frac{5 d x}{2}\right)+60 B \sin \left(3 c+\frac{5 d x}{2}\right)-51 C \sin \left(3 c+\frac{5 d x}{2}\right)+36 A \sin \left(4 c+\frac{5 d x}{2}\right)-36 B \sin \left(4 c+\frac{5 d x}{2}\right)+45 C \sin \left(4 c+\frac{5 d x}{2}\right)-12 A \sin \left(2 c+\frac{7 d x}{2}\right)+28 B \sin \left(2 c+\frac{7 d x}{2}\right)-19 C \sin \left(2 c+\frac{7 d x}{2}\right)+12 A \sin \left(3 c+\frac{7 d x}{2}\right)+4 B \sin \left(3 c+\frac{7 d x}{2}\right)+5 C \sin \left(3 c+\frac{7 d x}{2}\right)+12 A \sin \left(4 c+\frac{7 d x}{2}\right)-12 B \sin \left(4 c+\frac{7 d x}{2}\right)+21 C \sin \left(4 c+\frac{7 d x}{2}\right)+36 A \sin \left(5 c+\frac{7 d x}{2}\right)-36 B \sin \left(5 c+\frac{7 d x}{2}\right)+45 C \sin \left(5 c+\frac{7 d x}{2}\right)-48 A \sin \left(3 c+\frac{9 d x}{2}\right)+64 B \sin \left(3 c+\frac{9 d x}{2}\right)-64 C \sin \left(3 c+\frac{9 d x}{2}\right)-24 A \sin \left(4 c+\frac{9 d x}{2}\right)+40 B \sin \left(4 c+\frac{9 d x}{2}\right)-40 C \sin \left(4 c+\frac{9 d x}{2}\right)-24 A \sin \left(5 c+\frac{9 d x}{2}\right)+24 B \sin \left(5 c+\frac{9 d x}{2}\right)-24 C \sin \left(5 c+\frac{9 d x}{2}\right)\right) \sec ^3(c+d x)}{192 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{3 (4 A-4 B+5 C) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos (c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{3 (4 A-4 B+5 C) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) \cos (c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}","-\frac{(3 A-4 B+4 C) \tan ^3(c+d x)}{3 a d}-\frac{(3 A-4 B+4 C) \tan (c+d x)}{a d}+\frac{3 (4 A-4 B+5 C) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(4 A-4 B+5 C) \tan (c+d x) \sec ^3(c+d x)}{4 a d}+\frac{3 (4 A-4 B+5 C) \tan (c+d x) \sec (c+d x)}{8 a d}",1,"(-3*(4*A - 4*B + 5*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (3*(4*A - 4*B + 5*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-60*A*Sin[(d*x)/2] + 108*B*Sin[(d*x)/2] - 75*C*Sin[(d*x)/2] - 60*A*Sin[(3*d*x)/2] + 124*B*Sin[(3*d*x)/2] - 91*C*Sin[(3*d*x)/2] + 204*A*Sin[c - (d*x)/2] - 252*B*Sin[c - (d*x)/2] + 219*C*Sin[c - (d*x)/2] - 60*A*Sin[c + (d*x)/2] + 12*B*Sin[c + (d*x)/2] + 21*C*Sin[c + (d*x)/2] + 84*A*Sin[2*c + (d*x)/2] - 132*B*Sin[2*c + (d*x)/2] + 165*C*Sin[2*c + (d*x)/2] + 36*A*Sin[c + (3*d*x)/2] + 28*B*Sin[c + (3*d*x)/2] + 5*C*Sin[c + (3*d*x)/2] + 36*A*Sin[2*c + (3*d*x)/2] - 36*B*Sin[2*c + (3*d*x)/2] + 69*C*Sin[2*c + (3*d*x)/2] + 132*A*Sin[3*c + (3*d*x)/2] - 132*B*Sin[3*c + (3*d*x)/2] + 165*C*Sin[3*c + (3*d*x)/2] - 156*A*Sin[c + (5*d*x)/2] + 220*B*Sin[c + (5*d*x)/2] - 211*C*Sin[c + (5*d*x)/2] - 60*A*Sin[2*c + (5*d*x)/2] + 124*B*Sin[2*c + (5*d*x)/2] - 115*C*Sin[2*c + (5*d*x)/2] - 60*A*Sin[3*c + (5*d*x)/2] + 60*B*Sin[3*c + (5*d*x)/2] - 51*C*Sin[3*c + (5*d*x)/2] + 36*A*Sin[4*c + (5*d*x)/2] - 36*B*Sin[4*c + (5*d*x)/2] + 45*C*Sin[4*c + (5*d*x)/2] - 12*A*Sin[2*c + (7*d*x)/2] + 28*B*Sin[2*c + (7*d*x)/2] - 19*C*Sin[2*c + (7*d*x)/2] + 12*A*Sin[3*c + (7*d*x)/2] + 4*B*Sin[3*c + (7*d*x)/2] + 5*C*Sin[3*c + (7*d*x)/2] + 12*A*Sin[4*c + (7*d*x)/2] - 12*B*Sin[4*c + (7*d*x)/2] + 21*C*Sin[4*c + (7*d*x)/2] + 36*A*Sin[5*c + (7*d*x)/2] - 36*B*Sin[5*c + (7*d*x)/2] + 45*C*Sin[5*c + (7*d*x)/2] - 48*A*Sin[3*c + (9*d*x)/2] + 64*B*Sin[3*c + (9*d*x)/2] - 64*C*Sin[3*c + (9*d*x)/2] - 24*A*Sin[4*c + (9*d*x)/2] + 40*B*Sin[4*c + (9*d*x)/2] - 40*C*Sin[4*c + (9*d*x)/2] - 24*A*Sin[5*c + (9*d*x)/2] + 24*B*Sin[5*c + (9*d*x)/2] - 24*C*Sin[5*c + (9*d*x)/2]))/(192*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x]))","B",1
450,1,898,148,6.3190061,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{2 (2 A-3 B+3 C) \cos (c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{2 (2 A-3 B+3 C) \cos (c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-6 A \sin \left(\frac{d x}{2}\right)+6 B \sin \left(\frac{d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)+30 A \sin \left(\frac{3 d x}{2}\right)-27 B \sin \left(\frac{3 d x}{2}\right)+39 C \sin \left(\frac{3 d x}{2}\right)-12 A \sin \left(c-\frac{d x}{2}\right)+12 B \sin \left(c-\frac{d x}{2}\right)-24 C \sin \left(c-\frac{d x}{2}\right)-6 A \sin \left(c+\frac{d x}{2}\right)+6 B \sin \left(c+\frac{d x}{2}\right)-6 C \sin \left(c+\frac{d x}{2}\right)-24 A \sin \left(2 c+\frac{d x}{2}\right)+24 B \sin \left(2 c+\frac{d x}{2}\right)-24 C \sin \left(2 c+\frac{d x}{2}\right)+12 A \sin \left(c+\frac{3 d x}{2}\right)-9 B \sin \left(c+\frac{3 d x}{2}\right)+21 C \sin \left(c+\frac{3 d x}{2}\right)+12 A \sin \left(2 c+\frac{3 d x}{2}\right)-9 B \sin \left(2 c+\frac{3 d x}{2}\right)+9 C \sin \left(2 c+\frac{3 d x}{2}\right)-6 A \sin \left(3 c+\frac{3 d x}{2}\right)+9 B \sin \left(3 c+\frac{3 d x}{2}\right)-9 C \sin \left(3 c+\frac{3 d x}{2}\right)+6 A \sin \left(c+\frac{5 d x}{2}\right)-3 B \sin \left(c+\frac{5 d x}{2}\right)+7 C \sin \left(c+\frac{5 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+C \sin \left(2 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{5 d x}{2}\right)-3 C \sin \left(3 c+\frac{5 d x}{2}\right)-6 A \sin \left(4 c+\frac{5 d x}{2}\right)+9 B \sin \left(4 c+\frac{5 d x}{2}\right)-9 C \sin \left(4 c+\frac{5 d x}{2}\right)+12 A \sin \left(2 c+\frac{7 d x}{2}\right)-12 B \sin \left(2 c+\frac{7 d x}{2}\right)+16 C \sin \left(2 c+\frac{7 d x}{2}\right)+6 A \sin \left(3 c+\frac{7 d x}{2}\right)-6 B \sin \left(3 c+\frac{7 d x}{2}\right)+10 C \sin \left(3 c+\frac{7 d x}{2}\right)+6 A \sin \left(4 c+\frac{7 d x}{2}\right)-6 B \sin \left(4 c+\frac{7 d x}{2}\right)+6 C \sin \left(4 c+\frac{7 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}","\frac{(3 A-3 B+4 C) \tan ^3(c+d x)}{3 a d}+\frac{(3 A-3 B+4 C) \tan (c+d x)}{a d}-\frac{(2 A-3 B+3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}-\frac{(2 A-3 B+3 C) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(2*(2*A - 3*B + 3*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*(2*A - 3*B + 3*C)*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-6*A*Sin[(d*x)/2] + 6*B*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2] + 30*A*Sin[(3*d*x)/2] - 27*B*Sin[(3*d*x)/2] + 39*C*Sin[(3*d*x)/2] - 12*A*Sin[c - (d*x)/2] + 12*B*Sin[c - (d*x)/2] - 24*C*Sin[c - (d*x)/2] - 6*A*Sin[c + (d*x)/2] + 6*B*Sin[c + (d*x)/2] - 6*C*Sin[c + (d*x)/2] - 24*A*Sin[2*c + (d*x)/2] + 24*B*Sin[2*c + (d*x)/2] - 24*C*Sin[2*c + (d*x)/2] + 12*A*Sin[c + (3*d*x)/2] - 9*B*Sin[c + (3*d*x)/2] + 21*C*Sin[c + (3*d*x)/2] + 12*A*Sin[2*c + (3*d*x)/2] - 9*B*Sin[2*c + (3*d*x)/2] + 9*C*Sin[2*c + (3*d*x)/2] - 6*A*Sin[3*c + (3*d*x)/2] + 9*B*Sin[3*c + (3*d*x)/2] - 9*C*Sin[3*c + (3*d*x)/2] + 6*A*Sin[c + (5*d*x)/2] - 3*B*Sin[c + (5*d*x)/2] + 7*C*Sin[c + (5*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] + C*Sin[2*c + (5*d*x)/2] + 3*B*Sin[3*c + (5*d*x)/2] - 3*C*Sin[3*c + (5*d*x)/2] - 6*A*Sin[4*c + (5*d*x)/2] + 9*B*Sin[4*c + (5*d*x)/2] - 9*C*Sin[4*c + (5*d*x)/2] + 12*A*Sin[2*c + (7*d*x)/2] - 12*B*Sin[2*c + (7*d*x)/2] + 16*C*Sin[2*c + (7*d*x)/2] + 6*A*Sin[3*c + (7*d*x)/2] - 6*B*Sin[3*c + (7*d*x)/2] + 10*C*Sin[3*c + (7*d*x)/2] + 6*A*Sin[4*c + (7*d*x)/2] - 6*B*Sin[4*c + (7*d*x)/2] + 6*C*Sin[4*c + (7*d*x)/2]))/(24*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x]))","B",1
451,1,392,119,4.567104,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-4 \sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)-2 (2 A-2 B+3 C) \cos \left(\frac{1}{2} (c+d x)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 (2 A-2 B+3 C) \cos \left(\frac{1}{2} (c+d x)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 (B-C) \sin \left(\frac{d x}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (B-C) \sin \left(\frac{d x}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C \cos \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C \cos \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{a d (\sec (c+d x)+1) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(A-2 B+2 C) \tan (c+d x)}{a d}+\frac{(2 A-2 B+3 C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(2 A-2 B+3 C) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(Cos[(c + d*x)/2]*Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-2*(2*A - 2*B + 3*C)*Cos[(c + d*x)/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A - 2*B + 3*C)*Cos[(c + d*x)/2]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 4*(A - B + C)*Sec[c/2]*Sin[(d*x)/2] + (C*Cos[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*(B - C)*Cos[(c + d*x)/2]*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (C*Cos[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(B - C)*Cos[(c + d*x)/2]*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x]))","B",1
452,1,255,63,1.5128186,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{C \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-(B-C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{a d (\sec (c+d x)+1) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A-B+C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{(B-C) \tanh ^{-1}(\sin (c+d x))}{a d}+\frac{C \tan (c+d x)}{a d}",1,"(4*Cos[(c + d*x)/2]*Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((A - B + C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(-((B - C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])) + (C*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x]))","B",1
453,1,163,52,0.5078809,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]),x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) \left(A d x-C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(A-B+C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{A x}{a}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(4*Cos[(c + d*x)/2]*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(Cos[(c + d*x)/2]*(A*d*x - C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - (A - B + C)*Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
454,1,77,62,0.4195421,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) (d x (B-A)+A \sin (c+d x))\right)}{a d (\cos (c+d x)+1)}","\frac{(2 A-B+C) \sin (c+d x)}{a d}-\frac{(A-B+C) \sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x (A-B)}{a}",1,"(2*Cos[(c + d*x)/2]*((A - B + C)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((-A + B)*d*x + A*Sin[c + d*x])))/(a*d*(1 + Cos[c + d*x]))","A",1
455,1,213,108,0.580331,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(4 d x (3 A-2 B+2 C) \cos \left(c+\frac{d x}{2}\right)+4 d x (3 A-2 B+2 C) \cos \left(\frac{d x}{2}\right)-4 A \sin \left(c+\frac{d x}{2}\right)-3 A \sin \left(c+\frac{3 d x}{2}\right)-3 A \sin \left(2 c+\frac{3 d x}{2}\right)+A \sin \left(2 c+\frac{5 d x}{2}\right)+A \sin \left(3 c+\frac{5 d x}{2}\right)-20 A \sin \left(\frac{d x}{2}\right)+4 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+4 B \sin \left(2 c+\frac{3 d x}{2}\right)+20 B \sin \left(\frac{d x}{2}\right)-16 C \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","-\frac{(2 A-2 B+C) \sin (c+d x)}{a d}+\frac{(3 A-2 B+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A-B+C) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{x (3 A-2 B+2 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(4*(3*A - 2*B + 2*C)*d*x*Cos[(d*x)/2] + 4*(3*A - 2*B + 2*C)*d*x*Cos[c + (d*x)/2] - 20*A*Sin[(d*x)/2] + 20*B*Sin[(d*x)/2] - 16*C*Sin[(d*x)/2] - 4*A*Sin[c + (d*x)/2] + 4*B*Sin[c + (d*x)/2] - 3*A*Sin[c + (3*d*x)/2] + 4*B*Sin[c + (3*d*x)/2] - 3*A*Sin[2*c + (3*d*x)/2] + 4*B*Sin[2*c + (3*d*x)/2] + A*Sin[2*c + (5*d*x)/2] + A*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","A",1
456,1,307,139,1.1920007,"\int \frac{\cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-12 d x (3 A-3 B+2 C) \cos \left(c+\frac{d x}{2}\right)-12 d x (3 A-3 B+2 C) \cos \left(\frac{d x}{2}\right)+21 A \sin \left(c+\frac{d x}{2}\right)+18 A \sin \left(c+\frac{3 d x}{2}\right)+18 A \sin \left(2 c+\frac{3 d x}{2}\right)-2 A \sin \left(2 c+\frac{5 d x}{2}\right)-2 A \sin \left(3 c+\frac{5 d x}{2}\right)+A \sin \left(3 c+\frac{7 d x}{2}\right)+A \sin \left(4 c+\frac{7 d x}{2}\right)+69 A \sin \left(\frac{d x}{2}\right)-12 B \sin \left(c+\frac{d x}{2}\right)-9 B \sin \left(c+\frac{3 d x}{2}\right)-9 B \sin \left(2 c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{5 d x}{2}\right)-60 B \sin \left(\frac{d x}{2}\right)+12 C \sin \left(c+\frac{d x}{2}\right)+12 C \sin \left(c+\frac{3 d x}{2}\right)+12 C \sin \left(2 c+\frac{3 d x}{2}\right)+60 C \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","-\frac{(4 A-3 B+3 C) \sin ^3(c+d x)}{3 a d}+\frac{(4 A-3 B+3 C) \sin (c+d x)}{a d}-\frac{(3 A-3 B+2 C) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{x (3 A-3 B+2 C)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-12*(3*A - 3*B + 2*C)*d*x*Cos[(d*x)/2] - 12*(3*A - 3*B + 2*C)*d*x*Cos[c + (d*x)/2] + 69*A*Sin[(d*x)/2] - 60*B*Sin[(d*x)/2] + 60*C*Sin[(d*x)/2] + 21*A*Sin[c + (d*x)/2] - 12*B*Sin[c + (d*x)/2] + 12*C*Sin[c + (d*x)/2] + 18*A*Sin[c + (3*d*x)/2] - 9*B*Sin[c + (3*d*x)/2] + 12*C*Sin[c + (3*d*x)/2] + 18*A*Sin[2*c + (3*d*x)/2] - 9*B*Sin[2*c + (3*d*x)/2] + 12*C*Sin[2*c + (3*d*x)/2] - 2*A*Sin[2*c + (5*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] - 2*A*Sin[3*c + (5*d*x)/2] + 3*B*Sin[3*c + (5*d*x)/2] + A*Sin[3*c + (7*d*x)/2] + A*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","B",1
457,1,393,174,0.9873869,"\int \frac{\cos ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(72 d x (5 A-4 B+4 C) \cos \left(c+\frac{d x}{2}\right)+72 d x (5 A-4 B+4 C) \cos \left(\frac{d x}{2}\right)-168 A \sin \left(c+\frac{d x}{2}\right)-120 A \sin \left(c+\frac{3 d x}{2}\right)-120 A \sin \left(2 c+\frac{3 d x}{2}\right)+40 A \sin \left(2 c+\frac{5 d x}{2}\right)+40 A \sin \left(3 c+\frac{5 d x}{2}\right)-5 A \sin \left(3 c+\frac{7 d x}{2}\right)-5 A \sin \left(4 c+\frac{7 d x}{2}\right)+3 A \sin \left(4 c+\frac{9 d x}{2}\right)+3 A \sin \left(5 c+\frac{9 d x}{2}\right)-552 A \sin \left(\frac{d x}{2}\right)+168 B \sin \left(c+\frac{d x}{2}\right)+144 B \sin \left(c+\frac{3 d x}{2}\right)+144 B \sin \left(2 c+\frac{3 d x}{2}\right)-16 B \sin \left(2 c+\frac{5 d x}{2}\right)-16 B \sin \left(3 c+\frac{5 d x}{2}\right)+8 B \sin \left(3 c+\frac{7 d x}{2}\right)+8 B \sin \left(4 c+\frac{7 d x}{2}\right)+552 B \sin \left(\frac{d x}{2}\right)-96 C \sin \left(c+\frac{d x}{2}\right)-72 C \sin \left(c+\frac{3 d x}{2}\right)-72 C \sin \left(2 c+\frac{3 d x}{2}\right)+24 C \sin \left(2 c+\frac{5 d x}{2}\right)+24 C \sin \left(3 c+\frac{5 d x}{2}\right)-480 C \sin \left(\frac{d x}{2}\right)\right)}{192 a d (\cos (c+d x)+1)}","\frac{(4 A-4 B+3 C) \sin ^3(c+d x)}{3 a d}-\frac{(4 A-4 B+3 C) \sin (c+d x)}{a d}+\frac{(5 A-4 B+4 C) \sin (c+d x) \cos ^3(c+d x)}{4 a d}+\frac{3 (5 A-4 B+4 C) \sin (c+d x) \cos (c+d x)}{8 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 x (5 A-4 B+4 C)}{8 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(72*(5*A - 4*B + 4*C)*d*x*Cos[(d*x)/2] + 72*(5*A - 4*B + 4*C)*d*x*Cos[c + (d*x)/2] - 552*A*Sin[(d*x)/2] + 552*B*Sin[(d*x)/2] - 480*C*Sin[(d*x)/2] - 168*A*Sin[c + (d*x)/2] + 168*B*Sin[c + (d*x)/2] - 96*C*Sin[c + (d*x)/2] - 120*A*Sin[c + (3*d*x)/2] + 144*B*Sin[c + (3*d*x)/2] - 72*C*Sin[c + (3*d*x)/2] - 120*A*Sin[2*c + (3*d*x)/2] + 144*B*Sin[2*c + (3*d*x)/2] - 72*C*Sin[2*c + (3*d*x)/2] + 40*A*Sin[2*c + (5*d*x)/2] - 16*B*Sin[2*c + (5*d*x)/2] + 24*C*Sin[2*c + (5*d*x)/2] + 40*A*Sin[3*c + (5*d*x)/2] - 16*B*Sin[3*c + (5*d*x)/2] + 24*C*Sin[3*c + (5*d*x)/2] - 5*A*Sin[3*c + (7*d*x)/2] + 8*B*Sin[3*c + (7*d*x)/2] - 5*A*Sin[4*c + (7*d*x)/2] + 8*B*Sin[4*c + (7*d*x)/2] + 3*A*Sin[4*c + (9*d*x)/2] + 3*A*Sin[5*c + (9*d*x)/2]))/(192*a*d*(1 + Cos[c + d*x]))","B",1
458,1,1069,194,6.4525445,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{4 (4 A-7 B+10 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{4 (4 A-7 B+10 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^3(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-48 A \sin \left(\frac{d x}{2}\right)+45 B \sin \left(\frac{d x}{2}\right)-6 C \sin \left(\frac{d x}{2}\right)+132 A \sin \left(\frac{3 d x}{2}\right)-201 B \sin \left(\frac{3 d x}{2}\right)+310 C \sin \left(\frac{3 d x}{2}\right)-120 A \sin \left(c-\frac{d x}{2}\right)+195 B \sin \left(c-\frac{d x}{2}\right)-306 C \sin \left(c-\frac{d x}{2}\right)+48 A \sin \left(c+\frac{d x}{2}\right)-51 B \sin \left(c+\frac{d x}{2}\right)+42 C \sin \left(c+\frac{d x}{2}\right)-120 A \sin \left(2 c+\frac{d x}{2}\right)+189 B \sin \left(2 c+\frac{d x}{2}\right)-270 C \sin \left(2 c+\frac{d x}{2}\right)-8 A \sin \left(c+\frac{3 d x}{2}\right)-B \sin \left(c+\frac{3 d x}{2}\right)+50 C \sin \left(c+\frac{3 d x}{2}\right)+72 A \sin \left(2 c+\frac{3 d x}{2}\right)-81 B \sin \left(2 c+\frac{3 d x}{2}\right)+90 C \sin \left(2 c+\frac{3 d x}{2}\right)-68 A \sin \left(3 c+\frac{3 d x}{2}\right)+119 B \sin \left(3 c+\frac{3 d x}{2}\right)-170 C \sin \left(3 c+\frac{3 d x}{2}\right)+84 A \sin \left(c+\frac{5 d x}{2}\right)-129 B \sin \left(c+\frac{5 d x}{2}\right)+198 C \sin \left(c+\frac{5 d x}{2}\right)-9 B \sin \left(2 c+\frac{5 d x}{2}\right)+42 C \sin \left(2 c+\frac{5 d x}{2}\right)+48 A \sin \left(3 c+\frac{5 d x}{2}\right)-57 B \sin \left(3 c+\frac{5 d x}{2}\right)+66 C \sin \left(3 c+\frac{5 d x}{2}\right)-36 A \sin \left(4 c+\frac{5 d x}{2}\right)+63 B \sin \left(4 c+\frac{5 d x}{2}\right)-90 C \sin \left(4 c+\frac{5 d x}{2}\right)+48 A \sin \left(2 c+\frac{7 d x}{2}\right)-75 B \sin \left(2 c+\frac{7 d x}{2}\right)+114 C \sin \left(2 c+\frac{7 d x}{2}\right)+6 A \sin \left(3 c+\frac{7 d x}{2}\right)-15 B \sin \left(3 c+\frac{7 d x}{2}\right)+36 C \sin \left(3 c+\frac{7 d x}{2}\right)+30 A \sin \left(4 c+\frac{7 d x}{2}\right)-39 B \sin \left(4 c+\frac{7 d x}{2}\right)+48 C \sin \left(4 c+\frac{7 d x}{2}\right)-12 A \sin \left(5 c+\frac{7 d x}{2}\right)+21 B \sin \left(5 c+\frac{7 d x}{2}\right)-30 C \sin \left(5 c+\frac{7 d x}{2}\right)+20 A \sin \left(3 c+\frac{9 d x}{2}\right)-32 B \sin \left(3 c+\frac{9 d x}{2}\right)+48 C \sin \left(3 c+\frac{9 d x}{2}\right)+6 A \sin \left(4 c+\frac{9 d x}{2}\right)-12 B \sin \left(4 c+\frac{9 d x}{2}\right)+22 C \sin \left(4 c+\frac{9 d x}{2}\right)+14 A \sin \left(5 c+\frac{9 d x}{2}\right)-20 B \sin \left(5 c+\frac{9 d x}{2}\right)+26 C \sin \left(5 c+\frac{9 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}","\frac{(5 A-8 B+12 C) \tan ^3(c+d x)}{3 a^2 d}+\frac{(5 A-8 B+12 C) \tan (c+d x)}{a^2 d}-\frac{(4 A-7 B+10 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(4 A-7 B+10 C) \tan (c+d x) \sec ^3(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{(4 A-7 B+10 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(4*(4*A - 7*B + 10*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (4*(4*A - 7*B + 10*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-48*A*Sin[(d*x)/2] + 45*B*Sin[(d*x)/2] - 6*C*Sin[(d*x)/2] + 132*A*Sin[(3*d*x)/2] - 201*B*Sin[(3*d*x)/2] + 310*C*Sin[(3*d*x)/2] - 120*A*Sin[c - (d*x)/2] + 195*B*Sin[c - (d*x)/2] - 306*C*Sin[c - (d*x)/2] + 48*A*Sin[c + (d*x)/2] - 51*B*Sin[c + (d*x)/2] + 42*C*Sin[c + (d*x)/2] - 120*A*Sin[2*c + (d*x)/2] + 189*B*Sin[2*c + (d*x)/2] - 270*C*Sin[2*c + (d*x)/2] - 8*A*Sin[c + (3*d*x)/2] - B*Sin[c + (3*d*x)/2] + 50*C*Sin[c + (3*d*x)/2] + 72*A*Sin[2*c + (3*d*x)/2] - 81*B*Sin[2*c + (3*d*x)/2] + 90*C*Sin[2*c + (3*d*x)/2] - 68*A*Sin[3*c + (3*d*x)/2] + 119*B*Sin[3*c + (3*d*x)/2] - 170*C*Sin[3*c + (3*d*x)/2] + 84*A*Sin[c + (5*d*x)/2] - 129*B*Sin[c + (5*d*x)/2] + 198*C*Sin[c + (5*d*x)/2] - 9*B*Sin[2*c + (5*d*x)/2] + 42*C*Sin[2*c + (5*d*x)/2] + 48*A*Sin[3*c + (5*d*x)/2] - 57*B*Sin[3*c + (5*d*x)/2] + 66*C*Sin[3*c + (5*d*x)/2] - 36*A*Sin[4*c + (5*d*x)/2] + 63*B*Sin[4*c + (5*d*x)/2] - 90*C*Sin[4*c + (5*d*x)/2] + 48*A*Sin[2*c + (7*d*x)/2] - 75*B*Sin[2*c + (7*d*x)/2] + 114*C*Sin[2*c + (7*d*x)/2] + 6*A*Sin[3*c + (7*d*x)/2] - 15*B*Sin[3*c + (7*d*x)/2] + 36*C*Sin[3*c + (7*d*x)/2] + 30*A*Sin[4*c + (7*d*x)/2] - 39*B*Sin[4*c + (7*d*x)/2] + 48*C*Sin[4*c + (7*d*x)/2] - 12*A*Sin[5*c + (7*d*x)/2] + 21*B*Sin[5*c + (7*d*x)/2] - 30*C*Sin[5*c + (7*d*x)/2] + 20*A*Sin[3*c + (9*d*x)/2] - 32*B*Sin[3*c + (9*d*x)/2] + 48*C*Sin[3*c + (9*d*x)/2] + 6*A*Sin[4*c + (9*d*x)/2] - 12*B*Sin[4*c + (9*d*x)/2] + 22*C*Sin[4*c + (9*d*x)/2] + 14*A*Sin[5*c + (9*d*x)/2] - 20*B*Sin[5*c + (9*d*x)/2] + 26*C*Sin[5*c + (9*d*x)/2]))/(48*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","B",1
459,1,901,169,6.3314847,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\frac{4 (2 A-4 B+7 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{4 (2 A-4 B+7 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(20 A \sin \left(\frac{d x}{2}\right)-14 B \sin \left(\frac{d x}{2}\right)+14 C \sin \left(\frac{d x}{2}\right)-22 A \sin \left(\frac{3 d x}{2}\right)+64 B \sin \left(\frac{3 d x}{2}\right)-97 C \sin \left(\frac{3 d x}{2}\right)+36 A \sin \left(c-\frac{d x}{2}\right)-84 B \sin \left(c-\frac{d x}{2}\right)+126 C \sin \left(c-\frac{d x}{2}\right)-36 A \sin \left(c+\frac{d x}{2}\right)+42 B \sin \left(c+\frac{d x}{2}\right)-42 C \sin \left(c+\frac{d x}{2}\right)+20 A \sin \left(2 c+\frac{d x}{2}\right)-56 B \sin \left(2 c+\frac{d x}{2}\right)+98 C \sin \left(2 c+\frac{d x}{2}\right)+18 A \sin \left(c+\frac{3 d x}{2}\right)-6 B \sin \left(c+\frac{3 d x}{2}\right)+3 C \sin \left(c+\frac{3 d x}{2}\right)-22 A \sin \left(2 c+\frac{3 d x}{2}\right)+34 B \sin \left(2 c+\frac{3 d x}{2}\right)-37 C \sin \left(2 c+\frac{3 d x}{2}\right)+18 A \sin \left(3 c+\frac{3 d x}{2}\right)-36 B \sin \left(3 c+\frac{3 d x}{2}\right)+63 C \sin \left(3 c+\frac{3 d x}{2}\right)-18 A \sin \left(c+\frac{5 d x}{2}\right)+48 B \sin \left(c+\frac{5 d x}{2}\right)-75 C \sin \left(c+\frac{5 d x}{2}\right)+6 A \sin \left(2 c+\frac{5 d x}{2}\right)+6 B \sin \left(2 c+\frac{5 d x}{2}\right)-15 C \sin \left(2 c+\frac{5 d x}{2}\right)-18 A \sin \left(3 c+\frac{5 d x}{2}\right)+30 B \sin \left(3 c+\frac{5 d x}{2}\right)-39 C \sin \left(3 c+\frac{5 d x}{2}\right)+6 A \sin \left(4 c+\frac{5 d x}{2}\right)-12 B \sin \left(4 c+\frac{5 d x}{2}\right)+21 C \sin \left(4 c+\frac{5 d x}{2}\right)-8 A \sin \left(2 c+\frac{7 d x}{2}\right)+20 B \sin \left(2 c+\frac{7 d x}{2}\right)-32 C \sin \left(2 c+\frac{7 d x}{2}\right)+6 B \sin \left(3 c+\frac{7 d x}{2}\right)-12 C \sin \left(3 c+\frac{7 d x}{2}\right)-8 A \sin \left(4 c+\frac{7 d x}{2}\right)+14 B \sin \left(4 c+\frac{7 d x}{2}\right)-20 C \sin \left(4 c+\frac{7 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}","-\frac{2 (2 A-5 B+8 C) \tan (c+d x)}{3 a^2 d}+\frac{(2 A-4 B+7 C) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}-\frac{(2 A-5 B+8 C) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(2 A-4 B+7 C) \tan (c+d x) \sec (c+d x)}{2 a^2 d}-\frac{(A-B+C) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-4*(2*A - 4*B + 7*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (4*(2*A - 4*B + 7*C)*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(20*A*Sin[(d*x)/2] - 14*B*Sin[(d*x)/2] + 14*C*Sin[(d*x)/2] - 22*A*Sin[(3*d*x)/2] + 64*B*Sin[(3*d*x)/2] - 97*C*Sin[(3*d*x)/2] + 36*A*Sin[c - (d*x)/2] - 84*B*Sin[c - (d*x)/2] + 126*C*Sin[c - (d*x)/2] - 36*A*Sin[c + (d*x)/2] + 42*B*Sin[c + (d*x)/2] - 42*C*Sin[c + (d*x)/2] + 20*A*Sin[2*c + (d*x)/2] - 56*B*Sin[2*c + (d*x)/2] + 98*C*Sin[2*c + (d*x)/2] + 18*A*Sin[c + (3*d*x)/2] - 6*B*Sin[c + (3*d*x)/2] + 3*C*Sin[c + (3*d*x)/2] - 22*A*Sin[2*c + (3*d*x)/2] + 34*B*Sin[2*c + (3*d*x)/2] - 37*C*Sin[2*c + (3*d*x)/2] + 18*A*Sin[3*c + (3*d*x)/2] - 36*B*Sin[3*c + (3*d*x)/2] + 63*C*Sin[3*c + (3*d*x)/2] - 18*A*Sin[c + (5*d*x)/2] + 48*B*Sin[c + (5*d*x)/2] - 75*C*Sin[c + (5*d*x)/2] + 6*A*Sin[2*c + (5*d*x)/2] + 6*B*Sin[2*c + (5*d*x)/2] - 15*C*Sin[2*c + (5*d*x)/2] - 18*A*Sin[3*c + (5*d*x)/2] + 30*B*Sin[3*c + (5*d*x)/2] - 39*C*Sin[3*c + (5*d*x)/2] + 6*A*Sin[4*c + (5*d*x)/2] - 12*B*Sin[4*c + (5*d*x)/2] + 21*C*Sin[4*c + (5*d*x)/2] - 8*A*Sin[2*c + (7*d*x)/2] + 20*B*Sin[2*c + (7*d*x)/2] - 32*C*Sin[2*c + (7*d*x)/2] + 6*B*Sin[3*c + (7*d*x)/2] - 12*C*Sin[3*c + (7*d*x)/2] - 8*A*Sin[4*c + (7*d*x)/2] + 14*B*Sin[4*c + (7*d*x)/2] - 20*C*Sin[4*c + (7*d*x)/2]))/(24*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","B",1
460,1,312,112,2.1088811,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{c}{2}\right) (A-B+C) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)+2 \sec \left(\frac{c}{2}\right) (A-4 B+7 C) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{6 C \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-6 (B-2 C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A-B+4 C) \tan (c+d x)}{3 a^2 d}+\frac{(B-2 C) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(B-2 C) \tan (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(4*Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((A - B + C)*Sec[c/2]*Sin[(d*x)/2] + 2*(A - 4*B + 7*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]^3*(-6*(B - 2*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (6*C*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) + (A - B + C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)","B",1
461,1,219,81,0.8268174,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(\tan \left(\frac{c}{2}\right) (A-B+C) \cos \left(\frac{1}{2} (c+d x)\right)+\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)-2 \sec \left(\frac{c}{2}\right) (2 A+B-4 C) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 C \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A+2 B-5 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^2 d}+\frac{(A-B+C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-4*Cos[(c + d*x)/2]*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(6*C*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (A - B + C)*Sec[c/2]*Sin[(d*x)/2] - 2*(2*A + B - 4*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + (A - B + C)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
462,1,175,74,0.5126742,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(12 A \sin \left(c+\frac{d x}{2}\right)-10 A \sin \left(c+\frac{3 d x}{2}\right)+9 A d x \cos \left(c+\frac{d x}{2}\right)+3 A d x \cos \left(c+\frac{3 d x}{2}\right)+3 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 A \sin \left(\frac{d x}{2}\right)+9 A d x \cos \left(\frac{d x}{2}\right)-6 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+6 B \sin \left(\frac{d x}{2}\right)+2 C \sin \left(c+\frac{3 d x}{2}\right)+6 C \sin \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","-\frac{(4 A-B-2 C) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{A x}{a^2}-\frac{(A-B+C) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*A*d*x*Cos[(d*x)/2] + 9*A*d*x*Cos[c + (d*x)/2] + 3*A*d*x*Cos[c + (3*d*x)/2] + 3*A*d*x*Cos[2*c + (3*d*x)/2] - 18*A*Sin[(d*x)/2] + 6*B*Sin[(d*x)/2] + 6*C*Sin[(d*x)/2] + 12*A*Sin[c + (d*x)/2] - 6*B*Sin[c + (d*x)/2] - 10*A*Sin[c + (3*d*x)/2] + 4*B*Sin[c + (3*d*x)/2] + 2*C*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
463,1,279,100,0.8067927,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-18 d x (2 A-B) \cos \left(c+\frac{d x}{2}\right)-18 d x (2 A-B) \cos \left(\frac{d x}{2}\right)-30 A \sin \left(c+\frac{d x}{2}\right)+41 A \sin \left(c+\frac{3 d x}{2}\right)+9 A \sin \left(2 c+\frac{3 d x}{2}\right)+3 A \sin \left(2 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{5 d x}{2}\right)-12 A d x \cos \left(c+\frac{3 d x}{2}\right)-12 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+66 A \sin \left(\frac{d x}{2}\right)+24 B \sin \left(c+\frac{d x}{2}\right)-20 B \sin \left(c+\frac{3 d x}{2}\right)+6 B d x \cos \left(c+\frac{3 d x}{2}\right)+6 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-36 B \sin \left(\frac{d x}{2}\right)-12 C \sin \left(c+\frac{d x}{2}\right)+8 C \sin \left(c+\frac{3 d x}{2}\right)+12 C \sin \left(\frac{d x}{2}\right)\right)}{12 a^2 d (\cos (c+d x)+1)^2}","\frac{(10 A-4 B+C) \sin (c+d x)}{3 a^2 d}-\frac{(2 A-B) \sin (c+d x)}{a^2 d (\sec (c+d x)+1)}-\frac{x (2 A-B)}{a^2}-\frac{(A-B+C) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-18*(2*A - B)*d*x*Cos[(d*x)/2] - 18*(2*A - B)*d*x*Cos[c + (d*x)/2] - 12*A*d*x*Cos[c + (3*d*x)/2] + 6*B*d*x*Cos[c + (3*d*x)/2] - 12*A*d*x*Cos[2*c + (3*d*x)/2] + 6*B*d*x*Cos[2*c + (3*d*x)/2] + 66*A*Sin[(d*x)/2] - 36*B*Sin[(d*x)/2] + 12*C*Sin[(d*x)/2] - 30*A*Sin[c + (d*x)/2] + 24*B*Sin[c + (d*x)/2] - 12*C*Sin[c + (d*x)/2] + 41*A*Sin[c + (3*d*x)/2] - 20*B*Sin[c + (3*d*x)/2] + 8*C*Sin[c + (3*d*x)/2] + 9*A*Sin[2*c + (3*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] + 3*A*Sin[3*c + (5*d*x)/2]))/(12*a^2*d*(1 + Cos[c + d*x])^2)","B",1
464,1,377,156,1.5295997,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(36 d x (7 A-4 B+2 C) \cos \left(c+\frac{d x}{2}\right)+36 d x (7 A-4 B+2 C) \cos \left(\frac{d x}{2}\right)+147 A \sin \left(c+\frac{d x}{2}\right)-239 A \sin \left(c+\frac{3 d x}{2}\right)-63 A \sin \left(2 c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{5 d x}{2}\right)-15 A \sin \left(3 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{7 d x}{2}\right)+3 A \sin \left(4 c+\frac{7 d x}{2}\right)+84 A d x \cos \left(c+\frac{3 d x}{2}\right)+84 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 A \sin \left(\frac{d x}{2}\right)-120 B \sin \left(c+\frac{d x}{2}\right)+164 B \sin \left(c+\frac{3 d x}{2}\right)+36 B \sin \left(2 c+\frac{3 d x}{2}\right)+12 B \sin \left(2 c+\frac{5 d x}{2}\right)+12 B \sin \left(3 c+\frac{5 d x}{2}\right)-48 B d x \cos \left(c+\frac{3 d x}{2}\right)-48 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+264 B \sin \left(\frac{d x}{2}\right)+96 C \sin \left(c+\frac{d x}{2}\right)-80 C \sin \left(c+\frac{3 d x}{2}\right)+24 C d x \cos \left(c+\frac{3 d x}{2}\right)+24 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-144 C \sin \left(\frac{d x}{2}\right)\right)}{192 a^2 d}","-\frac{2 (8 A-5 B+2 C) \sin (c+d x)}{3 a^2 d}+\frac{(7 A-4 B+2 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(8 A-5 B+2 C) \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{x (7 A-4 B+2 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(36*(7*A - 4*B + 2*C)*d*x*Cos[(d*x)/2] + 36*(7*A - 4*B + 2*C)*d*x*Cos[c + (d*x)/2] + 84*A*d*x*Cos[c + (3*d*x)/2] - 48*B*d*x*Cos[c + (3*d*x)/2] + 24*C*d*x*Cos[c + (3*d*x)/2] + 84*A*d*x*Cos[2*c + (3*d*x)/2] - 48*B*d*x*Cos[2*c + (3*d*x)/2] + 24*C*d*x*Cos[2*c + (3*d*x)/2] - 381*A*Sin[(d*x)/2] + 264*B*Sin[(d*x)/2] - 144*C*Sin[(d*x)/2] + 147*A*Sin[c + (d*x)/2] - 120*B*Sin[c + (d*x)/2] + 96*C*Sin[c + (d*x)/2] - 239*A*Sin[c + (3*d*x)/2] + 164*B*Sin[c + (3*d*x)/2] - 80*C*Sin[c + (3*d*x)/2] - 63*A*Sin[2*c + (3*d*x)/2] + 36*B*Sin[2*c + (3*d*x)/2] - 15*A*Sin[2*c + (5*d*x)/2] + 12*B*Sin[2*c + (5*d*x)/2] - 15*A*Sin[3*c + (5*d*x)/2] + 12*B*Sin[3*c + (5*d*x)/2] + 3*A*Sin[3*c + (7*d*x)/2] + 3*A*Sin[4*c + (7*d*x)/2]))/(192*a^2*d)","B",1
465,1,473,185,1.8110429,"\int \frac{\cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(-36 d x (10 A-7 B+4 C) \cos \left(c+\frac{d x}{2}\right)-36 d x (10 A-7 B+4 C) \cos \left(\frac{d x}{2}\right)-156 A \sin \left(c+\frac{d x}{2}\right)+342 A \sin \left(c+\frac{3 d x}{2}\right)+118 A \sin \left(2 c+\frac{3 d x}{2}\right)+30 A \sin \left(2 c+\frac{5 d x}{2}\right)+30 A \sin \left(3 c+\frac{5 d x}{2}\right)-3 A \sin \left(3 c+\frac{7 d x}{2}\right)-3 A \sin \left(4 c+\frac{7 d x}{2}\right)+A \sin \left(4 c+\frac{9 d x}{2}\right)+A \sin \left(5 c+\frac{9 d x}{2}\right)-120 A d x \cos \left(c+\frac{3 d x}{2}\right)-120 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 A \sin \left(\frac{d x}{2}\right)+147 B \sin \left(c+\frac{d x}{2}\right)-239 B \sin \left(c+\frac{3 d x}{2}\right)-63 B \sin \left(2 c+\frac{3 d x}{2}\right)-15 B \sin \left(2 c+\frac{5 d x}{2}\right)-15 B \sin \left(3 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{7 d x}{2}\right)+3 B \sin \left(4 c+\frac{7 d x}{2}\right)+84 B d x \cos \left(c+\frac{3 d x}{2}\right)+84 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 B \sin \left(\frac{d x}{2}\right)-120 C \sin \left(c+\frac{d x}{2}\right)+164 C \sin \left(c+\frac{3 d x}{2}\right)+36 C \sin \left(2 c+\frac{3 d x}{2}\right)+12 C \sin \left(2 c+\frac{5 d x}{2}\right)+12 C \sin \left(3 c+\frac{5 d x}{2}\right)-48 C d x \cos \left(c+\frac{3 d x}{2}\right)-48 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+264 C \sin \left(\frac{d x}{2}\right)\right)}{192 a^2 d}","-\frac{(12 A-8 B+5 C) \sin ^3(c+d x)}{3 a^2 d}+\frac{(12 A-8 B+5 C) \sin (c+d x)}{a^2 d}-\frac{(10 A-7 B+4 C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(10 A-7 B+4 C) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{x (10 A-7 B+4 C)}{2 a^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(-36*(10*A - 7*B + 4*C)*d*x*Cos[(d*x)/2] - 36*(10*A - 7*B + 4*C)*d*x*Cos[c + (d*x)/2] - 120*A*d*x*Cos[c + (3*d*x)/2] + 84*B*d*x*Cos[c + (3*d*x)/2] - 48*C*d*x*Cos[c + (3*d*x)/2] - 120*A*d*x*Cos[2*c + (3*d*x)/2] + 84*B*d*x*Cos[2*c + (3*d*x)/2] - 48*C*d*x*Cos[2*c + (3*d*x)/2] + 516*A*Sin[(d*x)/2] - 381*B*Sin[(d*x)/2] + 264*C*Sin[(d*x)/2] - 156*A*Sin[c + (d*x)/2] + 147*B*Sin[c + (d*x)/2] - 120*C*Sin[c + (d*x)/2] + 342*A*Sin[c + (3*d*x)/2] - 239*B*Sin[c + (3*d*x)/2] + 164*C*Sin[c + (3*d*x)/2] + 118*A*Sin[2*c + (3*d*x)/2] - 63*B*Sin[2*c + (3*d*x)/2] + 36*C*Sin[2*c + (3*d*x)/2] + 30*A*Sin[2*c + (5*d*x)/2] - 15*B*Sin[2*c + (5*d*x)/2] + 12*C*Sin[2*c + (5*d*x)/2] + 30*A*Sin[3*c + (5*d*x)/2] - 15*B*Sin[3*c + (5*d*x)/2] + 12*C*Sin[3*c + (5*d*x)/2] - 3*A*Sin[3*c + (7*d*x)/2] + 3*B*Sin[3*c + (7*d*x)/2] - 3*A*Sin[4*c + (7*d*x)/2] + 3*B*Sin[4*c + (7*d*x)/2] + A*Sin[4*c + (9*d*x)/2] + A*Sin[5*c + (9*d*x)/2]))/(192*a^2*d)","B",1
466,1,1081,216,6.4702583,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{8 (2 A-6 B+13 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{8 (2 A-6 B+13 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^3(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(490 A \sin \left(\frac{d x}{2}\right)-870 B \sin \left(\frac{d x}{2}\right)+1235 C \sin \left(\frac{d x}{2}\right)-530 A \sin \left(\frac{3 d x}{2}\right)+1830 B \sin \left(\frac{3 d x}{2}\right)-3805 C \sin \left(\frac{3 d x}{2}\right)+654 A \sin \left(c-\frac{d x}{2}\right)-2094 B \sin \left(c-\frac{d x}{2}\right)+4329 C \sin \left(c-\frac{d x}{2}\right)-654 A \sin \left(c+\frac{d x}{2}\right)+1314 B \sin \left(c+\frac{d x}{2}\right)-1989 C \sin \left(c+\frac{d x}{2}\right)+490 A \sin \left(2 c+\frac{d x}{2}\right)-1650 B \sin \left(2 c+\frac{d x}{2}\right)+3575 C \sin \left(2 c+\frac{d x}{2}\right)+350 A \sin \left(c+\frac{3 d x}{2}\right)-450 B \sin \left(c+\frac{3 d x}{2}\right)+475 C \sin \left(c+\frac{3 d x}{2}\right)-530 A \sin \left(2 c+\frac{3 d x}{2}\right)+1230 B \sin \left(2 c+\frac{3 d x}{2}\right)-2005 C \sin \left(2 c+\frac{3 d x}{2}\right)+350 A \sin \left(3 c+\frac{3 d x}{2}\right)-1050 B \sin \left(3 c+\frac{3 d x}{2}\right)+2275 C \sin \left(3 c+\frac{3 d x}{2}\right)-378 A \sin \left(c+\frac{5 d x}{2}\right)+1278 B \sin \left(c+\frac{5 d x}{2}\right)-2673 C \sin \left(c+\frac{5 d x}{2}\right)+150 A \sin \left(2 c+\frac{5 d x}{2}\right)-90 B \sin \left(2 c+\frac{5 d x}{2}\right)-105 C \sin \left(2 c+\frac{5 d x}{2}\right)-378 A \sin \left(3 c+\frac{5 d x}{2}\right)+918 B \sin \left(3 c+\frac{5 d x}{2}\right)-1593 C \sin \left(3 c+\frac{5 d x}{2}\right)+150 A \sin \left(4 c+\frac{5 d x}{2}\right)-450 B \sin \left(4 c+\frac{5 d x}{2}\right)+975 C \sin \left(4 c+\frac{5 d x}{2}\right)-190 A \sin \left(2 c+\frac{7 d x}{2}\right)+630 B \sin \left(2 c+\frac{7 d x}{2}\right)-1325 C \sin \left(2 c+\frac{7 d x}{2}\right)+30 A \sin \left(3 c+\frac{7 d x}{2}\right)+60 B \sin \left(3 c+\frac{7 d x}{2}\right)-255 C \sin \left(3 c+\frac{7 d x}{2}\right)-190 A \sin \left(4 c+\frac{7 d x}{2}\right)+480 B \sin \left(4 c+\frac{7 d x}{2}\right)-875 C \sin \left(4 c+\frac{7 d x}{2}\right)+30 A \sin \left(5 c+\frac{7 d x}{2}\right)-90 B \sin \left(5 c+\frac{7 d x}{2}\right)+195 C \sin \left(5 c+\frac{7 d x}{2}\right)-44 A \sin \left(3 c+\frac{9 d x}{2}\right)+144 B \sin \left(3 c+\frac{9 d x}{2}\right)-304 C \sin \left(3 c+\frac{9 d x}{2}\right)+30 B \sin \left(4 c+\frac{9 d x}{2}\right)-90 C \sin \left(4 c+\frac{9 d x}{2}\right)-44 A \sin \left(5 c+\frac{9 d x}{2}\right)+114 B \sin \left(5 c+\frac{9 d x}{2}\right)-214 C \sin \left(5 c+\frac{9 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{240 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","-\frac{2 (11 A-36 B+76 C) \tan (c+d x)}{15 a^3 d}+\frac{(2 A-6 B+13 C) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}-\frac{(11 A-36 B+76 C) \tan (c+d x) \sec ^2(c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(2 A-6 B+13 C) \tan (c+d x) \sec (c+d x)}{2 a^3 d}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(A-6 B+11 C) \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-8*(2*A - 6*B + 13*C)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (8*(2*A - 6*B + 13*C)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(490*A*Sin[(d*x)/2] - 870*B*Sin[(d*x)/2] + 1235*C*Sin[(d*x)/2] - 530*A*Sin[(3*d*x)/2] + 1830*B*Sin[(3*d*x)/2] - 3805*C*Sin[(3*d*x)/2] + 654*A*Sin[c - (d*x)/2] - 2094*B*Sin[c - (d*x)/2] + 4329*C*Sin[c - (d*x)/2] - 654*A*Sin[c + (d*x)/2] + 1314*B*Sin[c + (d*x)/2] - 1989*C*Sin[c + (d*x)/2] + 490*A*Sin[2*c + (d*x)/2] - 1650*B*Sin[2*c + (d*x)/2] + 3575*C*Sin[2*c + (d*x)/2] + 350*A*Sin[c + (3*d*x)/2] - 450*B*Sin[c + (3*d*x)/2] + 475*C*Sin[c + (3*d*x)/2] - 530*A*Sin[2*c + (3*d*x)/2] + 1230*B*Sin[2*c + (3*d*x)/2] - 2005*C*Sin[2*c + (3*d*x)/2] + 350*A*Sin[3*c + (3*d*x)/2] - 1050*B*Sin[3*c + (3*d*x)/2] + 2275*C*Sin[3*c + (3*d*x)/2] - 378*A*Sin[c + (5*d*x)/2] + 1278*B*Sin[c + (5*d*x)/2] - 2673*C*Sin[c + (5*d*x)/2] + 150*A*Sin[2*c + (5*d*x)/2] - 90*B*Sin[2*c + (5*d*x)/2] - 105*C*Sin[2*c + (5*d*x)/2] - 378*A*Sin[3*c + (5*d*x)/2] + 918*B*Sin[3*c + (5*d*x)/2] - 1593*C*Sin[3*c + (5*d*x)/2] + 150*A*Sin[4*c + (5*d*x)/2] - 450*B*Sin[4*c + (5*d*x)/2] + 975*C*Sin[4*c + (5*d*x)/2] - 190*A*Sin[2*c + (7*d*x)/2] + 630*B*Sin[2*c + (7*d*x)/2] - 1325*C*Sin[2*c + (7*d*x)/2] + 30*A*Sin[3*c + (7*d*x)/2] + 60*B*Sin[3*c + (7*d*x)/2] - 255*C*Sin[3*c + (7*d*x)/2] - 190*A*Sin[4*c + (7*d*x)/2] + 480*B*Sin[4*c + (7*d*x)/2] - 875*C*Sin[4*c + (7*d*x)/2] + 30*A*Sin[5*c + (7*d*x)/2] - 90*B*Sin[5*c + (7*d*x)/2] + 195*C*Sin[5*c + (7*d*x)/2] - 44*A*Sin[3*c + (9*d*x)/2] + 144*B*Sin[3*c + (9*d*x)/2] - 304*C*Sin[3*c + (9*d*x)/2] + 30*B*Sin[4*c + (9*d*x)/2] - 90*C*Sin[4*c + (9*d*x)/2] - 44*A*Sin[5*c + (9*d*x)/2] + 114*B*Sin[5*c + (9*d*x)/2] - 214*C*Sin[5*c + (9*d*x)/2]))/(240*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","B",1
467,1,839,161,6.3832096,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{16 (3 C-B) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{16 (3 C-B) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-20 A \sin \left(\frac{d x}{2}\right)+160 B \sin \left(\frac{d x}{2}\right)-255 C \sin \left(\frac{d x}{2}\right)+22 A \sin \left(\frac{3 d x}{2}\right)-167 B \sin \left(\frac{3 d x}{2}\right)+567 C \sin \left(\frac{3 d x}{2}\right)-10 A \sin \left(c-\frac{d x}{2}\right)+170 B \sin \left(c-\frac{d x}{2}\right)-600 C \sin \left(c-\frac{d x}{2}\right)+10 A \sin \left(c+\frac{d x}{2}\right)-170 B \sin \left(c+\frac{d x}{2}\right)+375 C \sin \left(c+\frac{d x}{2}\right)-20 A \sin \left(2 c+\frac{d x}{2}\right)+160 B \sin \left(2 c+\frac{d x}{2}\right)-480 C \sin \left(2 c+\frac{d x}{2}\right)+75 B \sin \left(c+\frac{3 d x}{2}\right)-60 C \sin \left(c+\frac{3 d x}{2}\right)+22 A \sin \left(2 c+\frac{3 d x}{2}\right)-167 B \sin \left(2 c+\frac{3 d x}{2}\right)+402 C \sin \left(2 c+\frac{3 d x}{2}\right)+75 B \sin \left(3 c+\frac{3 d x}{2}\right)-225 C \sin \left(3 c+\frac{3 d x}{2}\right)+10 A \sin \left(c+\frac{5 d x}{2}\right)-95 B \sin \left(c+\frac{5 d x}{2}\right)+315 C \sin \left(c+\frac{5 d x}{2}\right)+15 B \sin \left(2 c+\frac{5 d x}{2}\right)+30 C \sin \left(2 c+\frac{5 d x}{2}\right)+10 A \sin \left(3 c+\frac{5 d x}{2}\right)-95 B \sin \left(3 c+\frac{5 d x}{2}\right)+240 C \sin \left(3 c+\frac{5 d x}{2}\right)+15 B \sin \left(4 c+\frac{5 d x}{2}\right)-45 C \sin \left(4 c+\frac{5 d x}{2}\right)+2 A \sin \left(2 c+\frac{7 d x}{2}\right)-22 B \sin \left(2 c+\frac{7 d x}{2}\right)+72 C \sin \left(2 c+\frac{7 d x}{2}\right)+15 C \sin \left(3 c+\frac{7 d x}{2}\right)+2 A \sin \left(4 c+\frac{7 d x}{2}\right)-22 B \sin \left(4 c+\frac{7 d x}{2}\right)+57 C \sin \left(4 c+\frac{7 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(2 A-7 B+27 C) \tan (c+d x)}{15 a^3 d}+\frac{(B-3 C) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(B-3 C) \tan (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(A-B+C) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(A+4 B-9 C) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(16*(-B + 3*C)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (16*(-B + 3*C)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-20*A*Sin[(d*x)/2] + 160*B*Sin[(d*x)/2] - 255*C*Sin[(d*x)/2] + 22*A*Sin[(3*d*x)/2] - 167*B*Sin[(3*d*x)/2] + 567*C*Sin[(3*d*x)/2] - 10*A*Sin[c - (d*x)/2] + 170*B*Sin[c - (d*x)/2] - 600*C*Sin[c - (d*x)/2] + 10*A*Sin[c + (d*x)/2] - 170*B*Sin[c + (d*x)/2] + 375*C*Sin[c + (d*x)/2] - 20*A*Sin[2*c + (d*x)/2] + 160*B*Sin[2*c + (d*x)/2] - 480*C*Sin[2*c + (d*x)/2] + 75*B*Sin[c + (3*d*x)/2] - 60*C*Sin[c + (3*d*x)/2] + 22*A*Sin[2*c + (3*d*x)/2] - 167*B*Sin[2*c + (3*d*x)/2] + 402*C*Sin[2*c + (3*d*x)/2] + 75*B*Sin[3*c + (3*d*x)/2] - 225*C*Sin[3*c + (3*d*x)/2] + 10*A*Sin[c + (5*d*x)/2] - 95*B*Sin[c + (5*d*x)/2] + 315*C*Sin[c + (5*d*x)/2] + 15*B*Sin[2*c + (5*d*x)/2] + 30*C*Sin[2*c + (5*d*x)/2] + 10*A*Sin[3*c + (5*d*x)/2] - 95*B*Sin[3*c + (5*d*x)/2] + 240*C*Sin[3*c + (5*d*x)/2] + 15*B*Sin[4*c + (5*d*x)/2] - 45*C*Sin[4*c + (5*d*x)/2] + 2*A*Sin[2*c + (7*d*x)/2] - 22*B*Sin[2*c + (7*d*x)/2] + 72*C*Sin[2*c + (7*d*x)/2] + 15*C*Sin[3*c + (7*d*x)/2] + 2*A*Sin[4*c + (7*d*x)/2] - 22*B*Sin[4*c + (7*d*x)/2] + 57*C*Sin[4*c + (7*d*x)/2]))/(60*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","B",1
468,1,277,132,1.6280456,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{\left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(240 C \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(5 (3 A+4 B-29 C) \sin \left(\frac{d x}{2}\right)-15 (A-5 C) \sin \left(c+\frac{d x}{2}\right)+15 A \sin \left(c+\frac{3 d x}{2}\right)+3 A \sin \left(2 c+\frac{5 d x}{2}\right)+10 B \sin \left(c+\frac{3 d x}{2}\right)+2 B \sin \left(2 c+\frac{5 d x}{2}\right)-95 C \sin \left(c+\frac{3 d x}{2}\right)+15 C \sin \left(2 c+\frac{3 d x}{2}\right)-22 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)\right)}{15 a^3 d (\cos (c+d x)+1)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(6 A+4 B-29 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(3 A+2 B-7 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"-1/15*((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(240*C*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Cos[(c + d*x)/2]*Sec[c/2]*(5*(3*A + 4*B - 29*C)*Sin[(d*x)/2] - 15*(A - 5*C)*Sin[c + (d*x)/2] + 15*A*Sin[c + (3*d*x)/2] + 10*B*Sin[c + (3*d*x)/2] - 95*C*Sin[c + (3*d*x)/2] + 15*C*Sin[2*c + (3*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] + 2*B*Sin[2*c + (5*d*x)/2] - 22*C*Sin[2*c + (5*d*x)/2])))/(a^3*d*(1 + Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
469,1,156,110,0.5540544,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(-15 (2 A+B) \sin \left(c+\frac{d x}{2}\right)+5 (8 A+3 B+4 C) \sin \left(\frac{d x}{2}\right)+20 A \sin \left(c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{3 d x}{2}\right)+7 A \sin \left(2 c+\frac{5 d x}{2}\right)+15 B \sin \left(c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+10 C \sin \left(c+\frac{3 d x}{2}\right)+2 C \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{240 a^3 d}","\frac{(2 A+3 B+7 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(A-C) \tan (c+d x)}{3 a d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(5*(8*A + 3*B + 4*C)*Sin[(d*x)/2] - 15*(2*A + B)*Sin[c + (d*x)/2] + 20*A*Sin[c + (3*d*x)/2] + 15*B*Sin[c + (3*d*x)/2] + 10*C*Sin[c + (3*d*x)/2] - 15*A*Sin[2*c + (3*d*x)/2] + 7*A*Sin[2*c + (5*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] + 2*C*Sin[2*c + (5*d*x)/2]))/(240*a^3*d)","A",1
470,1,289,115,0.9413449,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(270 A \sin \left(c+\frac{d x}{2}\right)-230 A \sin \left(c+\frac{3 d x}{2}\right)+90 A \sin \left(2 c+\frac{3 d x}{2}\right)-64 A \sin \left(2 c+\frac{5 d x}{2}\right)+150 A d x \cos \left(c+\frac{d x}{2}\right)+75 A d x \cos \left(c+\frac{3 d x}{2}\right)+75 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 A \sin \left(\frac{d x}{2}\right)+150 A d x \cos \left(\frac{d x}{2}\right)-60 B \sin \left(c+\frac{d x}{2}\right)+40 B \sin \left(c+\frac{3 d x}{2}\right)-30 B \sin \left(2 c+\frac{3 d x}{2}\right)+14 B \sin \left(2 c+\frac{5 d x}{2}\right)+80 B \sin \left(\frac{d x}{2}\right)-30 C \sin \left(c+\frac{d x}{2}\right)+30 C \sin \left(c+\frac{3 d x}{2}\right)+6 C \sin \left(2 c+\frac{5 d x}{2}\right)+30 C \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","-\frac{(22 A-2 B-3 C) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{A x}{a^3}-\frac{(7 A-2 B-3 C) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(150*A*d*x*Cos[(d*x)/2] + 150*A*d*x*Cos[c + (d*x)/2] + 75*A*d*x*Cos[c + (3*d*x)/2] + 75*A*d*x*Cos[2*c + (3*d*x)/2] + 15*A*d*x*Cos[2*c + (5*d*x)/2] + 15*A*d*x*Cos[3*c + (5*d*x)/2] - 370*A*Sin[(d*x)/2] + 80*B*Sin[(d*x)/2] + 30*C*Sin[(d*x)/2] + 270*A*Sin[c + (d*x)/2] - 60*B*Sin[c + (d*x)/2] - 30*C*Sin[c + (d*x)/2] - 230*A*Sin[c + (3*d*x)/2] + 40*B*Sin[c + (3*d*x)/2] + 30*C*Sin[c + (3*d*x)/2] + 90*A*Sin[2*c + (3*d*x)/2] - 30*B*Sin[2*c + (3*d*x)/2] - 64*A*Sin[2*c + (5*d*x)/2] + 14*B*Sin[2*c + (5*d*x)/2] + 6*C*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","B",1
471,1,419,141,1.7391807,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(-300 d x (3 A-B) \cos \left(c+\frac{d x}{2}\right)-300 d x (3 A-B) \cos \left(\frac{d x}{2}\right)-1125 A \sin \left(c+\frac{d x}{2}\right)+1215 A \sin \left(c+\frac{3 d x}{2}\right)-225 A \sin \left(2 c+\frac{3 d x}{2}\right)+363 A \sin \left(2 c+\frac{5 d x}{2}\right)+75 A \sin \left(3 c+\frac{5 d x}{2}\right)+15 A \sin \left(3 c+\frac{7 d x}{2}\right)+15 A \sin \left(4 c+\frac{7 d x}{2}\right)-450 A d x \cos \left(c+\frac{3 d x}{2}\right)-450 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-90 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-90 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+1755 A \sin \left(\frac{d x}{2}\right)+540 B \sin \left(c+\frac{d x}{2}\right)-460 B \sin \left(c+\frac{3 d x}{2}\right)+180 B \sin \left(2 c+\frac{3 d x}{2}\right)-128 B \sin \left(2 c+\frac{5 d x}{2}\right)+150 B d x \cos \left(c+\frac{3 d x}{2}\right)+150 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+30 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+30 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-740 B \sin \left(\frac{d x}{2}\right)-120 C \sin \left(c+\frac{d x}{2}\right)+80 C \sin \left(c+\frac{3 d x}{2}\right)-60 C \sin \left(2 c+\frac{3 d x}{2}\right)+28 C \sin \left(2 c+\frac{5 d x}{2}\right)+160 C \sin \left(\frac{d x}{2}\right)\right)}{960 a^3 d}","\frac{2 (36 A-11 B+C) \sin (c+d x)}{15 a^3 d}-\frac{(3 A-B) \sin (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{x (3 A-B)}{a^3}-\frac{(9 A-4 B-C) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(-300*(3*A - B)*d*x*Cos[(d*x)/2] - 300*(3*A - B)*d*x*Cos[c + (d*x)/2] - 450*A*d*x*Cos[c + (3*d*x)/2] + 150*B*d*x*Cos[c + (3*d*x)/2] - 450*A*d*x*Cos[2*c + (3*d*x)/2] + 150*B*d*x*Cos[2*c + (3*d*x)/2] - 90*A*d*x*Cos[2*c + (5*d*x)/2] + 30*B*d*x*Cos[2*c + (5*d*x)/2] - 90*A*d*x*Cos[3*c + (5*d*x)/2] + 30*B*d*x*Cos[3*c + (5*d*x)/2] + 1755*A*Sin[(d*x)/2] - 740*B*Sin[(d*x)/2] + 160*C*Sin[(d*x)/2] - 1125*A*Sin[c + (d*x)/2] + 540*B*Sin[c + (d*x)/2] - 120*C*Sin[c + (d*x)/2] + 1215*A*Sin[c + (3*d*x)/2] - 460*B*Sin[c + (3*d*x)/2] + 80*C*Sin[c + (3*d*x)/2] - 225*A*Sin[2*c + (3*d*x)/2] + 180*B*Sin[2*c + (3*d*x)/2] - 60*C*Sin[2*c + (3*d*x)/2] + 363*A*Sin[2*c + (5*d*x)/2] - 128*B*Sin[2*c + (5*d*x)/2] + 28*C*Sin[2*c + (5*d*x)/2] + 75*A*Sin[3*c + (5*d*x)/2] + 15*A*Sin[3*c + (7*d*x)/2] + 15*A*Sin[4*c + (7*d*x)/2]))/(960*a^3*d)","B",1
472,1,557,201,1.6429298,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(600 d x (13 A-6 B+2 C) \cos \left(c+\frac{d x}{2}\right)+600 d x (13 A-6 B+2 C) \cos \left(\frac{d x}{2}\right)+7560 A \sin \left(c+\frac{d x}{2}\right)-9230 A \sin \left(c+\frac{3 d x}{2}\right)+930 A \sin \left(2 c+\frac{3 d x}{2}\right)-2782 A \sin \left(2 c+\frac{5 d x}{2}\right)-750 A \sin \left(3 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{7 d x}{2}\right)-105 A \sin \left(4 c+\frac{7 d x}{2}\right)+15 A \sin \left(4 c+\frac{9 d x}{2}\right)+15 A \sin \left(5 c+\frac{9 d x}{2}\right)+3900 A d x \cos \left(c+\frac{3 d x}{2}\right)+3900 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 A \sin \left(\frac{d x}{2}\right)-4500 B \sin \left(c+\frac{d x}{2}\right)+4860 B \sin \left(c+\frac{3 d x}{2}\right)-900 B \sin \left(2 c+\frac{3 d x}{2}\right)+1452 B \sin \left(2 c+\frac{5 d x}{2}\right)+300 B \sin \left(3 c+\frac{5 d x}{2}\right)+60 B \sin \left(3 c+\frac{7 d x}{2}\right)+60 B \sin \left(4 c+\frac{7 d x}{2}\right)-1800 B d x \cos \left(c+\frac{3 d x}{2}\right)-1800 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-360 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-360 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+7020 B \sin \left(\frac{d x}{2}\right)+2160 C \sin \left(c+\frac{d x}{2}\right)-1840 C \sin \left(c+\frac{3 d x}{2}\right)+720 C \sin \left(2 c+\frac{3 d x}{2}\right)-512 C \sin \left(2 c+\frac{5 d x}{2}\right)+600 C d x \cos \left(c+\frac{3 d x}{2}\right)+600 C d x \cos \left(2 c+\frac{3 d x}{2}\right)+120 C d x \cos \left(2 c+\frac{5 d x}{2}\right)+120 C d x \cos \left(3 c+\frac{5 d x}{2}\right)-2960 C \sin \left(\frac{d x}{2}\right)\right)}{3840 a^3 d}","-\frac{2 (76 A-36 B+11 C) \sin (c+d x)}{15 a^3 d}+\frac{(13 A-6 B+2 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(76 A-36 B+11 C) \sin (c+d x) \cos (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{x (13 A-6 B+2 C)}{2 a^3}-\frac{(11 A-6 B+C) \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(600*(13*A - 6*B + 2*C)*d*x*Cos[(d*x)/2] + 600*(13*A - 6*B + 2*C)*d*x*Cos[c + (d*x)/2] + 3900*A*d*x*Cos[c + (3*d*x)/2] - 1800*B*d*x*Cos[c + (3*d*x)/2] + 600*C*d*x*Cos[c + (3*d*x)/2] + 3900*A*d*x*Cos[2*c + (3*d*x)/2] - 1800*B*d*x*Cos[2*c + (3*d*x)/2] + 600*C*d*x*Cos[2*c + (3*d*x)/2] + 780*A*d*x*Cos[2*c + (5*d*x)/2] - 360*B*d*x*Cos[2*c + (5*d*x)/2] + 120*C*d*x*Cos[2*c + (5*d*x)/2] + 780*A*d*x*Cos[3*c + (5*d*x)/2] - 360*B*d*x*Cos[3*c + (5*d*x)/2] + 120*C*d*x*Cos[3*c + (5*d*x)/2] - 12760*A*Sin[(d*x)/2] + 7020*B*Sin[(d*x)/2] - 2960*C*Sin[(d*x)/2] + 7560*A*Sin[c + (d*x)/2] - 4500*B*Sin[c + (d*x)/2] + 2160*C*Sin[c + (d*x)/2] - 9230*A*Sin[c + (3*d*x)/2] + 4860*B*Sin[c + (3*d*x)/2] - 1840*C*Sin[c + (3*d*x)/2] + 930*A*Sin[2*c + (3*d*x)/2] - 900*B*Sin[2*c + (3*d*x)/2] + 720*C*Sin[2*c + (3*d*x)/2] - 2782*A*Sin[2*c + (5*d*x)/2] + 1452*B*Sin[2*c + (5*d*x)/2] - 512*C*Sin[2*c + (5*d*x)/2] - 750*A*Sin[3*c + (5*d*x)/2] + 300*B*Sin[3*c + (5*d*x)/2] - 105*A*Sin[3*c + (7*d*x)/2] + 60*B*Sin[3*c + (7*d*x)/2] - 105*A*Sin[4*c + (7*d*x)/2] + 60*B*Sin[4*c + (7*d*x)/2] + 15*A*Sin[4*c + (9*d*x)/2] + 15*A*Sin[5*c + (9*d*x)/2]))/(3840*a^3*d)","B",1
473,1,655,237,2.6324908,"\int \frac{\cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(-600 d x (23 A-13 B+6 C) \cos \left(c+\frac{d x}{2}\right)-600 d x (23 A-13 B+6 C) \cos \left(\frac{d x}{2}\right)-11110 A \sin \left(c+\frac{d x}{2}\right)+15380 A \sin \left(c+\frac{3 d x}{2}\right)-380 A \sin \left(2 c+\frac{3 d x}{2}\right)+4777 A \sin \left(2 c+\frac{5 d x}{2}\right)+1625 A \sin \left(3 c+\frac{5 d x}{2}\right)+230 A \sin \left(3 c+\frac{7 d x}{2}\right)+230 A \sin \left(4 c+\frac{7 d x}{2}\right)-20 A \sin \left(4 c+\frac{9 d x}{2}\right)-20 A \sin \left(5 c+\frac{9 d x}{2}\right)+5 A \sin \left(5 c+\frac{11 d x}{2}\right)+5 A \sin \left(6 c+\frac{11 d x}{2}\right)-6900 A d x \cos \left(c+\frac{3 d x}{2}\right)-6900 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-1380 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-1380 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+20410 A \sin \left(\frac{d x}{2}\right)+7560 B \sin \left(c+\frac{d x}{2}\right)-9230 B \sin \left(c+\frac{3 d x}{2}\right)+930 B \sin \left(2 c+\frac{3 d x}{2}\right)-2782 B \sin \left(2 c+\frac{5 d x}{2}\right)-750 B \sin \left(3 c+\frac{5 d x}{2}\right)-105 B \sin \left(3 c+\frac{7 d x}{2}\right)-105 B \sin \left(4 c+\frac{7 d x}{2}\right)+15 B \sin \left(4 c+\frac{9 d x}{2}\right)+15 B \sin \left(5 c+\frac{9 d x}{2}\right)+3900 B d x \cos \left(c+\frac{3 d x}{2}\right)+3900 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 B \sin \left(\frac{d x}{2}\right)-4500 C \sin \left(c+\frac{d x}{2}\right)+4860 C \sin \left(c+\frac{3 d x}{2}\right)-900 C \sin \left(2 c+\frac{3 d x}{2}\right)+1452 C \sin \left(2 c+\frac{5 d x}{2}\right)+300 C \sin \left(3 c+\frac{5 d x}{2}\right)+60 C \sin \left(3 c+\frac{7 d x}{2}\right)+60 C \sin \left(4 c+\frac{7 d x}{2}\right)-1800 C d x \cos \left(c+\frac{3 d x}{2}\right)-1800 C d x \cos \left(2 c+\frac{3 d x}{2}\right)-360 C d x \cos \left(2 c+\frac{5 d x}{2}\right)-360 C d x \cos \left(3 c+\frac{5 d x}{2}\right)+7020 C \sin \left(\frac{d x}{2}\right)\right)}{3840 a^3 d}","-\frac{4 (34 A-19 B+9 C) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B+9 C) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B+6 C) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(23 A-13 B+6 C) \sin (c+d x) \cos ^2(c+d x)}{3 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{x (23 A-13 B+6 C)}{2 a^3}-\frac{(13 A-8 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(-600*(23*A - 13*B + 6*C)*d*x*Cos[(d*x)/2] - 600*(23*A - 13*B + 6*C)*d*x*Cos[c + (d*x)/2] - 6900*A*d*x*Cos[c + (3*d*x)/2] + 3900*B*d*x*Cos[c + (3*d*x)/2] - 1800*C*d*x*Cos[c + (3*d*x)/2] - 6900*A*d*x*Cos[2*c + (3*d*x)/2] + 3900*B*d*x*Cos[2*c + (3*d*x)/2] - 1800*C*d*x*Cos[2*c + (3*d*x)/2] - 1380*A*d*x*Cos[2*c + (5*d*x)/2] + 780*B*d*x*Cos[2*c + (5*d*x)/2] - 360*C*d*x*Cos[2*c + (5*d*x)/2] - 1380*A*d*x*Cos[3*c + (5*d*x)/2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] - 360*C*d*x*Cos[3*c + (5*d*x)/2] + 20410*A*Sin[(d*x)/2] - 12760*B*Sin[(d*x)/2] + 7020*C*Sin[(d*x)/2] - 11110*A*Sin[c + (d*x)/2] + 7560*B*Sin[c + (d*x)/2] - 4500*C*Sin[c + (d*x)/2] + 15380*A*Sin[c + (3*d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] + 4860*C*Sin[c + (3*d*x)/2] - 380*A*Sin[2*c + (3*d*x)/2] + 930*B*Sin[2*c + (3*d*x)/2] - 900*C*Sin[2*c + (3*d*x)/2] + 4777*A*Sin[2*c + (5*d*x)/2] - 2782*B*Sin[2*c + (5*d*x)/2] + 1452*C*Sin[2*c + (5*d*x)/2] + 1625*A*Sin[3*c + (5*d*x)/2] - 750*B*Sin[3*c + (5*d*x)/2] + 300*C*Sin[3*c + (5*d*x)/2] + 230*A*Sin[3*c + (7*d*x)/2] - 105*B*Sin[3*c + (7*d*x)/2] + 60*C*Sin[3*c + (7*d*x)/2] + 230*A*Sin[4*c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] + 60*C*Sin[4*c + (7*d*x)/2] - 20*A*Sin[4*c + (9*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2] - 20*A*Sin[5*c + (9*d*x)/2] + 15*B*Sin[5*c + (9*d*x)/2] + 5*A*Sin[5*c + (11*d*x)/2] + 5*A*Sin[6*c + (11*d*x)/2]))/(3840*a^3*d)","B",1
474,1,1322,254,6.4853923,"\int \frac{\sec ^5(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\frac{16 (2 A-8 B+21 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{16 (2 A-8 B+21 C) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{16 C \sec (c) \sec ^4(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (d x) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{16 \sec (c) \sec ^3(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (C \sin (c)+2 B \sin (d x)-8 C \sin (d x)) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{32 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(160 A \sin \left(\frac{d x}{2}\right)-559 B \sin \left(\frac{d x}{2}\right)+1308 C \sin \left(\frac{d x}{2}\right)\right) \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{16 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(55 A \sin \left(\frac{c}{2}\right)-139 B \sin \left(\frac{c}{2}\right)+258 C \sin \left(\frac{c}{2}\right)\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{16 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(55 A \sin \left(\frac{d x}{2}\right)-139 B \sin \left(\frac{d x}{2}\right)+258 C \sin \left(\frac{d x}{2}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{8 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(10 A \sin \left(\frac{c}{2}\right)-17 B \sin \left(\frac{c}{2}\right)+24 C \sin \left(\frac{c}{2}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{8 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(10 A \sin \left(\frac{d x}{2}\right)-17 B \sin \left(\frac{d x}{2}\right)+24 C \sin \left(\frac{d x}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{4 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)+C \sin \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{4 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}","-\frac{8 (20 A-83 B+216 C) \tan (c+d x)}{105 a^4 d}+\frac{(2 A-8 B+21 C) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}-\frac{(10 A-52 B+129 C) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{4 (20 A-83 B+216 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(2 A-8 B+21 C) \tan (c+d x) \sec (c+d x)}{2 a^4 d}-\frac{(A-B+C) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{(B-2 C) \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3}",1,"(-16*(2*A - 8*B + 21*C)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (16*(2*A - 8*B + 21*C)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (4*Cos[c/2 + (d*x)/2]^2*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*Sin[c/2] - B*Sin[c/2] + C*Sin[c/2]))/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (8*Cos[c/2 + (d*x)/2]^4*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(10*A*Sin[c/2] - 17*B*Sin[c/2] + 24*C*Sin[c/2]))/(35*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (16*Cos[c/2 + (d*x)/2]^6*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(55*A*Sin[c/2] - 139*B*Sin[c/2] + 258*C*Sin[c/2]))/(105*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (4*Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (8*Cos[c/2 + (d*x)/2]^3*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(10*A*Sin[(d*x)/2] - 17*B*Sin[(d*x)/2] + 24*C*Sin[(d*x)/2]))/(35*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (16*Cos[c/2 + (d*x)/2]^5*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(55*A*Sin[(d*x)/2] - 139*B*Sin[(d*x)/2] + 258*C*Sin[(d*x)/2]))/(105*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (32*Cos[c/2 + (d*x)/2]^7*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(160*A*Sin[(d*x)/2] - 559*B*Sin[(d*x)/2] + 1308*C*Sin[(d*x)/2]))/(105*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (16*C*Cos[c/2 + (d*x)/2]^8*Sec[c]*Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[d*x])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (16*Cos[c/2 + (d*x)/2]^8*Sec[c]*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(C*Sin[c] + 2*B*Sin[d*x] - 8*C*Sin[d*x]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","B",1
475,1,1208,204,6.4002393,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{32 (4 C-B) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{32 (4 C-B) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{32 C \sec (c) \sec ^3(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (d x) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{32 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(6 A \sin \left(\frac{d x}{2}\right)-160 B \sin \left(\frac{d x}{2}\right)+559 C \sin \left(\frac{d x}{2}\right)\right) \cos ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{16 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(6 A \sin \left(\frac{c}{2}\right)-55 B \sin \left(\frac{c}{2}\right)+139 C \sin \left(\frac{c}{2}\right)\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{16 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(6 A \sin \left(\frac{d x}{2}\right)-55 B \sin \left(\frac{d x}{2}\right)+139 C \sin \left(\frac{d x}{2}\right)\right) \cos ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{8 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(3 A \sin \left(\frac{c}{2}\right)-10 B \sin \left(\frac{c}{2}\right)+17 C \sin \left(\frac{c}{2}\right)\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{8 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(3 A \sin \left(\frac{d x}{2}\right)-10 B \sin \left(\frac{d x}{2}\right)+17 C \sin \left(\frac{d x}{2}\right)\right) \cos ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{4 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(A \sin \left(\frac{c}{2}\right)-B \sin \left(\frac{c}{2}\right)+C \sin \left(\frac{c}{2}\right)\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{4 \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}","\frac{(6 A-55 B+244 C) \tan (c+d x)}{105 a^4 d}+\frac{(3 A+25 B-88 C) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{(B-4 C) \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(B-4 C) \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{(2 A+5 B-12 C) \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(32*(-B + 4*C)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (32*(-B + 4*C)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (4*Cos[c/2 + (d*x)/2]^2*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*Sin[c/2] - B*Sin[c/2] + C*Sin[c/2]))/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (8*Cos[c/2 + (d*x)/2]^4*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(3*A*Sin[c/2] - 10*B*Sin[c/2] + 17*C*Sin[c/2]))/(35*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (16*Cos[c/2 + (d*x)/2]^6*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*A*Sin[c/2] - 55*B*Sin[c/2] + 139*C*Sin[c/2]))/(105*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (4*Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (8*Cos[c/2 + (d*x)/2]^3*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(3*A*Sin[(d*x)/2] - 10*B*Sin[(d*x)/2] + 17*C*Sin[(d*x)/2]))/(35*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (16*Cos[c/2 + (d*x)/2]^5*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*A*Sin[(d*x)/2] - 55*B*Sin[(d*x)/2] + 139*C*Sin[(d*x)/2]))/(105*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (32*Cos[c/2 + (d*x)/2]^7*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*A*Sin[(d*x)/2] - 160*B*Sin[(d*x)/2] + 559*C*Sin[(d*x)/2]))/(105*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (32*C*Cos[c/2 + (d*x)/2]^8*Sec[c]*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[d*x])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","B",1
476,1,335,173,2.7361291,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\frac{\left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(6720 C \cos ^8\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(70 (2 A+3 B-49 C) \sin \left(\frac{d x}{2}\right)-70 (2 A-31 C) \sin \left(c+\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)+56 A \sin \left(2 c+\frac{5 d x}{2}\right)+8 A \sin \left(3 c+\frac{7 d x}{2}\right)+126 B \sin \left(c+\frac{3 d x}{2}\right)+42 B \sin \left(2 c+\frac{5 d x}{2}\right)+6 B \sin \left(3 c+\frac{7 d x}{2}\right)-2625 C \sin \left(c+\frac{3 d x}{2}\right)+735 C \sin \left(2 c+\frac{3 d x}{2}\right)-1015 C \sin \left(2 c+\frac{5 d x}{2}\right)+105 C \sin \left(3 c+\frac{5 d x}{2}\right)-160 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)\right)}{210 a^4 d (\cos (c+d x)+1)^4 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(16 A+12 B-215 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(8 A+6 B-55 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a^4 d}-\frac{(A-B+C) \tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{(4 A+3 B-10 C) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"-1/210*((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(6720*C*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Cos[(c + d*x)/2]*Sec[c/2]*(70*(2*A + 3*B - 49*C)*Sin[(d*x)/2] - 70*(2*A - 31*C)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 126*B*Sin[c + (3*d*x)/2] - 2625*C*Sin[c + (3*d*x)/2] + 735*C*Sin[2*c + (3*d*x)/2] + 56*A*Sin[2*c + (5*d*x)/2] + 42*B*Sin[2*c + (5*d*x)/2] - 1015*C*Sin[2*c + (5*d*x)/2] + 105*C*Sin[3*c + (5*d*x)/2] + 8*A*Sin[3*c + (7*d*x)/2] + 6*B*Sin[3*c + (7*d*x)/2] - 160*C*Sin[3*c + (7*d*x)/2])))/(a^4*d*(1 + Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
477,1,200,148,0.7010423,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(-35 (5 A+4 B) \sin \left(c+\frac{d x}{2}\right)+70 (4 A+2 B+3 C) \sin \left(\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)-105 A \sin \left(2 c+\frac{3 d x}{2}\right)+91 A \sin \left(2 c+\frac{5 d x}{2}\right)+13 A \sin \left(3 c+\frac{7 d x}{2}\right)+168 B \sin \left(c+\frac{3 d x}{2}\right)+56 B \sin \left(2 c+\frac{5 d x}{2}\right)+8 B \sin \left(3 c+\frac{7 d x}{2}\right)+126 C \sin \left(c+\frac{3 d x}{2}\right)+42 C \sin \left(2 c+\frac{5 d x}{2}\right)+6 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{6720 a^4 d}","\frac{(8 A+13 B+36 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}+\frac{(23 A-2 B-54 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{(6 A+B-8 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(70*(4*A + 2*B + 3*C)*Sin[(d*x)/2] - 35*(5*A + 4*B)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 168*B*Sin[c + (3*d*x)/2] + 126*C*Sin[c + (3*d*x)/2] - 105*A*Sin[2*c + (3*d*x)/2] + 91*A*Sin[2*c + (5*d*x)/2] + 56*B*Sin[2*c + (5*d*x)/2] + 42*C*Sin[2*c + (5*d*x)/2] + 13*A*Sin[3*c + (7*d*x)/2] + 8*B*Sin[3*c + (7*d*x)/2] + 6*C*Sin[3*c + (7*d*x)/2]))/(6720*a^4*d)","A",1
478,1,231,154,0.7870573,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(-35 (18 A+5 B+4 C) \sin \left(c+\frac{d x}{2}\right)+70 (9 A+4 B+2 C) \sin \left(\frac{d x}{2}\right)+441 A \sin \left(c+\frac{3 d x}{2}\right)-315 A \sin \left(2 c+\frac{3 d x}{2}\right)+147 A \sin \left(2 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{5 d x}{2}\right)+36 A \sin \left(3 c+\frac{7 d x}{2}\right)+168 B \sin \left(c+\frac{3 d x}{2}\right)-105 B \sin \left(2 c+\frac{3 d x}{2}\right)+91 B \sin \left(2 c+\frac{5 d x}{2}\right)+13 B \sin \left(3 c+\frac{7 d x}{2}\right)+168 C \sin \left(c+\frac{3 d x}{2}\right)+56 C \sin \left(2 c+\frac{5 d x}{2}\right)+8 C \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{6720 a^4 d}","\frac{(6 A+8 B+13 C) \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{(6 A+8 B+13 C) \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{(8 A-B-6 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(70*(9*A + 4*B + 2*C)*Sin[(d*x)/2] - 35*(18*A + 5*B + 4*C)*Sin[c + (d*x)/2] + 441*A*Sin[c + (3*d*x)/2] + 168*B*Sin[c + (3*d*x)/2] + 168*C*Sin[c + (3*d*x)/2] - 315*A*Sin[2*c + (3*d*x)/2] - 105*B*Sin[2*c + (3*d*x)/2] + 147*A*Sin[2*c + (5*d*x)/2] + 91*B*Sin[2*c + (5*d*x)/2] + 56*C*Sin[2*c + (5*d*x)/2] - 105*A*Sin[3*c + (5*d*x)/2] + 36*A*Sin[3*c + (7*d*x)/2] + 13*B*Sin[3*c + (7*d*x)/2] + 8*C*Sin[3*c + (7*d*x)/2]))/(6720*a^4*d)","A",1
479,1,405,148,1.3969679,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(8260 A \sin \left(c+\frac{d x}{2}\right)-7140 A \sin \left(c+\frac{3 d x}{2}\right)+3780 A \sin \left(2 c+\frac{3 d x}{2}\right)-2800 A \sin \left(2 c+\frac{5 d x}{2}\right)+840 A \sin \left(3 c+\frac{5 d x}{2}\right)-520 A \sin \left(3 c+\frac{7 d x}{2}\right)+3675 A d x \cos \left(c+\frac{d x}{2}\right)+2205 A d x \cos \left(c+\frac{3 d x}{2}\right)+2205 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+735 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+735 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+105 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+105 A d x \cos \left(4 c+\frac{7 d x}{2}\right)-9940 A \sin \left(\frac{d x}{2}\right)+3675 A d x \cos \left(\frac{d x}{2}\right)-1260 B \sin \left(c+\frac{d x}{2}\right)+882 B \sin \left(c+\frac{3 d x}{2}\right)-630 B \sin \left(2 c+\frac{3 d x}{2}\right)+294 B \sin \left(2 c+\frac{5 d x}{2}\right)-210 B \sin \left(3 c+\frac{5 d x}{2}\right)+72 B \sin \left(3 c+\frac{7 d x}{2}\right)+1260 B \sin \left(\frac{d x}{2}\right)-350 C \sin \left(c+\frac{d x}{2}\right)+336 C \sin \left(c+\frac{3 d x}{2}\right)-210 C \sin \left(2 c+\frac{3 d x}{2}\right)+182 C \sin \left(2 c+\frac{5 d x}{2}\right)+26 C \sin \left(3 c+\frac{7 d x}{2}\right)+560 C \sin \left(\frac{d x}{2}\right)\right)}{13440 a^4 d}","-\frac{2 (80 A-3 B-4 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(55 A-6 B-8 C) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{A x}{a^4}-\frac{(10 A-3 B-4 C) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B+C) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(3675*A*d*x*Cos[(d*x)/2] + 3675*A*d*x*Cos[c + (d*x)/2] + 2205*A*d*x*Cos[c + (3*d*x)/2] + 2205*A*d*x*Cos[2*c + (3*d*x)/2] + 735*A*d*x*Cos[2*c + (5*d*x)/2] + 735*A*d*x*Cos[3*c + (5*d*x)/2] + 105*A*d*x*Cos[3*c + (7*d*x)/2] + 105*A*d*x*Cos[4*c + (7*d*x)/2] - 9940*A*Sin[(d*x)/2] + 1260*B*Sin[(d*x)/2] + 560*C*Sin[(d*x)/2] + 8260*A*Sin[c + (d*x)/2] - 1260*B*Sin[c + (d*x)/2] - 350*C*Sin[c + (d*x)/2] - 7140*A*Sin[c + (3*d*x)/2] + 882*B*Sin[c + (3*d*x)/2] + 336*C*Sin[c + (3*d*x)/2] + 3780*A*Sin[2*c + (3*d*x)/2] - 630*B*Sin[2*c + (3*d*x)/2] - 210*C*Sin[2*c + (3*d*x)/2] - 2800*A*Sin[2*c + (5*d*x)/2] + 294*B*Sin[2*c + (5*d*x)/2] + 182*C*Sin[2*c + (5*d*x)/2] + 840*A*Sin[3*c + (5*d*x)/2] - 210*B*Sin[3*c + (5*d*x)/2] - 520*A*Sin[3*c + (7*d*x)/2] + 72*B*Sin[3*c + (7*d*x)/2] + 26*C*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)","B",1
480,1,567,176,2.2702025,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(-7350 d x (4 A-B) \cos \left(c+\frac{d x}{2}\right)-7350 d x (4 A-B) \cos \left(\frac{d x}{2}\right)-46130 A \sin \left(c+\frac{d x}{2}\right)+46116 A \sin \left(c+\frac{3 d x}{2}\right)-18060 A \sin \left(2 c+\frac{3 d x}{2}\right)+19292 A \sin \left(2 c+\frac{5 d x}{2}\right)-2100 A \sin \left(3 c+\frac{5 d x}{2}\right)+3791 A \sin \left(3 c+\frac{7 d x}{2}\right)+735 A \sin \left(4 c+\frac{7 d x}{2}\right)+105 A \sin \left(4 c+\frac{9 d x}{2}\right)+105 A \sin \left(5 c+\frac{9 d x}{2}\right)-17640 A d x \cos \left(c+\frac{3 d x}{2}\right)-17640 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-5880 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-5880 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-840 A d x \cos \left(3 c+\frac{7 d x}{2}\right)-840 A d x \cos \left(4 c+\frac{7 d x}{2}\right)+60830 A \sin \left(\frac{d x}{2}\right)+16520 B \sin \left(c+\frac{d x}{2}\right)-14280 B \sin \left(c+\frac{3 d x}{2}\right)+7560 B \sin \left(2 c+\frac{3 d x}{2}\right)-5600 B \sin \left(2 c+\frac{5 d x}{2}\right)+1680 B \sin \left(3 c+\frac{5 d x}{2}\right)-1040 B \sin \left(3 c+\frac{7 d x}{2}\right)+4410 B d x \cos \left(c+\frac{3 d x}{2}\right)+4410 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+1470 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+1470 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+210 B d x \cos \left(3 c+\frac{7 d x}{2}\right)+210 B d x \cos \left(4 c+\frac{7 d x}{2}\right)-19880 B \sin \left(\frac{d x}{2}\right)-2520 C \sin \left(c+\frac{d x}{2}\right)+1764 C \sin \left(c+\frac{3 d x}{2}\right)-1260 C \sin \left(2 c+\frac{3 d x}{2}\right)+588 C \sin \left(2 c+\frac{5 d x}{2}\right)-420 C \sin \left(3 c+\frac{5 d x}{2}\right)+144 C \sin \left(3 c+\frac{7 d x}{2}\right)+2520 C \sin \left(\frac{d x}{2}\right)\right)}{26880 a^4 d}","\frac{2 (332 A-80 B+3 C) \sin (c+d x)}{105 a^4 d}-\frac{(88 A-25 B-3 C) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{x (4 A-B)}{a^4}-\frac{(12 A-5 B-2 C) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(-7350*(4*A - B)*d*x*Cos[(d*x)/2] - 7350*(4*A - B)*d*x*Cos[c + (d*x)/2] - 17640*A*d*x*Cos[c + (3*d*x)/2] + 4410*B*d*x*Cos[c + (3*d*x)/2] - 17640*A*d*x*Cos[2*c + (3*d*x)/2] + 4410*B*d*x*Cos[2*c + (3*d*x)/2] - 5880*A*d*x*Cos[2*c + (5*d*x)/2] + 1470*B*d*x*Cos[2*c + (5*d*x)/2] - 5880*A*d*x*Cos[3*c + (5*d*x)/2] + 1470*B*d*x*Cos[3*c + (5*d*x)/2] - 840*A*d*x*Cos[3*c + (7*d*x)/2] + 210*B*d*x*Cos[3*c + (7*d*x)/2] - 840*A*d*x*Cos[4*c + (7*d*x)/2] + 210*B*d*x*Cos[4*c + (7*d*x)/2] + 60830*A*Sin[(d*x)/2] - 19880*B*Sin[(d*x)/2] + 2520*C*Sin[(d*x)/2] - 46130*A*Sin[c + (d*x)/2] + 16520*B*Sin[c + (d*x)/2] - 2520*C*Sin[c + (d*x)/2] + 46116*A*Sin[c + (3*d*x)/2] - 14280*B*Sin[c + (3*d*x)/2] + 1764*C*Sin[c + (3*d*x)/2] - 18060*A*Sin[2*c + (3*d*x)/2] + 7560*B*Sin[2*c + (3*d*x)/2] - 1260*C*Sin[2*c + (3*d*x)/2] + 19292*A*Sin[2*c + (5*d*x)/2] - 5600*B*Sin[2*c + (5*d*x)/2] + 588*C*Sin[2*c + (5*d*x)/2] - 2100*A*Sin[3*c + (5*d*x)/2] + 1680*B*Sin[3*c + (5*d*x)/2] - 420*C*Sin[3*c + (5*d*x)/2] + 3791*A*Sin[3*c + (7*d*x)/2] - 1040*B*Sin[3*c + (7*d*x)/2] + 144*C*Sin[3*c + (7*d*x)/2] + 735*A*Sin[4*c + (7*d*x)/2] + 105*A*Sin[4*c + (9*d*x)/2] + 105*A*Sin[5*c + (9*d*x)/2]))/(26880*a^4*d)","B",1
481,1,345,239,5.3847254,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(210 \cos ^7\left(\frac{1}{2} (c+d x)\right) (4 (B-4 A) \sin (c+d x)+2 d x (21 A-8 B+2 C)+A \sin (2 (c+d x)))+4 \tan \left(\frac{c}{2}\right) (447 A-286 B+160 C) \cos ^5\left(\frac{1}{2} (c+d x)\right)-6 \tan \left(\frac{c}{2}\right) (39 A-32 B+25 C) \cos ^3\left(\frac{1}{2} (c+d x)\right)+15 \tan \left(\frac{c}{2}\right) (A-B+C) \cos \left(\frac{1}{2} (c+d x)\right)+15 \sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right)-8 \sec \left(\frac{c}{2}\right) (1653 A-764 B+260 C) \sin \left(\frac{d x}{2}\right) \cos ^6\left(\frac{1}{2} (c+d x)\right)+4 \sec \left(\frac{c}{2}\right) (447 A-286 B+160 C) \sin \left(\frac{d x}{2}\right) \cos ^4\left(\frac{1}{2} (c+d x)\right)-6 \sec \left(\frac{c}{2}\right) (39 A-32 B+25 C) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{105 a^4 d (\cos (c+d x)+1)^4 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{8 (216 A-83 B+20 C) \sin (c+d x)}{105 a^4 d}+\frac{(21 A-8 B+2 C) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B+20 C) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(129 A-52 B+10 C) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{x (21 A-8 B+2 C)}{2 a^4}-\frac{(A-B+C) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}-\frac{(2 A-B) \sin (c+d x) \cos (c+d x)}{5 a d (a \sec (c+d x)+a)^3}",1,"(4*Cos[(c + d*x)/2]*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(15*(A - B + C)*Sec[c/2]*Sin[(d*x)/2] - 6*(39*A - 32*B + 25*C)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + 4*(447*A - 286*B + 160*C)*Cos[(c + d*x)/2]^4*Sec[c/2]*Sin[(d*x)/2] - 8*(1653*A - 764*B + 260*C)*Cos[(c + d*x)/2]^6*Sec[c/2]*Sin[(d*x)/2] + 210*Cos[(c + d*x)/2]^7*(2*(21*A - 8*B + 2*C)*d*x + 4*(-4*A + B)*Sin[c + d*x] + A*Sin[2*(c + d*x)]) + 15*(A - B + C)*Cos[(c + d*x)/2]*Tan[c/2] - 6*(39*A - 32*B + 25*C)*Cos[(c + d*x)/2]^3*Tan[c/2] + 4*(447*A - 286*B + 160*C)*Cos[(c + d*x)/2]^5*Tan[c/2]))/(105*a^4*d*(1 + Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
482,1,185,239,1.7181288,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} ((2871 A+3322 B+3020 C) \cos (c+d x)+13 (99 A+88 B+80 C) \cos (2 (c+d x))+1287 A \cos (3 (c+d x))+198 A \cos (4 (c+d x))+198 A \cos (5 (c+d x))+1089 A+1144 B \cos (3 (c+d x))+176 B \cos (4 (c+d x))+176 B \cos (5 (c+d x))+968 B+1040 C \cos (3 (c+d x))+160 C \cos (4 (c+d x))+160 C \cos (5 (c+d x))+1510 C)}{3465 d}","\frac{2 a (99 A+88 B+80 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (99 A+88 B+80 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 a d}-\frac{8 (99 A+88 B+80 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{4 a (99 A+88 B+80 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (11 B+C) \tan (c+d x) \sec ^4(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}",1,"((1089*A + 968*B + 1510*C + (2871*A + 3322*B + 3020*C)*Cos[c + d*x] + 13*(99*A + 88*B + 80*C)*Cos[2*(c + d*x)] + 1287*A*Cos[3*(c + d*x)] + 1144*B*Cos[3*(c + d*x)] + 1040*C*Cos[3*(c + d*x)] + 198*A*Cos[4*(c + d*x)] + 176*B*Cos[4*(c + d*x)] + 160*C*Cos[4*(c + d*x)] + 198*A*Cos[5*(c + d*x)] + 176*B*Cos[5*(c + d*x)] + 160*C*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3465*d)","A",1
483,1,153,193,1.8502661,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} (2 (63 A+99 B+88 C) \cos (c+d x)+11 (21 A+18 B+16 C) \cos (2 (c+d x))+42 A \cos (3 (c+d x))+42 A \cos (4 (c+d x))+189 A+36 B \cos (3 (c+d x))+36 B \cos (4 (c+d x))+162 B+32 C \cos (3 (c+d x))+32 C \cos (4 (c+d x))+214 C)}{315 d}","\frac{2 (21 A+18 B+16 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac{4 (21 A+18 B+16 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a (21 A+18 B+16 C) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (9 B+C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}",1,"((189*A + 162*B + 214*C + 2*(63*A + 99*B + 88*C)*Cos[c + d*x] + 11*(21*A + 18*B + 16*C)*Cos[2*(c + d*x)] + 42*A*Cos[3*(c + d*x)] + 36*B*Cos[3*(c + d*x)] + 32*C*Cos[3*(c + d*x)] + 42*A*Cos[4*(c + d*x)] + 36*B*Cos[4*(c + d*x)] + 32*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(315*d)","A",1
484,1,119,147,1.3431779,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} (3 (35 A+42 B+36 C) \cos (c+d x)+(35 A+28 B+24 C) \cos (2 (c+d x))+35 A \cos (3 (c+d x))+35 A+28 B \cos (3 (c+d x))+28 B+24 C \cos (3 (c+d x))+54 C)}{105 d}","\frac{2 (35 A-14 B+18 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (35 A+49 B+27 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 B+C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{7 d}",1,"((35*A + 28*B + 54*C + 3*(35*A + 42*B + 36*C)*Cos[c + d*x] + (35*A + 28*B + 24*C)*Cos[2*(c + d*x)] + 35*A*Cos[3*(c + d*x)] + 28*B*Cos[3*(c + d*x)] + 24*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(105*d)","A",1
485,1,83,104,0.8445394,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} ((15 A+10 B+8 C) \cos (2 (c+d x))+15 A+2 (5 B+4 C) \cos (c+d x)+10 B+14 C)}{15 d}","\frac{2 a (15 A+5 B+7 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (5 B-2 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}",1,"((15*A + 10*B + 14*C + 2*(5*B + 4*C)*Cos[c + d*x] + (15*A + 10*B + 8*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d)","A",1
486,1,101,100,0.8338946,"\int \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((3 B+2 C) \cos (c+d x)+C)\right)}{3 d}","\frac{2 \sqrt{a} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a (3 B+C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(C + (3*B + 2*C)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
487,1,94,98,0.4081373,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (A+2 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (A \cos (c+d x)+2 C)\right)}{2 d}","\frac{\sqrt{a} (A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a (A-2 C) \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(A + 2*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*C + A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
488,1,113,117,0.5046218,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (3 A+4 B+8 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 A \cos (c+d x)+3 A+4 B)\right)}{8 d}","\frac{\sqrt{a} (3 A+4 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a (A+4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(3*A + 4*B + 8*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(3*A + 4*B + 2*A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
489,1,152,163,0.4956284,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 A \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+2 B \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)+C \left(\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{a (5 A+6 B+8 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a (A+6 B) \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"((C*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]]) + 2*B*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 2*A*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","C",1
490,1,90,209,0.2512838,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(A \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+B \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+C \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)\right)}{d}","\frac{a (35 A+40 B+48 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (35 A+40 B+48 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a (35 A+40 B+48 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a (A+8 B) \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"(2*(C*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]] + B*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]] + A*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
491,1,185,243,2.0529351,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} ((12441 A+12386 B+12684 C) \cos (c+d x)+(4422 A+4862 B+4368 C) \cos (2 (c+d x))+5577 A \cos (3 (c+d x))+858 A \cos (4 (c+d x))+858 A \cos (5 (c+d x))+3564 A+4862 B \cos (3 (c+d x))+748 B \cos (4 (c+d x))+748 B \cos (5 (c+d x))+4114 B+4368 C \cos (3 (c+d x))+672 C \cos (4 (c+d x))+672 C \cos (5 (c+d x))+4956 C)}{6930 d}","\frac{2 a^2 (99 A+110 B+84 C) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (429 A+374 B+336 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (429 A+374 B+336 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}-\frac{4 a (429 A+374 B+336 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{2 a (11 B+3 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}",1,"(a*(3564*A + 4114*B + 4956*C + (12441*A + 12386*B + 12684*C)*Cos[c + d*x] + (4422*A + 4862*B + 4368*C)*Cos[2*(c + d*x)] + 5577*A*Cos[3*(c + d*x)] + 4862*B*Cos[3*(c + d*x)] + 4368*C*Cos[3*(c + d*x)] + 858*A*Cos[4*(c + d*x)] + 748*B*Cos[4*(c + d*x)] + 672*C*Cos[4*(c + d*x)] + 858*A*Cos[5*(c + d*x)] + 748*B*Cos[5*(c + d*x)] + 672*C*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(6930*d)","A",1
492,1,152,187,2.1724595,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} ((567 A+648 B+748 C) \cos (c+d x)+(882 A+858 B+748 C) \cos (2 (c+d x))+189 A \cos (3 (c+d x))+189 A \cos (4 (c+d x))+693 A+156 B \cos (3 (c+d x))+156 B \cos (4 (c+d x))+702 B+136 C \cos (3 (c+d x))+136 C \cos (4 (c+d x))+752 C)}{630 d}","\frac{8 a^2 (63 A+57 B+47 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (63 A-18 B+22 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{315 d}+\frac{2 a (63 A+57 B+47 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 (3 B+C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{21 a d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}",1,"(a*(693*A + 702*B + 752*C + (567*A + 648*B + 748*C)*Cos[c + d*x] + (882*A + 858*B + 748*C)*Cos[2*(c + d*x)] + 189*A*Cos[3*(c + d*x)] + 156*B*Cos[3*(c + d*x)] + 136*C*Cos[3*(c + d*x)] + 189*A*Cos[4*(c + d*x)] + 156*B*Cos[4*(c + d*x)] + 136*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(630*d)","A",1
493,1,120,144,1.6470604,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} ((525 A+462 B+468 C) \cos (c+d x)+2 (35 A+63 B+52 C) \cos (2 (c+d x))+175 A \cos (3 (c+d x))+70 A+126 B \cos (3 (c+d x))+126 B+104 C \cos (3 (c+d x))+164 C)}{210 d}","\frac{8 a^2 (35 A+21 B+19 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (35 A+21 B+19 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 (7 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}",1,"(a*(70*A + 126*B + 164*C + (525*A + 462*B + 468*C)*Cos[c + d*x] + 2*(35*A + 63*B + 52*C)*Cos[2*(c + d*x)] + 175*A*Cos[3*(c + d*x)] + 126*B*Cos[3*(c + d*x)] + 104*C*Cos[3*(c + d*x)])*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d)","A",1
494,1,132,142,1.389096,"\int (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((15 A+25 B+18 C) \cos (2 (c+d x))+15 A+2 (5 B+9 C) \cos (c+d x)+25 B+24 C)+30 \sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{30 d}","\frac{2 a^{3/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (15 A+20 B+12 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (5 B+3 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*(30*Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(15*A + 25*B + 24*C + 2*(5*B + 9*C)*Cos[c + d*x] + (15*A + 25*B + 18*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(30*d)","A",1
495,1,115,144,2.1303649,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)-1} (\sec (c+d x) (3 A \cos (2 (c+d x))+3 A+4 C)+4 (3 B+5 C))+6 (3 A+2 B) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{6 d \sqrt{\sec (c+d x)-1}}","\frac{a^{3/2} (3 A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}-\frac{a^2 (3 A-6 B-8 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{a (3 A-2 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{d}",1,"(a*Sqrt[a*(1 + Sec[c + d*x])]*(6*(3*A + 2*B)*ArcTan[Sqrt[-1 + Sec[c + d*x]]] + Sqrt[-1 + Sec[c + d*x]]*(4*(3*B + 5*C) + (3*A + 4*C + 3*A*Cos[2*(c + d*x)])*Sec[c + d*x]))*Tan[(c + d*x)/2])/(6*d*Sqrt[-1 + Sec[c + d*x]])","A",1
496,1,117,157,0.7795215,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (7 A+12 B+8 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) ((7 A+4 B) \cos (c+d x)+A \cos (2 (c+d x))+A+8 C)\right)}{8 d}","\frac{a^{3/2} (7 A+12 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (5 A+4 B-8 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}-\frac{a (A-4 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(7*A + 12*B + 8*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(A + 8*C + (7*A + 4*B)*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
497,1,124,165,1.6530999,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\cos (c+d x) \sqrt{\sec (c+d x)-1} (2 (11 A+6 B) \cos (c+d x)+4 A \cos (2 (c+d x))+37 A+42 B+24 C)+(33 A+42 B+72 C) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{24 d \sqrt{\sec (c+d x)-1}}","\frac{a^{3/2} (11 A+14 B+24 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (19 A+30 B+24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a (A+2 B) \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a*((33*A + 42*B + 72*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]] + Cos[c + d*x]*(37*A + 42*B + 24*C + 2*(11*A + 6*B)*Cos[c + d*x] + 4*A*Cos[2*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(24*d*Sqrt[-1 + Sec[c + d*x]])","A",1
498,1,157,215,1.746244,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (75 A+88 B+112 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (93 A+88 B+48 C) \cos (c+d x)+4 (15 A+8 B) \cos (2 (c+d x))+12 A \cos (3 (c+d x))+285 A+296 B+336 C)\right)}{384 d}","\frac{a^{3/2} (75 A+88 B+112 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (75 A+88 B+112 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (39 A+56 B+48 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a (3 A+8 B) \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(75*A + 88*B + 112*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (285*A + 296*B + 336*C + 2*(93*A + 88*B + 48*C)*Cos[c + d*x] + 4*(15*A + 8*B)*Cos[2*(c + d*x)] + 12*A*Cos[3*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(384*d)","A",1
499,1,182,263,2.7326648,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(15 \sqrt{2} (133 A+150 B+176 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (1007 A+930 B+880 C) \cos (c+d x)+4 (181 A+150 B+80 C) \cos (2 (c+d x))+228 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+2671 A+120 B \cos (3 (c+d x))+2850 B+2960 C)\right)}{3840 d}","\frac{a^{3/2} (133 A+150 B+176 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (67 A+90 B+80 C) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (133 A+150 B+176 C) \sin (c+d x) \cos (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a (3 A+10 B) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(133*A + 150*B + 176*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (2671*A + 2850*B + 2960*C + 2*(1007*A + 930*B + 880*C)*Cos[c + d*x] + 4*(181*A + 150*B + 80*C)*Cos[2*(c + d*x)] + 228*A*Cos[3*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
500,1,222,294,2.18048,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \sqrt{a (\sec (c+d x)+1)} (70 (4576 A+5083 B+5552 C) \cos (c+d x)+14 (32747 A+31850 B+30334 C) \cos (2 (c+d x))+141570 A \cos (3 (c+d x))+156585 A \cos (4 (c+d x))+20878 A \cos (5 (c+d x))+20878 A \cos (6 (c+d x))+322751 A+138450 B \cos (3 (c+d x))+138450 B \cos (4 (c+d x))+18460 B \cos (5 (c+d x))+18460 B \cos (6 (c+d x))+325910 B+125520 C \cos (3 (c+d x))+125520 C \cos (4 (c+d x))+16736 C \cos (5 (c+d x))+16736 C \cos (6 (c+d x))+343612 C)}{180180 d}","\frac{2 a^3 (2717 A+2522 B+2224 C) \tan (c+d x) \sec ^3(c+d x)}{9009 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (10439 A+9230 B+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (143 A+182 B+136 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}-\frac{4 a^2 (10439 A+9230 B+8368 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{45045 d}+\frac{2 a (10439 A+9230 B+8368 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{15015 d}+\frac{2 a (13 B+5 C) \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d}",1,"(a^2*(322751*A + 325910*B + 343612*C + 70*(4576*A + 5083*B + 5552*C)*Cos[c + d*x] + 14*(32747*A + 31850*B + 30334*C)*Cos[2*(c + d*x)] + 141570*A*Cos[3*(c + d*x)] + 138450*B*Cos[3*(c + d*x)] + 125520*C*Cos[3*(c + d*x)] + 156585*A*Cos[4*(c + d*x)] + 138450*B*Cos[4*(c + d*x)] + 125520*C*Cos[4*(c + d*x)] + 20878*A*Cos[5*(c + d*x)] + 18460*B*Cos[5*(c + d*x)] + 16736*C*Cos[5*(c + d*x)] + 20878*A*Cos[6*(c + d*x)] + 18460*B*Cos[6*(c + d*x)] + 16736*C*Cos[6*(c + d*x)])*Sec[c + d*x]^6*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(180180*d)","A",1
501,1,188,229,1.6651053,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \sqrt{a (\sec (c+d x)+1)} ((49830 A+49654 B+50140 C) \cos (c+d x)+4 (4290 A+4642 B+4615 C) \cos (2 (c+d x))+22935 A \cos (3 (c+d x))+3795 A \cos (4 (c+d x))+3795 A \cos (5 (c+d x))+13365 A+20878 B \cos (3 (c+d x))+3212 B \cos (4 (c+d x))+3212 B \cos (5 (c+d x))+15356 B+18460 C \cos (3 (c+d x))+2840 C \cos (4 (c+d x))+2840 C \cos (5 (c+d x))+18140 C)}{13860 d}","\frac{64 a^3 (165 A+143 B+125 C) \tan (c+d x)}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (165 A+143 B+125 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{2 (99 A-22 B+26 C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{693 d}+\frac{2 a (165 A+143 B+125 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}+\frac{2 (11 B+5 C) \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{99 a d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d}",1,"(a^2*(13365*A + 15356*B + 18140*C + (49830*A + 49654*B + 50140*C)*Cos[c + d*x] + 4*(4290*A + 4642*B + 4615*C)*Cos[2*(c + d*x)] + 22935*A*Cos[3*(c + d*x)] + 20878*B*Cos[3*(c + d*x)] + 18460*C*Cos[3*(c + d*x)] + 3795*A*Cos[4*(c + d*x)] + 3212*B*Cos[4*(c + d*x)] + 2840*C*Cos[4*(c + d*x)] + 3795*A*Cos[5*(c + d*x)] + 3212*B*Cos[5*(c + d*x)] + 2840*C*Cos[5*(c + d*x)])*Sec[c + d*x]^5*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(13860*d)","A",1
502,1,156,184,2.4529395,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \sqrt{a (\sec (c+d x)+1)} (2 (882 A+1215 B+1396 C) \cos (c+d x)+4 (966 A+870 B+803 C) \cos (2 (c+d x))+588 A \cos (3 (c+d x))+903 A \cos (4 (c+d x))+2961 A+690 B \cos (3 (c+d x))+690 B \cos (4 (c+d x))+2790 B+584 C \cos (3 (c+d x))+584 C \cos (4 (c+d x))+2908 C)}{1260 d}","\frac{64 a^3 (21 A+15 B+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (21 A+15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a (21 A+15 B+13 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 (9 B-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}",1,"(a^2*(2961*A + 2790*B + 2908*C + 2*(882*A + 1215*B + 1396*C)*Cos[c + d*x] + 4*(966*A + 870*B + 803*C)*Cos[2*(c + d*x)] + 588*A*Cos[3*(c + d*x)] + 690*B*Cos[3*(c + d*x)] + 584*C*Cos[3*(c + d*x)] + 903*A*Cos[4*(c + d*x)] + 690*B*Cos[4*(c + d*x)] + 584*C*Cos[4*(c + d*x)])*Sec[c + d*x]^4*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d)","A",1
503,1,170,182,2.4437198,"\int (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((840 A+987 B+930 C) \cos (c+d x)+2 (35 A+98 B+115 C) \cos (2 (c+d x))+280 A \cos (3 (c+d x))+70 A+301 B \cos (3 (c+d x))+196 B+230 C \cos (3 (c+d x))+290 C)+420 \sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{7}{2}}(c+d x)\right)}{420 d}","\frac{2 a^{5/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (245 A+224 B+160 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (35 A+56 B+40 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (7 B+5 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^3*Sqrt[a*(1 + Sec[c + d*x])]*(420*Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(7/2) + 2*(70*A + 196*B + 290*C + (840*A + 987*B + 930*C)*Cos[c + d*x] + 2*(35*A + 98*B + 115*C)*Cos[2*(c + d*x)] + 280*A*Cos[3*(c + d*x)] + 301*B*Cos[3*(c + d*x)] + 230*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(420*d)","A",1
504,1,209,184,2.0089617,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos (c+d x) (a (\sec (c+d x)+1))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{\cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) ((45 A+40 B+112 C) \cos (c+d x)+4 (15 A+40 B+43 C) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+60 A+160 B+196 C)}{(\cos (c+d x)+1)^2}+\frac{60 (5 A+2 B) \sin (c+d x) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)}{\sqrt{\sec (c+d x)-1} (\sec (c+d x)+1)^3}\right)}{30 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a^{5/2} (5 A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 (15 A+70 B+64 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (15 A-10 B-16 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}-\frac{a (5 A-2 C) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{d}",1,"(Cos[c + d*x]*(a*(1 + Sec[c + d*x]))^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((60*(5*A + 2*B)*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Sin[c + d*x])/(Sqrt[-1 + Sec[c + d*x]]*(1 + Sec[c + d*x])^3) + (Cos[c + d*x]*(60*A + 160*B + 196*C + (45*A + 40*B + 112*C)*Cos[c + d*x] + 4*(15*A + 40*B + 43*C)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(1 + Cos[c + d*x])^2))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
505,1,152,197,1.2063669,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(6 \sqrt{2} (19 A+20 B+8 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+2 \sin \left(\frac{1}{2} (c+d x)\right) ((9 A+48 B+128 C) \cos (c+d x)+3 (11 A+4 B) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+33 A+12 B+16 C)\right)}{48 d}","\frac{a^{5/2} (19 A+20 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (27 A-12 B-56 C) \sin (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (A-4 B-8 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}-\frac{a (3 A-4 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{6 d}+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{5/2}}{2 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(19*A + 20*B + 8*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(33*A + 12*B + 16*C + (9*A + 48*B + 128*C)*Cos[c + d*x] + 3*(11*A + 4*B)*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
506,1,140,207,1.8047497,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{\sec (c+d x)-1} (3 (27 A+22 B+8 C) \cos (c+d x)+(17 A+6 B) \cos (2 (c+d x))+2 A \cos (3 (c+d x))+17 A+6 B+48 C)+3 (25 A+38 B+40 C) \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{24 d \sqrt{\sec (c+d x)-1}}","\frac{a^{5/2} (25 A+38 B+40 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (49 A+54 B-24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (3 A+2 B-8 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a (5 A+6 B) \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d}",1,"(a^2*(3*(25*A + 38*B + 40*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]] + (17*A + 6*B + 48*C + 3*(27*A + 22*B + 8*C)*Cos[c + d*x] + (17*A + 6*B)*Cos[2*(c + d*x)] + 2*A*Cos[3*(c + d*x)])*Sqrt[-1 + Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(24*d*Sqrt[-1 + Sec[c + d*x]])","A",1
507,1,156,215,2.3326371,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (163 A+200 B+304 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} ((362 A+272 B+96 C) \cos (c+d x)+4 (23 A+8 B) \cos (2 (c+d x))+12 A \cos (3 (c+d x))+581 A+632 B+528 C)\right)}{384 d}","\frac{a^{5/2} (163 A+200 B+304 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (299 A+392 B+432 C) \sin (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (17 A+24 B+16 C) \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d}+\frac{a (5 A+8 B) \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(163*A + 200*B + 304*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(581*A + 632*B + 528*C + (362*A + 272*B + 96*C)*Cos[c + d*x] + 4*(23*A + 8*B)*Cos[2*(c + d*x)] + 12*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(384*d)","A",1
508,1,183,261,2.6282398,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(15 \sqrt{2} (283 A+326 B+400 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) ((3874 A+3620 B+2720 C) \cos (c+d x)+4 (331 A+230 B+80 C) \cos (2 (c+d x))+348 A \cos (3 (c+d x))+48 A \cos (4 (c+d x))+5521 A+120 B \cos (3 (c+d x))+5810 B+6320 C)\right)}{3840 d}","\frac{a^{5/2} (283 A+326 B+400 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (787 A+950 B+1040 C) \sin (c+d x) \cos (c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (79 A+110 B+80 C) \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{240 d}+\frac{a (A+2 B) \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(283*A + 326*B + 400*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (5521*A + 5810*B + 6320*C + (3874*A + 3620*B + 2720*C)*Cos[c + d*x] + 4*(331*A + 230*B + 80*C)*Cos[2*(c + d*x)] + 348*A*Cos[3*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 48*A*Cos[4*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3840*d)","A",1
509,1,217,311,3.4114775,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(15 \sqrt{2} (1015 A+1132 B+1304 C) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+\left(\sin \left(\frac{3}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (2 (8085 A+7748 B+7240 C) \cos (c+d x)+4 (1575 A+1324 B+920 C) \cos (2 (c+d x))+2140 A \cos (3 (c+d x))+560 A \cos (4 (c+d x))+80 A \cos (5 (c+d x))+20965 A+1392 B \cos (3 (c+d x))+192 B \cos (4 (c+d x))+22084 B+480 C \cos (3 (c+d x))+23240 C)\right)}{15360 d}","\frac{a^{5/2} (1015 A+1132 B+1304 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (545 A+628 B+680 C) \sin (c+d x) \cos ^2(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1015 A+1132 B+1304 C) \sin (c+d x) \cos (c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (115 A+156 B+120 C) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{480 d}+\frac{a (5 A+12 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(1015*A + 1132*B + 1304*C)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + (20965*A + 22084*B + 23240*C + 2*(8085*A + 7748*B + 7240*C)*Cos[c + d*x] + 4*(1575*A + 1324*B + 920*C)*Cos[2*(c + d*x)] + 2140*A*Cos[3*(c + d*x)] + 1392*B*Cos[3*(c + d*x)] + 480*C*Cos[3*(c + d*x)] + 560*A*Cos[4*(c + d*x)] + 192*B*Cos[4*(c + d*x)] + 80*A*Cos[5*(c + d*x)])*(-Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(15360*d)","A",1
510,1,7186,254,30.2649886,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{2 (21 A-3 B+19 C) \tan (c+d x) \sec ^2(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (21 A-93 B+29 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 a d}+\frac{4 (147 A-111 B+143 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (9 B-C) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}",1,"Result too large to show","C",0
511,1,2322,208,11.3771575,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (35 A-7 B+31 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 a d}-\frac{4 (35 A-49 B+37 C) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 B-C) \tan (c+d x) \sec ^2(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"(4*Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*(-1/3*(A*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^(7/2) + (2*B*Sin[(c + d*x)/2])/(7*(1 - 2*Sin[(c + d*x)/2]^2)^(7/2)) + ((A - B + C)*Csc[(c + d*x)/2]^9*(363825*Sin[(c + d*x)/2]^2 - 4729725*Sin[(c + d*x)/2]^4 + 26785605*Sin[(c + d*x)/2]^6 - 86790165*Sin[(c + d*x)/2]^8 + 177677808*Sin[(c + d*x)/2]^10 - 239283044*Sin[(c + d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12 + 213120160*Sin[(c + d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^14 - 121497024*Sin[(c + d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^16 + 40125184*Sin[(c + d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^18 - 5840384*Sin[(c + d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^2*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^4*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^6*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^8*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^10*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^12*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^14*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^16*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^18*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^20*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12*(-6 + 5*Sin[(c + d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12*(103 - 164*Sin[(c + d*x)/2]^2 + 70*Sin[(c + d*x)/2]^4)))/(40425*(1 - 2*Sin[(c + d*x)/2]^2)^(9/2)*(-1 + 2*Sin[(c + d*x)/2]^2)) + (4*B*((3*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^(5/2) + 4*(Sin[(c + d*x)/2]/(1 - 2*Sin[(c + d*x)/2]^2)^(3/2) + (2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2])))/35 + (A*((5*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^(7/2) + 2*((3*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^(5/2) + 4*(Sin[(c + d*x)/2]/(1 - 2*Sin[(c + d*x)/2]^2)^(3/2) + (2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]))))/105))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","C",0
512,1,1666,164,8.3149196,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)} \left(-\frac{(A-B+C) \left(440 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right) \sin ^{16}\left(\frac{1}{2} (c+d x)\right)+69120 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^{16}\left(\frac{1}{2} (c+d x)\right)-42048 \sin ^{16}\left(\frac{1}{2} (c+d x)\right)-1500 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right) \sin ^{14}\left(\frac{1}{2} (c+d x)\right)-414720 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^{14}\left(\frac{1}{2} (c+d x)\right)+226656 \sin ^{14}\left(\frac{1}{2} (c+d x)\right)+1770 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right) \sin ^{12}\left(\frac{1}{2} (c+d x)\right)+1080000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^{12}\left(\frac{1}{2} (c+d x)\right)-518760 \sin ^{12}\left(\frac{1}{2} (c+d x)\right)-710 \, _2F_1\left(2,\frac{9}{2};\frac{11}{2};\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right) \sin ^{10}\left(\frac{1}{2} (c+d x)\right)-40 \cos ^6\left(\frac{1}{2} (c+d x)\right) \, _4F_3\left(2,2,2,\frac{9}{2};1,1,\frac{11}{2};\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right) \sin ^{10}\left(\frac{1}{2} (c+d x)\right)+60 \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{9}{2};1,\frac{11}{2};\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right) \left(4 \sin ^2\left(\frac{1}{2} (c+d x)\right)-5\right) \sin ^{10}\left(\frac{1}{2} (c+d x)\right)-1598400 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^{10}\left(\frac{1}{2} (c+d x)\right)+655812 \sin ^{10}\left(\frac{1}{2} (c+d x)\right)+1458000 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^8\left(\frac{1}{2} (c+d x)\right)-486630 \sin ^8\left(\frac{1}{2} (c+d x)\right)-833760 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^6\left(\frac{1}{2} (c+d x)\right)+210105 \sin ^6\left(\frac{1}{2} (c+d x)\right)+291060 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^4\left(\frac{1}{2} (c+d x)\right)-48825 \sin ^4\left(\frac{1}{2} (c+d x)\right)-56700 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}} \sin ^2\left(\frac{1}{2} (c+d x)\right)+4725 \sin ^2\left(\frac{1}{2} (c+d x)\right)+4725 \tanh ^{-1}\left(\sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \sqrt{\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1}}\right) \csc ^7\left(\frac{1}{2} (c+d x)\right)}{675 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{7/2} \left(2 \sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right)}+\frac{8}{15} B \left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2}}\right)+\frac{1}{30} A \left(\frac{3 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{5/2}}+4 \left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}+\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2}}\right)\right)-\frac{A \sin \left(\frac{1}{2} (c+d x)\right)}{2 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{5/2}}+\frac{2 B \sin \left(\frac{1}{2} (c+d x)\right)}{5 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{5/2}}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (15 A-10 B+14 C) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (5 B-C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 a d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(4*Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*(-1/2*(A*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^(5/2) + (2*B*Sin[(c + d*x)/2])/(5*(1 - 2*Sin[(c + d*x)/2]^2)^(5/2)) + (8*B*(Sin[(c + d*x)/2]/(1 - 2*Sin[(c + d*x)/2]^2)^(3/2) + (2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]))/15 - ((A - B + C)*Csc[(c + d*x)/2]^7*(4725*Sin[(c + d*x)/2]^2 - 48825*Sin[(c + d*x)/2]^4 + 210105*Sin[(c + d*x)/2]^6 - 486630*Sin[(c + d*x)/2]^8 + 655812*Sin[(c + d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^10 - 518760*Sin[(c + d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12 + 226656*Sin[(c + d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^14 - 42048*Sin[(c + d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^2*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^4*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^6*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^8*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^10*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^12*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^14*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^16*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^10*(-5 + 4*Sin[(c + d*x)/2]^2)))/(675*(1 - 2*Sin[(c + d*x)/2]^2)^(7/2)*(-1 + 2*Sin[(c + d*x)/2]^2)) + (A*((3*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^(5/2) + 4*(Sin[(c + d*x)/2]/(1 - 2*Sin[(c + d*x)/2]^2)^(3/2) + (2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2])))/30))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","C",0
513,1,628,118,7.10462,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{4 \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)} \cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{(A-B+C) \csc ^5\left(\frac{1}{2} (c+d x)\right) \left(-12 \sin ^8\left(\frac{1}{2} (c+d x)\right) \cos ^4\left(\frac{1}{2} (c+d x)\right) \, _3F_2\left(2,2,\frac{7}{2};1,\frac{9}{2};-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)-12 \left(3 \sin ^4\left(\frac{1}{2} (c+d x)\right)-7 \sin ^2\left(\frac{1}{2} (c+d x)\right)+4\right) \sin ^8\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(2,\frac{7}{2};\frac{9}{2};-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)+7 \sqrt{-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}} \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^3 \left(8 \sin ^4\left(\frac{1}{2} (c+d x)\right)-20 \sin ^2\left(\frac{1}{2} (c+d x)\right)+15\right) \left(\left(3-7 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right) \sqrt{-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}-3 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right) \tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}\right)\right)\right)}{63 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{7/2}}-\frac{4 A \sin ^3\left(\frac{1}{2} (c+d x)\right)}{3 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2}}+\frac{4 B \sin \left(\frac{1}{2} (c+d x)\right)}{3 \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}+\frac{2 B \sin \left(\frac{1}{2} (c+d x)\right)}{3 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2}}\right)}{d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 (3 B-2 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 a d}",1,"(4*Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*((2*B*Sin[(c + d*x)/2])/(3*(1 - 2*Sin[(c + d*x)/2]^2)^(3/2)) - (4*A*Sin[(c + d*x)/2]^3)/(3*(1 - 2*Sin[(c + d*x)/2]^2)^(3/2)) + (4*B*Sin[(c + d*x)/2])/(3*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]) + ((A - B + C)*Csc[(c + d*x)/2]^5*(-12*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 7/2}, {1, 9/2}, -(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))]*Sin[(c + d*x)/2]^8 - 12*Hypergeometric2F1[2, 7/2, 9/2, -(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))]*Sin[(c + d*x)/2]^8*(4 - 7*Sin[(c + d*x)/2]^2 + 3*Sin[(c + d*x)/2]^4) + 7*Sqrt[-(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))]*(1 - 2*Sin[(c + d*x)/2]^2)^3*(15 - 20*Sin[(c + d*x)/2]^2 + 8*Sin[(c + d*x)/2]^4)*((3 - 7*Sin[(c + d*x)/2]^2)*Sqrt[-(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))] - 3*ArcTanh[Sqrt[-(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))]]*(1 - 2*Sin[(c + d*x)/2]^2))))/(63*(1 - 2*Sin[(c + d*x)/2]^2)^(7/2))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","C",0
514,1,123,118,1.44343,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(-(A-B+C) \sqrt{\cos (c+d x)} \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+\sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 C \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]*(Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] - (A - B + C)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]]*Sqrt[Cos[c + d*x]] + 2*C*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
515,1,120,120,1.122293,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \left(-\sqrt{2} (A-B+C) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)+(A-2 B) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)+A (\cos (c+d x)-1)\right)}{d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{A \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"((A*(-1 + Cos[c + d*x]) + (A - 2*B)*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Sqrt[-1 + Sec[c + d*x]] - Sqrt[2]*(A - B + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]])*Sin[c + d*x])/(d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
516,1,16835,169,27.586516,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{(7 A-4 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"Result too large to show","C",0
517,1,161,213,0.9005387,"\int \frac{\cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} \left(-2 (A-6 B) \cos (c+d x)+3 (7 A-2 B+8 C)+8 A \cos ^2(c+d x)\right)-3 (9 A-14 B+8 C) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+24 \sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{24 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(7 A-2 B+8 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{(9 A-14 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(A-6 B) \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"((-3*(9*A - 14*B + 8*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 24*Sqrt[2]*(A - B + C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(3*(7*A - 2*B + 8*C) - 2*(A - 6*B)*Cos[c + d*x] + 8*A*Cos[c + d*x]^2)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(24*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
518,1,178,259,1.1017685,"\int \frac{\cos ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} \left(2 (43 A-8 B+48 C) \cos (c+d x)-8 (A-8 B) \cos ^2(c+d x)+48 A \cos ^3(c+d x)-63 A+168 B-48 C\right)+3 (107 A-72 B+112 C) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-192 \sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{192 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{(21 A-56 B+16 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{(107 A-72 B+112 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 \sqrt{a} d}-\frac{\sqrt{2} (A-B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(43 A-8 B+48 C) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-8 B) \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}",1,"((3*(107*A - 72*B + 112*C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 192*Sqrt[2]*(A - B + C)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-63*A + 168*B - 48*C + 2*(43*A - 8*B + 48*C)*Cos[c + d*x] - 8*(A - 8*B)*Cos[c + d*x]^2 + 48*A*Cos[c + d*x]^3)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(192*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
519,1,2746,277,11.5242881,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{(11 A-15 B+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(245 A-273 B+397 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{210 a^2 d}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(7 A-7 B+11 C) \tan (c+d x) \sec ^3(c+d x)}{14 a d \sqrt{a \sec (c+d x)+a}}-\frac{(35 A-63 B+67 C) \tan (c+d x) \sec ^2(c+d x)}{70 a d \sqrt{a \sec (c+d x)+a}}-\frac{(455 A-651 B+799 C) \tan (c+d x)}{105 a d \sqrt{a \sec (c+d x)+a}}",1,"(4*Cos[(c + d*x)/2]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*((4*A*Sin[(c + d*x)/2])/(7*(1 - 2*Sin[(c + d*x)/2]^2)^(7/2)) - ((A - B + C)*(1 - 2*Sin[(c + d*x)/2]))/(28*(1 + Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(7/2)) + ((A - B + C)*(1 + 2*Sin[(c + d*x)/2]))/(28*(1 - Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(7/2)) - ((A - B + C)*(315*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]] + (5 + 3*Sin[(c + d*x)/2])/((1 - Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(5/2)) - (11 + 17*Sin[(c + d*x)/2])/((1 - Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(3/2)) + (61 + 71*Sin[(c + d*x)/2])/((1 - Sin[(c + d*x)/2])*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[(c + d*x)/2]^2])/(1 - Sin[(c + d*x)/2])))/70 + ((A - B + C)*(315*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]] + (5 - 3*Sin[(c + d*x)/2])/((1 + Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(5/2)) - (11 - 17*Sin[(c + d*x)/2])/((1 + Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(3/2)) + (61 - 71*Sin[(c + d*x)/2])/((1 + Sin[(c + d*x)/2])*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]) + (193*Sqrt[1 - 2*Sin[(c + d*x)/2]^2])/(1 + Sin[(c + d*x)/2])))/70 - ((7*A - 3*B - C)*Csc[(c + d*x)/2]^9*(363825*Sin[(c + d*x)/2]^2 - 4729725*Sin[(c + d*x)/2]^4 + 26785605*Sin[(c + d*x)/2]^6 - 86790165*Sin[(c + d*x)/2]^8 + 177677808*Sin[(c + d*x)/2]^10 - 239283044*Sin[(c + d*x)/2]^12 + 52080*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12 + 213120160*Sin[(c + d*x)/2]^14 - 168280*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^14 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^14 - 121497024*Sin[(c + d*x)/2]^16 + 212520*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^16 + 3360*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^16 + 40125184*Sin[(c + d*x)/2]^18 - 124320*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^18 - 2240*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^18 - 5840384*Sin[(c + d*x)/2]^20 + 28000*Hypergeometric2F1[2, 11/2, 13/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^20 + 560*HypergeometricPFQ[{2, 2, 2, 2, 11/2}, {1, 1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^20 + 363825*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 5336100*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^2*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 34636140*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^4*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 131060160*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^6*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 320535600*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^8*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 530671680*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^10*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 604296000*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^12*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 468948480*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^14*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 237726720*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^16*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 70963200*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^18*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 9461760*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^20*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 1120*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 11/2}, {1, 1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12*(-6 + 5*Sin[(c + d*x)/2]^2) + 280*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 11/2}, {1, 13/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12*(103 - 164*Sin[(c + d*x)/2]^2 + 70*Sin[(c + d*x)/2]^4)))/(80850*(1 - 2*Sin[(c + d*x)/2]^2)^(9/2)*(-1 + 2*Sin[(c + d*x)/2]^2)) + (8*A*((3*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^(5/2) + 4*(Sin[(c + d*x)/2]/(1 - 2*Sin[(c + d*x)/2]^2)^(3/2) + (2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2])))/35))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","C",0
520,1,2025,229,8.4797274,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(7 A-11 B+15 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(15 A-35 B+39 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{30 a^2 d}-\frac{(A-B+C) \tan (c+d x) \sec ^3(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(5 A-5 B+9 C) \tan (c+d x) \sec ^2(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{(45 A-65 B+93 C) \tan (c+d x)}{15 a d \sqrt{a \sec (c+d x)+a}}",1,"(4*Cos[(c + d*x)/2]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*((4*A*Sin[(c + d*x)/2])/(5*(1 - 2*Sin[(c + d*x)/2]^2)^(5/2)) - ((A - B + C)*(1 - 2*Sin[(c + d*x)/2]))/(20*(1 + Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(5/2)) + ((A - B + C)*(1 + 2*Sin[(c + d*x)/2]))/(20*(1 - Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(5/2)) + (16*A*(Sin[(c + d*x)/2]/(1 - 2*Sin[(c + d*x)/2]^2)^(3/2) + (2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]))/15 - ((A - B + C)*(-105*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]] + (4 + 3*Sin[(c + d*x)/2])/((1 - Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(3/2)) - (19 + 29*Sin[(c + d*x)/2])/((1 - Sin[(c + d*x)/2])*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]) - (67*Sqrt[1 - 2*Sin[(c + d*x)/2]^2])/(1 - Sin[(c + d*x)/2])))/30 + ((A - B + C)*(-105*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]] + (4 - 3*Sin[(c + d*x)/2])/((1 + Sin[(c + d*x)/2])*(1 - 2*Sin[(c + d*x)/2]^2)^(3/2)) - (19 - 29*Sin[(c + d*x)/2])/((1 + Sin[(c + d*x)/2])*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]) - (67*Sqrt[1 - 2*Sin[(c + d*x)/2]^2])/(1 + Sin[(c + d*x)/2])))/30 + ((7*A - 3*B - C)*Csc[(c + d*x)/2]^7*(4725*Sin[(c + d*x)/2]^2 - 48825*Sin[(c + d*x)/2]^4 + 210105*Sin[(c + d*x)/2]^6 - 486630*Sin[(c + d*x)/2]^8 + 655812*Sin[(c + d*x)/2]^10 - 710*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^10 - 40*Cos[(c + d*x)/2]^6*HypergeometricPFQ[{2, 2, 2, 9/2}, {1, 1, 11/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^10 - 518760*Sin[(c + d*x)/2]^12 + 1770*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^12 + 226656*Sin[(c + d*x)/2]^14 - 1500*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^14 - 42048*Sin[(c + d*x)/2]^16 + 440*Hypergeometric2F1[2, 9/2, 11/2, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^16 + 4725*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 56700*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^2*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 291060*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^4*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 833760*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^6*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 1458000*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^8*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 1598400*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^10*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 1080000*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^12*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] - 414720*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^14*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 69120*ArcTanh[Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]]*Sin[(c + d*x)/2]^16*Sqrt[Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)] + 60*Cos[(c + d*x)/2]^4*HypergeometricPFQ[{2, 2, 9/2}, {1, 11/2}, Sin[(c + d*x)/2]^2/(-1 + 2*Sin[(c + d*x)/2]^2)]*Sin[(c + d*x)/2]^10*(-5 + 4*Sin[(c + d*x)/2]^2)))/(1350*(1 - 2*Sin[(c + d*x)/2]^2)^(7/2)*(-1 + 2*Sin[(c + d*x)/2]^2))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","C",0
521,1,7119,181,25.7580443,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{(3 A-7 B+11 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(3 A-3 B+7 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{6 a^2 d}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(3 A-9 B+13 C) \tan (c+d x)}{3 a d \sqrt{a \sec (c+d x)+a}}",1,"Result too large to show","C",0
522,1,748,120,6.4898718,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{4 \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)} \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{(7 A-3 B-C) \sin \left(\frac{1}{2} (c+d x)\right) \left(\frac{2 \sin ^2\left(\frac{1}{2} (c+d x)\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \, _2F_1\left(2,\frac{5}{2};\frac{7}{2};-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}+5 \sqrt{-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}} \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^2 \left(3-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right) \csc ^4\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}-\tanh ^{-1}\left(\sqrt{-\frac{\sin ^2\left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}\right)\right)\right)}{10 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2}}+\frac{(A-B+C) \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}{1-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{(A-B+C) \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+\frac{(A-B+C) \left(2 \sin \left(\frac{1}{2} (c+d x)\right)+1\right)}{4 \left(1-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}-\frac{(A-B+C) \left(1-2 \sin \left(\frac{1}{2} (c+d x)\right)\right)}{4 \left(\sin \left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}+\frac{3}{2} (A-B+C) \tan ^{-1}\left(\frac{1-2 \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}\right)-\frac{3}{2} (A-B+C) \tan ^{-1}\left(\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)+1}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}\right)+\frac{4 A \sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}}\right)}{d \sqrt{\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{(A+3 B-7 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(A-B+C) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 C \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}",1,"(4*Cos[(c + d*x)/2]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*((3*(A - B + C)*ArcTan[(1 - 2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]])/2 - (3*(A - B + C)*ArcTan[(1 + 2*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2]])/2 + (4*A*Sin[(c + d*x)/2])/Sqrt[1 - 2*Sin[(c + d*x)/2]^2] - ((A - B + C)*(1 - 2*Sin[(c + d*x)/2]))/(4*(1 + Sin[(c + d*x)/2])*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]) + ((A - B + C)*(1 + 2*Sin[(c + d*x)/2]))/(4*(1 - Sin[(c + d*x)/2])*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]) + ((A - B + C)*Sqrt[1 - 2*Sin[(c + d*x)/2]^2])/(1 - Sin[(c + d*x)/2]) - ((A - B + C)*Sqrt[1 - 2*Sin[(c + d*x)/2]^2])/(1 + Sin[(c + d*x)/2]) + ((7*A - 3*B - C)*Sin[(c + d*x)/2]*((2*Cos[(c + d*x)/2]^2*Hypergeometric2F1[2, 5/2, 7/2, -(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))]*Sin[(c + d*x)/2]^2)/(1 - 2*Sin[(c + d*x)/2]^2) + 5*Csc[(c + d*x)/2]^4*Sqrt[-(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))]*(1 - 2*Sin[(c + d*x)/2]^2)^2*(3 - 2*Sin[(c + d*x)/2]^2)*(-ArcTanh[Sqrt[-(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))]] + Sqrt[-(Sin[(c + d*x)/2]^2/(1 - 2*Sin[(c + d*x)/2]^2))])))/(10*(1 - 2*Sin[(c + d*x)/2]^2)^(3/2))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","C",0
523,1,16084,131,28.4742961,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(5 A-B-3 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
524,1,179,173,2.4086233,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(2 \sin ^2\left(\frac{1}{2} (c+d x)\right) (2 A \cos (c+d x)+3 A-B+C)+\sqrt{2} (9 A-5 B+C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)-4 (3 A-2 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)\right)}{2 a d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(9 A-5 B+C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(3 A-2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(3 A-B+C) \sin (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"-1/2*((-4*(3*A - 2*B)*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + Sqrt[2]*(9*A - 5*B + C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] + 2*(3*A - B + C + 2*A*Cos[c + d*x])*Sin[(c + d*x)/2]^2)*Tan[(c + d*x)/2])/(a*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
525,1,17639,232,28.1729989,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{(19 A-12 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(13 A-9 B+5 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(7 A-6 B+2 C) \sin (c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}+\frac{(2 A-B+C) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
526,1,221,284,3.0690525,"\int \frac{\cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \left(-4 \sin ^2\left(\frac{1}{2} (c+d x)\right) ((43 A-18 B+24 C) \cos (c+d x)-3 (A-2 B) \cos (2 (c+d x))+2 A \cos (3 (c+d x))+60 A-36 B+36 C)+12 (47 A-38 B+24 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)-24 \sqrt{2} (17 A-13 B+9 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)\right)}{48 a d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","-\frac{(47 A-38 B+24 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 a^{3/2} d}+\frac{(17 A-13 B+9 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(21 A-14 B+12 C) \sin (c+d x)}{8 a d \sqrt{a \sec (c+d x)+a}}+\frac{(5 A-3 B+3 C) \sin (c+d x) \cos ^2(c+d x)}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(13 A-12 B+6 C) \sin (c+d x) \cos (c+d x)}{12 a d \sqrt{a \sec (c+d x)+a}}",1,"((12*(47*A - 38*B + 24*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] - 24*Sqrt[2]*(17*A - 13*B + 9*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sqrt[-1 + Sec[c + d*x]] - 4*(60*A - 36*B + 36*C + (43*A - 18*B + 24*C)*Cos[c + d*x] - 3*(A - 2*B)*Cos[2*(c + d*x)] + 2*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]^2)*Tan[(c + d*x)/2])/(48*a*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
527,1,7237,277,25.5002176,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{(75 A-163 B+283 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(195 A-475 B+787 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{240 a^3 d}+\frac{(45 A-85 B+157 C) \tan (c+d x) \sec ^2(c+d x)}{80 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(465 A-985 B+1729 C) \tan (c+d x)}{120 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^4(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(5 A-13 B+21 C) \tan (c+d x) \sec ^3(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
528,1,7197,227,25.9829911,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{(19 A-75 B+163 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(15 A-39 B+95 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}-\frac{(21 A-93 B+197 C) \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(A-9 B+17 C) \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
529,1,7172,179,25.5681665,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{(5 A+19 B-75 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(A-B+9 C) \tan (c+d x)}{4 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(3 A+5 B-13 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
530,1,7163,137,25.3898692,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{(3 A+5 B+19 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(7 A+B-9 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
531,1,16171,171,28.0456955,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{(43 A-3 B-5 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-3 B-5 C) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
532,1,181,217,5.3855818,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{-2 \tan ^3\left(\frac{1}{2} (c+d x)\right) ((55 A-15 B+7 C) \cos (c+d x)+8 A \cos (2 (c+d x))+43 A-11 B+3 C)-\sqrt{2} (115 A-43 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\frac{\sqrt{\sec (c+d x)-1}}{\sqrt{2}}\right)+32 (5 A-2 B) \sin (c+d x) \sqrt{\sec (c+d x)-1} \tan ^{-1}\left(\sqrt{\sec (c+d x)-1}\right)}{32 a^2 d (\cos (c+d x)-1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(115 A-43 B+3 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(5 A-2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(35 A-11 B+3 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(15 A-7 B-C) \sin (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(32*(5*A - 2*B)*ArcTan[Sqrt[-1 + Sec[c + d*x]]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x] - Sqrt[2]*(115*A - 43*B + 3*C)*ArcTan[Sqrt[-1 + Sec[c + d*x]]/Sqrt[2]]*Sqrt[-1 + Sec[c + d*x]]*Sin[c + d*x] - 2*(43*A - 11*B + 3*C + (55*A - 15*B + 7*C)*Cos[c + d*x] + 8*A*Cos[2*(c + d*x)])*Tan[(c + d*x)/2]^3)/(32*a^2*d*(-1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
533,1,17717,280,28.8213126,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{(39 A-20 B+8 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{5/2} d}-\frac{(219 A-115 B+43 C) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(63 A-35 B+11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(31 A-15 B+7 C) \sin (c+d x) \cos (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(19 A-11 B+3 C) \sin (c+d x) \cos (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \cos (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
534,1,527,217,6.6032652,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a \csc (c) e^{-i d x} \cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) e^{2 i d x} (5 A+3 (B+C)) \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left(35 A \left(3 e^{i (c+d x)}+e^{2 i (c+d x)}+3 e^{3 i (c+d x)}-1\right) \left(1+e^{2 i (c+d x)}\right)^2+7 B \left(3 e^{i (c+d x)}-5 e^{2 i (c+d x)}+27 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+33 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+9 e^{7 i (c+d x)}-5\right)+C \left(21 e^{i (c+d x)}-85 e^{2 i (c+d x)}+189 e^{3 i (c+d x)}+85 e^{4 i (c+d x)}+231 e^{5 i (c+d x)}+25 e^{6 i (c+d x)}+63 e^{7 i (c+d x)}-25\right)\right)}{\left(1+e^{2 i (c+d x)}\right)^3}+10 \sin (c) e^{i d x} (7 A+7 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{105 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 a (7 A+7 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 a (5 A+3 (B+C)) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (7 A+7 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (5 A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(2*a*Cos[c + d*x]^2*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(7*Sqrt[2]*(5*A + 3*(B + C))*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(35*A*(1 + E^((2*I)*(c + d*x)))^2*(-1 + 3*E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + 3*E^((3*I)*(c + d*x))) + 7*B*(-5 + 3*E^(I*(c + d*x)) - 5*E^((2*I)*(c + d*x)) + 27*E^((3*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) + 33*E^((5*I)*(c + d*x)) + 5*E^((6*I)*(c + d*x)) + 9*E^((7*I)*(c + d*x))) + C*(-25 + 21*E^(I*(c + d*x)) - 85*E^((2*I)*(c + d*x)) + 189*E^((3*I)*(c + d*x)) + 85*E^((4*I)*(c + d*x)) + 231*E^((5*I)*(c + d*x)) + 25*E^((6*I)*(c + d*x)) + 63*E^((7*I)*(c + d*x))))*Sqrt[Sec[c + d*x]])/(E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3) + 10*(7*A + 7*B + 5*C)*E^(I*d*x)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*Sin[c]))/(105*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","C",1
535,1,366,181,6.308146,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 a e^{-i c} \left(-1+e^{2 i c}\right) \csc (c) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left((5 A+5 B+3 C) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-5 i (3 A+B+C) \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-15 A e^{i (c+d x)}-30 A e^{3 i (c+d x)}-15 A e^{5 i (c+d x)}-15 B e^{i (c+d x)}-30 B e^{3 i (c+d x)}-5 B e^{4 i (c+d x)}-15 B e^{5 i (c+d x)}+5 B-3 C e^{i (c+d x)}-24 C e^{3 i (c+d x)}-5 C e^{4 i (c+d x)}-9 C e^{5 i (c+d x)}+5 C\right)}{15 d \left(1+e^{2 i (c+d x)}\right)^2 \sec ^{\frac{3}{2}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 a (5 A+5 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(2*a*(-1 + E^((2*I)*c))*Csc[c]*(5*B + 5*C - 15*A*E^(I*(c + d*x)) - 15*B*E^(I*(c + d*x)) - 3*C*E^(I*(c + d*x)) - 30*A*E^((3*I)*(c + d*x)) - 30*B*E^((3*I)*(c + d*x)) - 24*C*E^((3*I)*(c + d*x)) - 5*B*E^((4*I)*(c + d*x)) - 5*C*E^((4*I)*(c + d*x)) - 15*A*E^((5*I)*(c + d*x)) - 15*B*E^((5*I)*(c + d*x)) - 9*C*E^((5*I)*(c + d*x)) - (5*I)*(3*A + B + C)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A + 5*B + 3*C)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*c)*(1 + E^((2*I)*(c + d*x)))^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(3/2))","C",1
536,1,208,143,1.8426721,"\int \frac{(a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-i (A-B-C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 (3 A+3 B+C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 i A \cos (2 (c+d x))+3 i A+3 B \sin (2 (c+d x))-3 i B \cos (2 (c+d x))-3 i B+2 C \sin (c+d x)+3 C \sin (2 (c+d x))-3 i C \cos (2 (c+d x))-3 i C\right)}{3 d}","\frac{2 a (3 A+3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(a*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((3*I)*A - (3*I)*B - (3*I)*C + (3*I)*A*Cos[2*(c + d*x)] - (3*I)*B*Cos[2*(c + d*x)] - (3*I)*C*Cos[2*(c + d*x)] + 2*(3*A + 3*B + C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - I*(A - B - C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*C*Sin[c + d*x] + 3*B*Sin[2*(c + d*x)] + 3*C*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
537,1,183,138,1.8910048,"\int \frac{(a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2 i (A+B-C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 (A+3 (B+C)) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \sin (2 (c+d x))+6 i A \cos (c+d x)+6 i B \cos (c+d x)+6 C \sin (c+d x)-6 i C \cos (c+d x)\right)}{3 d}","\frac{2 a (A+3 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a C \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((6*I)*A*Cos[c + d*x] + (6*I)*B*Cos[c + d*x] - (6*I)*C*Cos[c + d*x] + 2*(A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*I)*(A + B - C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 6*C*Sin[c + d*x] + A*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
538,1,177,146,1.8171971,"\int \frac{(a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-4 i (3 A+5 (B+C)) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) (10 (A+B) \sin (c+d x)+6 i (3 A+5 (B+C))+3 A \sin (2 (c+d x)))+20 (A+B+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 a (A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(20*(A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4*I)*(3*A + 5*(B + C))*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((6*I)*(3*A + 5*(B + C)) + 10*(A + B)*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)])))/(30*d*E^(I*d*x))","C",1
539,1,201,182,2.2035341,"\int \frac{(a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i (3 A+3 B+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) (5 (23 A+28 (B+C)) \sin (c+d x)+42 (A+B) \sin (2 (c+d x))+84 i (3 A+3 B+5 C)+15 A \sin (3 (c+d x)))+40 (5 A+7 (B+C)) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 d}","\frac{2 a (5 A+7 (B+C)) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+7 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (3 (A+B)+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(40*(5*A + 7*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(3*A + 3*B + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((84*I)*(3*A + 3*B + 5*C) + 5*(23*A + 28*(B + C))*Sin[c + d*x] + 42*(A + B)*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)])))/(420*d*E^(I*d*x))","C",1
540,1,231,215,2.7224286,"\int \frac{(a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (7 A+9 (B+C)) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) (30 (23 A+23 B+28 C) \sin (c+d x)+14 (19 A+18 (B+C)) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+1176 i A+90 B \sin (3 (c+d x))+1512 i B+1512 i C)+240 (5 A+5 B+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{2520 d}","\frac{2 a (7 A+9 (B+C)) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 (A+B)+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 (A+B)+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (7 A+9 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (A+B) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(5*A + 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(7*A + 9*(B + C))*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((1176*I)*A + (1512*I)*B + (1512*I)*C + 30*(23*A + 23*B + 28*C)*Sin[c + d*x] + 14*(19*A + 18*(B + C))*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(2520*d*E^(I*d*x))","C",1
541,1,1270,291,7.2008876,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{8 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{45 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{10 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{9 d}+\frac{\sec (c) (7 C \sin (c)+9 B \sin (d x)+18 C \sin (d x)) \sec ^3(c+d x)}{63 d}+\frac{\sec (c) (45 B \sin (c)+90 C \sin (c)+63 A \sin (d x)+126 B \sin (d x)+112 C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac{\sec (c) (63 A \sin (c)+126 B \sin (c)+112 C \sin (c)+210 A \sin (d x)+180 B \sin (d x)+150 C \sin (d x)) \sec (c+d x)}{315 d}+\frac{2 (12 A+9 B+8 C) \cos (d x) \csc (c)}{15 d}+\frac{2 (7 A+6 B+5 C) \tan (c)}{21 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (12 A+9 B+8 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (7 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (12 A+9 B+8 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (9 B+4 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{63 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d}",1,"(4*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (8*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (4*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (10*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(12*A + 9*B + 8*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(9*d) + (Sec[c]*Sec[c + d*x]^3*(7*C*Sin[c] + 9*B*Sin[d*x] + 18*C*Sin[d*x]))/(63*d) + (Sec[c]*Sec[c + d*x]^2*(45*B*Sin[c] + 90*C*Sin[c] + 63*A*Sin[d*x] + 126*B*Sin[d*x] + 112*C*Sin[d*x]))/(315*d) + (Sec[c]*Sec[c + d*x]*(63*A*Sin[c] + 126*B*Sin[c] + 112*C*Sin[c] + 210*A*Sin[d*x] + 180*B*Sin[d*x] + 150*C*Sin[d*x]))/(315*d) + (2*(7*A + 6*B + 5*C)*Tan[c])/(21*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",0
542,1,1216,255,7.04917,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^4(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{7 d}+\frac{\sec (c) (5 C \sin (c)+7 B \sin (d x)+14 C \sin (d x)) \sec ^2(c+d x)}{35 d}+\frac{\sec (c) (21 B \sin (c)+42 C \sin (c)+35 A \sin (d x)+70 B \sin (d x)+60 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{2 (5 A+4 B+3 C) \cos (d x) \csc (c)}{5 d}+\frac{(7 A+14 B+12 C) \tan (c)}{21 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}","\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+4 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (14 A+7 B+6 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (5 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 B+4 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{35 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (2*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (4*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(5*A + 4*B + 3*C)*Cos[d*x]*Csc[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(7*d) + (Sec[c]*Sec[c + d*x]^2*(5*C*Sin[c] + 7*B*Sin[d*x] + 14*C*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]*(21*B*Sin[c] + 42*C*Sin[c] + 35*A*Sin[d*x] + 70*B*Sin[d*x] + 60*C*Sin[d*x]))/(105*d) + ((7*A + 14*B + 12*C)*Tan[c])/(21*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",1
543,1,265,214,3.4503325,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 e^{-i d x} \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(40 (3 A+2 B+C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 A \sin (c+d x)+15 A \sin (3 (c+d x))+2 i (5 B+4 C) e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+30 B \sin (c+d x)+10 B \sin (2 (c+d x))+30 B \sin (3 (c+d x))-90 i B \cos (c+d x)-30 i B \cos (3 (c+d x))+36 C \sin (c+d x)+20 C \sin (2 (c+d x))+24 C \sin (3 (c+d x))-72 i C \cos (c+d x)-24 i C \cos (3 (c+d x))\right)}{30 d}","\frac{2 a^2 (15 A+25 B+17 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (3 A+2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (5 B+4 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)}{15 d}-\frac{4 a^2 (5 B+4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}{5 d}",1,"(a^2*Sec[c + d*x]^(5/2)*(Cos[d*x] + I*Sin[d*x])*((-90*I)*B*Cos[c + d*x] - (72*I)*C*Cos[c + d*x] - (30*I)*B*Cos[3*(c + d*x)] - (24*I)*C*Cos[3*(c + d*x)] + 40*(3*A + 2*B + C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + ((2*I)*(5*B + 4*C)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 15*A*Sin[c + d*x] + 30*B*Sin[c + d*x] + 36*C*Sin[c + d*x] + 10*B*Sin[2*(c + d*x)] + 20*C*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)] + 30*B*Sin[3*(c + d*x)] + 24*C*Sin[3*(c + d*x)]))/(30*d*E^(I*d*x))","C",1
544,1,209,208,2.4777327,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(8 (2 A+3 B+2 C) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-4 i (A-C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+A \sin (c+d x)+A \sin (3 (c+d x))+12 i A \cos (2 (c+d x))+12 i A+6 B \sin (2 (c+d x))+4 C \sin (c+d x)+12 C \sin (2 (c+d x))-12 i C \cos (2 (c+d x))-12 i C\right)}{6 d}","-\frac{2 a^2 (A-3 B-5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (2 A+3 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (A-C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)}{3 d}+\frac{4 a^2 (A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}}",1,"(a^2*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((12*I)*A - (12*I)*C + (12*I)*A*Cos[2*(c + d*x)] - (12*I)*C*Cos[2*(c + d*x)] + 8*(2*A + 3*B + 2*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - (4*I)*(A - C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + A*Sin[c + d*x] + 4*C*Sin[c + d*x] + 6*B*Sin[2*(c + d*x)] + 12*C*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(6*d*E^(I*d*x))","C",1
545,1,187,214,2.0705375,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(40 (A+2 B+3 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-8 i (4 A+5 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+3 A \sin (c+d x)+20 A \sin (2 (c+d x))+3 A \sin (3 (c+d x))+96 i A \cos (c+d x)+10 B \sin (2 (c+d x))+120 i B \cos (c+d x)+60 C \sin (c+d x)\right)}{30 d}","-\frac{2 a^2 (7 A+5 B-15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^2 (A+2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (4 A+5 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (4 A+5 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*((96*I)*A*Cos[c + d*x] + (120*I)*B*Cos[c + d*x] + 40*(A + 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (8*I)*(4*A + 5*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 3*A*Sin[c + d*x] + 60*C*Sin[c + d*x] + 20*A*Sin[2*(c + d*x)] + 10*B*Sin[2*(c + d*x)] + 3*A*Sin[3*(c + d*x)]))/(30*d)","C",1
546,1,189,219,2.24157,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(-112 i (3 A+4 B+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) (5 (51 A+28 (2 B+C)) \sin (c+d x)+42 (2 A+B) \sin (2 (c+d x))+15 A \sin (3 (c+d x))+504 i A+672 i B+840 i C)+80 (6 A+7 (B+2 C)) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 d}","\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (6 A+7 B+14 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (3 A+4 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (4 A+7 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(80*(6*A + 7*(B + 2*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(3*A + 4*B + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((504*I)*A + (672*I)*B + (840*I)*C + 5*(51*A + 28*(2*B + C))*Sin[c + d*x] + 42*(2*A + B)*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)])))/(420*d)","C",1
547,1,234,255,3.537529,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (8 A+9 B+12 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (46 A+51 B+56 C) \sin (c+d x)+14 (37 A+36 B+18 C) \sin (2 (c+d x))+180 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+2688 i A+90 B \sin (3 (c+d x))+3024 i B+4032 i C)+240 (5 A+6 B+7 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (5 A+6 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (8 A+9 B+12 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (4 A+9 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(8*A + 9*B + 12*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((2688*I)*A + (3024*I)*B + (4032*I)*C + 30*(46*A + 51*B + 56*C)*Sin[c + d*x] + 14*(37*A + 36*B + 18*C)*Sin[2*(c + d*x)] + 180*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
548,1,270,291,4.6900547,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2464 i (7 A+8 B+9 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (941 A+22 (46 B+51 C)) \sin (c+d x)+308 (38 A+37 B+36 C) \sin (2 (c+d x))+4545 A \sin (3 (c+d x))+1540 A \sin (4 (c+d x))+315 A \sin (5 (c+d x))+51744 i A+3960 B \sin (3 (c+d x))+770 B \sin (4 (c+d x))+59136 i B+1980 C \sin (3 (c+d x))+66528 i C)+480 (50 A+55 B+66 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{27720 d}","\frac{4 a^2 (7 A+8 B+9 C) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (89 A+121 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (50 A+55 B+66 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (50 A+55 B+66 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (7 A+8 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (4 A+11 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(480*(50*A + 55*B + 66*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(7*A + 8*B + 9*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((51744*I)*A + (59136*I)*B + (66528*I)*C + 30*(941*A + 22*(46*B + 51*C))*Sin[c + d*x] + 308*(38*A + 37*B + 36*C)*Sin[2*(c + d*x)] + 4545*A*Sin[3*(c + d*x)] + 3960*B*Sin[3*(c + d*x)] + 1980*C*Sin[3*(c + d*x)] + 1540*A*Sin[4*(c + d*x)] + 770*B*Sin[4*(c + d*x)] + 315*A*Sin[5*(c + d*x)])))/(27720*d*E^(I*d*x))","C",1
549,1,1324,343,7.4085316,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{7 A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{17 B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{45 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{13 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}+\frac{11 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}+\frac{5 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{11 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{22 d}+\frac{\sec (c) (9 C \sin (c)+11 B \sin (d x)+33 C \sin (d x)) \sec ^4(c+d x)}{198 d}+\frac{\sec (c) (77 B \sin (c)+231 C \sin (c)+99 A \sin (d x)+297 B \sin (d x)+378 C \sin (d x)) \sec ^3(c+d x)}{1386 d}+\frac{\sec (c) (495 A \sin (c)+1485 B \sin (c)+1890 C \sin (c)+2079 A \sin (d x)+2618 B \sin (d x)+2310 C \sin (d x)) \sec ^2(c+d x)}{6930 d}+\frac{\sec (c) (2079 A \sin (c)+2618 B \sin (c)+2310 C \sin (c)+4290 A \sin (d x)+3630 B \sin (d x)+3150 C \sin (d x)) \sec (c+d x)}{6930 d}+\frac{(21 A+17 B+15 C) \cos (d x) \csc (c)}{15 d}+\frac{(143 A+121 B+105 C) \tan (c)}{231 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}","\frac{4 a^3 (264 A+253 B+210 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{1155 d}+\frac{4 a^3 (143 A+121 B+105 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{231 d}+\frac{2 (99 A+143 B+105 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (21 A+17 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (143 A+121 B+105 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (21 A+17 B+15 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (11 B+6 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{99 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^3}{11 d}",1,"(7*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (17*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (13*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (11*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (5*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(11*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((21*A + 17*B + 15*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(22*d) + (Sec[c]*Sec[c + d*x]^4*(9*C*Sin[c] + 11*B*Sin[d*x] + 33*C*Sin[d*x]))/(198*d) + (Sec[c]*Sec[c + d*x]^3*(77*B*Sin[c] + 231*C*Sin[c] + 99*A*Sin[d*x] + 297*B*Sin[d*x] + 378*C*Sin[d*x]))/(1386*d) + (Sec[c]*Sec[c + d*x]^2*(495*A*Sin[c] + 1485*B*Sin[c] + 1890*C*Sin[c] + 2079*A*Sin[d*x] + 2618*B*Sin[d*x] + 2310*C*Sin[d*x]))/(6930*d) + (Sec[c]*Sec[c + d*x]*(2079*A*Sin[c] + 2618*B*Sin[c] + 2310*C*Sin[c] + 4290*A*Sin[d*x] + 3630*B*Sin[d*x] + 3150*C*Sin[d*x]))/(6930*d) + ((143*A + 121*B + 105*C)*Tan[c])/(231*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",0
550,1,1267,307,7.2548316,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{7 B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{17 C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos ^5(c+d x) \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{45 \sqrt{2} d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}+\frac{13 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}+\frac{11 C \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}+\frac{(\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{18 d}+\frac{\sec (c) (7 C \sin (c)+9 B \sin (d x)+27 C \sin (d x)) \sec ^3(c+d x)}{126 d}+\frac{\sec (c) (45 B \sin (c)+135 C \sin (c)+63 A \sin (d x)+189 B \sin (d x)+238 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac{\sec (c) (63 A \sin (c)+189 B \sin (c)+238 C \sin (c)+315 A \sin (d x)+390 B \sin (d x)+330 C \sin (d x)) \sec (c+d x)}{630 d}+\frac{(27 A+21 B+17 C) \cos (d x) \csc (c)}{15 d}+\frac{(21 A+26 B+22 C) \tan (c)}{42 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}","\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{315 d}+\frac{4 a^3 (27 A+21 B+17 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (21 A+13 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (27 A+21 B+17 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (3 B+2 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{21 a d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}{9 d}",1,"(3*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (7*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (17*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^5*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*Sqrt[2]*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (13*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (11*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((27*A + 21*B + 17*C)*Cos[d*x]*Csc[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^3*(7*C*Sin[c] + 9*B*Sin[d*x] + 27*C*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^2*(45*B*Sin[c] + 135*C*Sin[c] + 63*A*Sin[d*x] + 189*B*Sin[d*x] + 238*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(63*A*Sin[c] + 189*B*Sin[c] + 238*C*Sin[c] + 315*A*Sin[d*x] + 390*B*Sin[d*x] + 330*C*Sin[d*x]))/(630*d) + ((21*A + 26*B + 22*C)*Tan[c])/(42*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",0
551,1,359,271,5.2672192,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 e^{-i d x} \sec ^{\frac{7}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(14 i (5 A+9 B+7 C) e^{-2 i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{7/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (35 A+21 B+13 C) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+70 A \sin (c+d x)+630 A \sin (2 (c+d x))+70 A \sin (3 (c+d x))+315 A \sin (4 (c+d x))-840 i A \cos (2 (c+d x))-210 i A \cos (4 (c+d x))-630 i A+210 B \sin (c+d x)+840 B \sin (2 (c+d x))+210 B \sin (3 (c+d x))+378 B \sin (4 (c+d x))-1512 i B \cos (2 (c+d x))-378 i B \cos (4 (c+d x))-1134 i B+380 C \sin (c+d x)+840 C \sin (2 (c+d x))+260 C \sin (3 (c+d x))+294 C \sin (4 (c+d x))-1176 i C \cos (2 (c+d x))-294 i C \cos (4 (c+d x))-882 i C\right)}{420 d}","\frac{4 a^3 (140 A+147 B+106 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (5 A+9 B+7 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (35 A+21 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (5 A+9 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (7 B+6 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)^2}{35 a d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}{7 d}",1,"(a^3*Sec[c + d*x]^(7/2)*(Cos[d*x] + I*Sin[d*x])*((-630*I)*A - (1134*I)*B - (882*I)*C - (840*I)*A*Cos[2*(c + d*x)] - (1512*I)*B*Cos[2*(c + d*x)] - (1176*I)*C*Cos[2*(c + d*x)] - (210*I)*A*Cos[4*(c + d*x)] - (378*I)*B*Cos[4*(c + d*x)] - (294*I)*C*Cos[4*(c + d*x)] + 80*(35*A + 21*B + 13*C)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + ((14*I)*(5*A + 9*B + 7*C)*(1 + E^((2*I)*(c + d*x)))^(7/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^((2*I)*(c + d*x)) + 70*A*Sin[c + d*x] + 210*B*Sin[c + d*x] + 380*C*Sin[c + d*x] + 630*A*Sin[2*(c + d*x)] + 840*B*Sin[2*(c + d*x)] + 840*C*Sin[2*(c + d*x)] + 70*A*Sin[3*(c + d*x)] + 210*B*Sin[3*(c + d*x)] + 260*C*Sin[3*(c + d*x)] + 315*A*Sin[4*(c + d*x)] + 378*B*Sin[4*(c + d*x)] + 294*C*Sin[4*(c + d*x)]))/(420*d*E^(I*d*x))","C",1
552,1,316,271,4.7076638,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^3 e^{-i d x} \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-4 i (5 A-5 B-9 C) e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (5 A+5 B+3 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+30 A \sin (c+d x)+10 A \sin (2 (c+d x))+30 A \sin (3 (c+d x))+5 A \sin (4 (c+d x))+180 i A \cos (c+d x)+60 i A \cos (3 (c+d x))+90 B \sin (c+d x)+20 B \sin (2 (c+d x))+90 B \sin (3 (c+d x))-180 i B \cos (c+d x)-60 i B \cos (3 (c+d x))+132 C \sin (c+d x)+60 C \sin (2 (c+d x))+108 C \sin (3 (c+d x))-324 i C \cos (c+d x)-108 i C \cos (3 (c+d x))\right)}{60 d}","\frac{4 a^3 (5 A+20 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 (5 A-5 B-9 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (5 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A-5 B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{2 (5 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 \sec (c+d x)+a^2\right)^2}{15 a d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d \sqrt{\sec (c+d x)}}",1,"(a^3*Sec[c + d*x]^(5/2)*(Cos[d*x] + I*Sin[d*x])*((180*I)*A*Cos[c + d*x] - (180*I)*B*Cos[c + d*x] - (324*I)*C*Cos[c + d*x] + (60*I)*A*Cos[3*(c + d*x)] - (60*I)*B*Cos[3*(c + d*x)] - (108*I)*C*Cos[3*(c + d*x)] + 80*(5*A + 5*B + 3*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] - ((4*I)*(5*A - 5*B - 9*C)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 30*A*Sin[c + d*x] + 90*B*Sin[c + d*x] + 132*C*Sin[c + d*x] + 10*A*Sin[2*(c + d*x)] + 20*B*Sin[2*(c + d*x)] + 60*C*Sin[2*(c + d*x)] + 30*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 108*C*Sin[3*(c + d*x)] + 5*A*Sin[4*(c + d*x)]))/(60*d*E^(I*d*x))","C",1
553,1,275,270,2.9622387,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{a^3 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-8 i (9 A+5 B-5 C) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (3 A+5 (B+C)) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+30 A \sin (c+d x)+6 A \sin (2 (c+d x))+30 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+216 i A \cos (2 (c+d x))+216 i A+10 B \sin (c+d x)+60 B \sin (2 (c+d x))+10 B \sin (3 (c+d x))+120 i B \cos (2 (c+d x))+120 i B+40 C \sin (c+d x)+180 C \sin (2 (c+d x))-120 i C \cos (2 (c+d x))-120 i C\right)}{60 d}","-\frac{4 a^3 (6 A-5 B-20 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}-\frac{2 (9 A+5 B-5 C) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (3 A+5 (B+C)) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (9 A+5 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+5 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^3*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((216*I)*A + (120*I)*B - (120*I)*C + (216*I)*A*Cos[2*(c + d*x)] + (120*I)*B*Cos[2*(c + d*x)] - (120*I)*C*Cos[2*(c + d*x)] + 80*(3*A + 5*(B + C))*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - (8*I)*(9*A + 5*B - 5*C)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 30*A*Sin[c + d*x] + 10*B*Sin[c + d*x] + 40*C*Sin[c + d*x] + 6*A*Sin[2*(c + d*x)] + 60*B*Sin[2*(c + d*x)] + 180*C*Sin[2*(c + d*x)] + 30*A*Sin[3*(c + d*x)] + 10*B*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(60*d*E^(I*d*x))","C",1
554,1,266,271,2.9800536,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (7 A+9 B+5 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+80 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 A \sin (c+d x)+550 A \sin (2 (c+d x))+126 A \sin (3 (c+d x))+15 A \sin (4 (c+d x))+2352 i A \cos (c+d x)+42 B \sin (c+d x)+420 B \sin (2 (c+d x))+42 B \sin (3 (c+d x))+3024 i B \cos (c+d x)+840 C \sin (c+d x)+140 C \sin (2 (c+d x))+1680 i C \cos (c+d x)\right)}{420 d}","-\frac{4 a^3 (41 A+42 B-35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d}+\frac{2 (7 A+9 B+5 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+21 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (7 A+9 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (6 A+7 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((2352*I)*A*Cos[c + d*x] + (3024*I)*B*Cos[c + d*x] + (1680*I)*C*Cos[c + d*x] + 80*(13*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(7*A + 9*B + 5*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 126*A*Sin[c + d*x] + 42*B*Sin[c + d*x] + 840*C*Sin[c + d*x] + 550*A*Sin[2*(c + d*x)] + 420*B*Sin[2*(c + d*x)] + 140*C*Sin[2*(c + d*x)] + 126*A*Sin[3*(c + d*x)] + 42*B*Sin[3*(c + d*x)] + 15*A*Sin[4*(c + d*x)]))/(420*d*E^(I*d*x))","C",1
555,1,214,271,2.9496909,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(-224 i (17 A+21 B+27 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) (30 (97 A+107 B+84 C) \sin (c+d x)+14 (73 A+54 B+18 C) \sin (2 (c+d x))+270 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+5712 i A+90 B \sin (3 (c+d x))+7056 i B+9072 i C)+480 (11 A+13 B+21 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{2520 d}","\frac{2 (73 A+99 B+63 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (11 A+13 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (17 A+21 B+27 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (2 A+3 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{21 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(480*(11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (224*I)*(17*A + 21*B + 27*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((5712*I)*A + (7056*I)*B + (9072*I)*C + 30*(97*A + 107*B + 84*C)*Sin[c + d*x] + 14*(73*A + 54*B + 18*C)*Sin[2*(c + d*x)] + 270*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(2520*d)","C",1
556,1,246,307,5.0211667,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(-2464 i (15 A+17 B+21 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (1953 A+2134 B+2354 C) \sin (c+d x)+308 (75 A+73 B+54 C) \sin (2 (c+d x))+8505 A \sin (3 (c+d x))+2310 A \sin (4 (c+d x))+315 A \sin (5 (c+d x))+110880 i A+5940 B \sin (3 (c+d x))+770 B \sin (4 (c+d x))+125664 i B+1980 C \sin (3 (c+d x))+155232 i C)+480 (105 A+121 B+143 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{27720 d}","\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (105 A+121 B+143 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (15 A+17 B+21 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (6 A+11 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{99 a d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(480*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(15*A + 17*B + 21*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((110880*I)*A + (125664*I)*B + (155232*I)*C + 30*(1953*A + 2134*B + 2354*C)*Sin[c + d*x] + 308*(75*A + 73*B + 54*C)*Sin[2*(c + d*x)] + 8505*A*Sin[3*(c + d*x)] + 5940*B*Sin[3*(c + d*x)] + 1980*C*Sin[3*(c + d*x)] + 2310*A*Sin[4*(c + d*x)] + 770*B*Sin[4*(c + d*x)] + 315*A*Sin[5*(c + d*x)])))/(27720*d)","C",1
557,1,300,343,6.161018,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-4928 i (175 A+195 B+221 C) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (780 (1811 A+1953 B+2134 C) \sin (c+d x)+77 (7825 A+7800 B+7592 C) \sin (2 (c+d x))+251550 A \sin (3 (c+d x))+90860 A \sin (4 (c+d x))+24570 A \sin (5 (c+d x))+3465 A \sin (6 (c+d x))+2587200 i A+221130 B \sin (3 (c+d x))+60060 B \sin (4 (c+d x))+8190 B \sin (5 (c+d x))+2882880 i B+154440 C \sin (3 (c+d x))+20020 C \sin (4 (c+d x))+3267264 i C)+12480 (95 A+105 B+121 C) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{720720 d}","\frac{4 a^3 (175 A+195 B+221 C) \sin (c+d x)}{585 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (236 A+273 B+286 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (145 A+195 B+143 C) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (95 A+105 B+121 C) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (95 A+105 B+121 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (175 A+195 B+221 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{2 (6 A+13 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{143 a d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^3}{13 d \sec ^{\frac{11}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(12480*(95*A + 105*B + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4928*I)*(175*A + 195*B + 221*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((2587200*I)*A + (2882880*I)*B + (3267264*I)*C + 780*(1811*A + 1953*B + 2134*C)*Sin[c + d*x] + 77*(7825*A + 7800*B + 7592*C)*Sin[2*(c + d*x)] + 251550*A*Sin[3*(c + d*x)] + 221130*B*Sin[3*(c + d*x)] + 154440*C*Sin[3*(c + d*x)] + 90860*A*Sin[4*(c + d*x)] + 60060*B*Sin[4*(c + d*x)] + 20020*C*Sin[4*(c + d*x)] + 24570*A*Sin[5*(c + d*x)] + 8190*B*Sin[5*(c + d*x)] + 3465*A*Sin[6*(c + d*x)])))/(720720*d*E^(I*d*x))","C",1
558,1,1307,250,8.0573888,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{7 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{10 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}-\frac{10 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{8 C \sec (c) \sin (d x) \sec ^2(c+d x)}{5 d}+\frac{8 \sec (c) (3 C \sin (c)+5 B \sin (d x)-5 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{6 (5 A-5 B+7 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}-\frac{4 (-5 \cos (c) B-2 B+2 C+3 A \cos (c)+5 C \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(5 A-5 B+7 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}-\frac{(3 A-5 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (5 A-5 B+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}-\frac{(3 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A-5 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (7*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (10*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (10*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((6*(5*A - 5*B + 7*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(5*d) + (8*Sec[c]*Sec[c + d*x]*(3*C*Sin[c] + 5*B*Sin[d*x] - 5*C*Sin[d*x]))/(15*d) - (4*(-2*B + 2*C + 3*A*Cos[c] - 5*B*Cos[c] + 5*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",1
559,1,1261,205,7.4883384,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}-\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{10 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 (A-3 B+3 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{4 \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{8 C \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{4 (5 \cos (c) C+2 C+3 A \cos (c)-3 B \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{(A-3 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(3 A-3 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-1/3*(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (2*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (10*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(A - 3*B + 3*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (4*(2*C + 3*A*Cos[c] - 3*B*Cos[c] + 5*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",0
560,1,1224,162,6.8534284,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}-\frac{2 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 (A-B+3 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}-\frac{4 \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}\right)}{d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}+\frac{(A-B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (2*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (2*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(A - B + 3*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (4*(A - B + C)*Tan[c/2])/d))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",1
561,1,1243,133,6.5540443,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{2 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 (\cos (2 c) A+2 A-B+C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{4 \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{8 A \cos (c) \sin (d x)}{d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}-\frac{(A-B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"-((Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x]))) + (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (2*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(2*A - B + C + A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*A*Cos[c]*Sin[d*x])/d + (4*(A - B + C)*Tan[c/2])/d))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",0
562,1,1287,174,6.6967148,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{10 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}-\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{2 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 (\cos (2 c) A+2 A-2 B+C-B \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}-\frac{4 \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{4 A \cos (2 d x) \sin (2 c)}{3 d}-\frac{8 (A-B) \cos (c) \sin (d x)}{d}+\frac{4 A \cos (2 c) \sin (2 d x)}{3 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","\frac{(5 A-3 B+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(3 A-3 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (10*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (2*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(2*A - 2*B + C + A*Cos[2*c] - B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*A*Cos[2*d*x]*Sin[2*c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (8*(A - B)*Cos[c]*Sin[d*x])/d + (4*A*Cos[2*c]*Sin[2*d*x])/(3*d) - (4*(A - B + C)*Tan[c/2])/d))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",1
563,1,1350,214,6.7936258,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","-\frac{7 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{10 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{10 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}-\frac{2 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{(33 \cos (2 c) A+51 A-40 B+40 C-20 B \cos (2 c)+20 C \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{10 d}+\frac{4 \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}\right)}{d}-\frac{4 (A-B) \cos (2 d x) \sin (2 c)}{3 d}+\frac{2 A \cos (3 d x) \sin (3 c)}{5 d}+\frac{2 (33 A-20 B+20 C) \cos (c) \sin (d x)}{5 d}-\frac{4 (A-B) \cos (2 c) \sin (2 d x)}{3 d}+\frac{2 A \cos (3 c) \sin (3 d x)}{5 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{(7 A-5 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(5 A-5 B+3 C) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(5 A-5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (7 A-5 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(-7*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (10*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (10*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (2*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/10*((51*A - 40*B + 40*C + 33*A*Cos[2*c] - 20*B*Cos[2*c] + 20*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d - (4*(A - B)*Cos[2*d*x]*Sin[2*c])/(3*d) + (2*A*Cos[3*d*x]*Sin[3*c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (2*(33*A - 20*B + 20*C)*Cos[c]*Sin[d*x])/(5*d) - (4*(A - B)*Cos[2*c]*Sin[2*d*x])/(3*d) + (2*A*Cos[3*c]*Sin[3*d*x])/(5*d) + (4*(A - B + C)*Tan[c/2])/d))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",0
564,1,1406,250,7.0436181,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])),x]","\frac{7 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{7 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \cos (c+d x) \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{30 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}-\frac{10 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{10 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{(33 \cos (2 c) A+51 A-51 B+40 C-33 B \cos (2 c)+20 C \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{10 d}-\frac{4 \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{2 (27 A-14 B+14 C) \cos (2 d x) \sin (2 c)}{21 d}-\frac{2 (A-B) \cos (3 d x) \sin (3 c)}{5 d}+\frac{A \cos (4 d x) \sin (4 c)}{7 d}-\frac{2 (33 A-33 B+20 C) \cos (c) \sin (d x)}{5 d}+\frac{2 (27 A-14 B+14 C) \cos (2 c) \sin (2 d x)}{21 d}-\frac{2 (A-B) \cos (3 c) \sin (3 d x)}{5 d}+\frac{A \cos (4 c) \sin (4 d x)}{7 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (\sec (c+d x) a+a)}","-\frac{(A-B+C) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac{(7 A-7 B+5 C) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(9 A-7 B+7 C) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (9 A-7 B+7 C) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{5 (9 A-7 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (7 A-7 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(7*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (7*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (30*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) - (10*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (10*C*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((51*A - 51*B + 40*C + 33*A*Cos[2*c] - 33*B*Cos[2*c] + 20*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(10*d) + (2*(27*A - 14*B + 14*C)*Cos[2*d*x]*Sin[2*c])/(21*d) - (2*(A - B)*Cos[3*d*x]*Sin[3*c])/(5*d) + (A*Cos[4*d*x]*Sin[4*c])/(7*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (2*(33*A - 33*B + 20*C)*Cos[c]*Sin[d*x])/(5*d) + (2*(27*A - 14*B + 14*C)*Cos[2*c]*Sin[2*d*x])/(21*d) - (2*(A - B)*Cos[3*c]*Sin[3*d*x])/(5*d) + (A*Cos[4*c]*Sin[4*d*x])/(7*d) - (4*(A - B + C)*Tan[c/2])/d))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","C",0
565,1,1347,251,7.8921618,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{8 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{14 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{8 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{20 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{40 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)-5 B \sin \left(\frac{d x}{2}\right)+8 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A-4 B+7 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{16 C \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{8 (10 \cos (c) C+2 C+2 A \cos (c)-5 B \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}","-\frac{(A-4 B+7 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(2 A-5 B+10 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}-\frac{(A-4 B+7 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{(2 A-5 B+10 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (14*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (20*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (40*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(A - 4*B + 7*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - 5*B*Sin[(d*x)/2] + 8*C*Sin[(d*x)/2]))/(3*d) + (16*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (8*(2*C + 2*A*Cos[c] - 5*B*Cos[c] + 10*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
566,1,567,207,4.2945863,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-2 \sqrt{\sec (c+d x)} \left(\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) (A-B+C) \sec ^2\left(\frac{1}{2} (c+d x)\right)-2 \sec \left(\frac{c}{2}\right) (A+2 B-5 C) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-2 \tan \left(\frac{c}{2}\right) (A+2 B-5 C)+6 (B-4 C) \csc (c) \cos (d x)\right)+4 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+8 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+8 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-20 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A+2 B-5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(A+2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(B-4 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{(B-4 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (8*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 4*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 8*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 20*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 2*Sqrt[Sec[c + d*x]]*(6*(B - 4*C)*Cos[d*x]*Csc[c] - 2*(A + 2*B - 5*C)*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + (A - B + C)*Sec[c/2]*Sec[(c + d*x)/2]^3*Sin[(d*x)/2] - 2*(A + 2*B - 5*C)*Tan[c/2] + (A - B + C)*Sec[(c + d*x)/2]^2*Tan[c/2])))/(3*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)","C",1
567,1,1097,173,6.8502599,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{2 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{8 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{4 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{8 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(4 A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)-2 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}-\frac{8 (4 A-B-2 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}","\frac{(2 A+B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}-\frac{(A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (4*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(A - C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(4*A*Sin[(d*x)/2] - B*Sin[(d*x)/2] - 2*C*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*(4*A - B - 2*C)*Tan[c/2])/(3*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
568,1,1114,184,6.9085417,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{8 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{2 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{20 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{8 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)-4 B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (\cos (2 c) A+3 A-B) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{16 A \cos (c) \sin (d x)}{d}+\frac{8 (7 A-4 B+C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}","-\frac{(5 A-2 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{(5 A-2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"(-8*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (2*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (20*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (8*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (4*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(3*A - B + A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(7*A*Sin[(d*x)/2] - 4*B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (16*A*Cos[c]*Sin[d*x])/d + (8*(7*A - 4*B + C)*Tan[c/2])/(3*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
569,1,762,220,7.038011,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{2 \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-2 \sqrt{\sec (c+d x)} \left(\sec \left(\frac{c}{2}\right) (A-B+C) \sin \left(\frac{d x}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right)+\tan \left(\frac{c}{2}\right) (A-B+C) \sec ^2\left(\frac{1}{2} (c+d x)\right)-2 \sec \left(\frac{c}{2}\right) (10 A-7 B+4 C) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x) ((2 A-B) \cos (2 c)+5 A-3 B+C)-2 \tan \left(\frac{c}{2}\right) (10 A-7 B+4 C)-12 (2 A-B) \cos (c) \sin (d x)+2 A \sin (2 c) \cos (2 d x)+2 A \cos (2 c) \sin (2 d x)\right)-14 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-40 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+8 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+20 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-2 \sqrt{2} C \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-8 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(10 A-5 B+2 C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(7 A-4 B+C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(7 A-4 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-14*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (8*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - 40*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 20*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 8*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 2*Sqrt[Sec[c + d*x]]*(3*(5*A - 3*B + C + (2*A - B)*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2] + 2*A*Cos[2*d*x]*Sin[2*c] - 2*(10*A - 7*B + 4*C)*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + (A - B + C)*Sec[c/2]*Sec[(c + d*x)/2]^3*Sin[(d*x)/2] - 12*(2*A - B)*Cos[c]*Sin[d*x] + 2*A*Cos[2*c]*Sin[2*d*x] - 2*(10*A - 7*B + 4*C)*Tan[c/2] + (A - B + C)*Sec[(c + d*x)/2]^2*Tan[c/2])))/(3*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(1 + Sec[c + d*x])^2)","C",0
570,1,1442,254,7.2590893,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{112 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{14 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{8 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{20 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{40 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{20 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(13 A \sin \left(\frac{d x}{2}\right)-10 B \sin \left(\frac{d x}{2}\right)+7 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{(73 \cos (2 c) A+151 A-100 B+60 C-40 B \cos (2 c)+20 C \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{8 (2 A-B) \cos (2 d x) \sin (2 c)}{3 d}+\frac{4 A \cos (3 d x) \sin (3 c)}{5 d}+\frac{4 (73 A-40 B+20 C) \cos (c) \sin (d x)}{5 d}-\frac{8 (2 A-B) \cos (2 c) \sin (2 d x)}{3 d}+\frac{4 A \cos (3 c) \sin (3 d x)}{5 d}+\frac{8 (13 A-10 B+7 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}","-\frac{(3 A-2 B+C) \sin (c+d x)}{a^2 d \sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}+\frac{(56 A-35 B+20 C) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (3 A-2 B+C) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{5 (3 A-2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(56 A-35 B+20 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-B+C) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(-112*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (14*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (8*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (20*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (40*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (20*C*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/5*((151*A - 100*B + 60*C + 73*A*Cos[2*c] - 40*B*Cos[2*c] + 20*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d - (8*(2*A - B)*Cos[2*d*x]*Sin[2*c])/(3*d) + (4*A*Cos[3*d*x]*Sin[3*c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(13*A*Sin[(d*x)/2] - 10*B*Sin[(d*x)/2] + 7*C*Sin[(d*x)/2]))/(3*d) + (4*(73*A - 40*B + 20*C)*Cos[c]*Sin[d*x])/(5*d) - (8*(2*A - B)*Cos[2*c]*Sin[2*d*x])/(3*d) + (4*A*Cos[3*c]*Sin[3*d*x])/(5*d) + (8*(13*A - 10*B + 7*C)*Tan[c/2])/(3*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
571,1,1462,308,8.7986343,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{6 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{98 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{238 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{52 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{44 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)-8 B \sin \left(\frac{d x}{2}\right)+13 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (3 A-8 B+13 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)-13 B \sin \left(\frac{d x}{2}\right)+29 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (9 A-49 B+119 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{32 C \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{8 (33 \cos (c) C+4 C+3 A \cos (c)-13 B \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","-\frac{(9 A-49 B+119 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}-\frac{(9 A-49 B+119 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(3 A-13 B+33 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-49 B+119 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2}",1,"(-6*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (98*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (238*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (44*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(9*A - 49*B + 119*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(3*A*Sin[(d*x)/2] - 8*B*Sin[(d*x)/2] + 13*C*Sin[(d*x)/2]))/(15*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] - 13*B*Sin[(d*x)/2] + 29*C*Sin[(d*x)/2]))/(3*d) + (32*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (8*(4*C + 3*A*Cos[c] - 13*B*Cos[c] + 33*C*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (8*(3*A - 8*B + 13*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
572,1,1430,269,7.487678,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{6 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{98 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{52 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)+3 B \sin \left(\frac{d x}{2}\right)-8 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (2 A+3 B-8 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+3 B \sin \left(\frac{d x}{2}\right)-13 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A+9 B-49 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{8 (-A-3 B+13 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(A+3 B-13 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(A+9 B-49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(A+3 B-13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A+9 B-49 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(2 A+3 B-8 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (6*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (98*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(A + 9*B - 49*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 3*B*Sin[(d*x)/2] - 13*C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(2*A*Sin[(d*x)/2] + 3*B*Sin[(d*x)/2] - 8*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*(-A - 3*B + 13*C)*Tan[c/2])/(3*d) + (8*(2*A + 3*B - 8*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
573,1,1425,231,7.1578094,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{2 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{2 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{6 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)-2 B \sin \left(\frac{d x}{2}\right)-3 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (7 A-2 B-3 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+B \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-B-9 C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{8 (A+B+3 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(A-B-9 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-B-9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(4 A+B-6 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (2*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (6*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(A - B - 9*C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(7*A*Sin[(d*x)/2] - 2*B*Sin[(d*x)/2] - 3*C*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + B*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/(3*d) + (8*(A + B + 3*C)*Tan[c/2])/(3*d) - (8*(7*A - 2*B - 3*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
574,1,1431,231,7.26959,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{6 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{2 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(12 A \sin \left(\frac{d x}{2}\right)-7 B \sin \left(\frac{d x}{2}\right)+2 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (12 A-7 B+2 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (9 A+B-C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{8 (9 A-B-C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(3 A+B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A+B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A+B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(6 A-B-4 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(6*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (2*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(9*A + B - C)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(12*A*Sin[(d*x)/2] - 7*B*Sin[(d*x)/2] + 2*C*Sin[(d*x)/2]))/(15*d) - (8*(9*A - B - C)*Tan[c/2])/(3*d) + (8*(12*A - 7*B + 2*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
575,1,1449,241,7.4292329,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\frac{98 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{6 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{52 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(17 A \sin \left(\frac{d x}{2}\right)-12 B \sin \left(\frac{d x}{2}\right)+7 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (17 A-12 B+7 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(23 A \sin \left(\frac{d x}{2}\right)-9 B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (10 \cos (2 c) A+39 A-9 B-C) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{32 A \cos (c) \sin (d x)}{d}+\frac{8 (23 A-9 B+C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","-\frac{(13 A-3 B-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(13 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(49 A-9 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(8 A-3 B-2 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(-98*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (6*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (2*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(39*A - 9*B - C + 10*A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(23*A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(17*A*Sin[(d*x)/2] - 12*B*Sin[(d*x)/2] + 7*C*Sin[(d*x)/2]))/(15*d) + (32*A*Cos[c]*Sin[d*x])/d + (8*(23*A - 9*B + C)*Tan[c/2])/(3*d) - (8*(17*A - 12*B + 7*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
576,1,1497,274,7.6978729,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\frac{238 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{98 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{6 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{44 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{52 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{4 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(22 A \sin \left(\frac{d x}{2}\right)-17 B \sin \left(\frac{d x}{2}\right)+12 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (22 A-17 B+12 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(43 A \sin \left(\frac{d x}{2}\right)-23 B \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (30 \cos (2 c) A+89 A-39 B+9 C-10 B \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{16 A \cos (2 d x) \sin (2 c)}{3 d}-\frac{32 (3 A-B) \cos (c) \sin (d x)}{d}+\frac{16 A \cos (2 c) \sin (2 d x)}{3 d}-\frac{8 (43 A-23 B+9 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(33 A-13 B+3 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(119 A-49 B+9 C) \sin (c+d x)}{30 d \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(119 A-49 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(238*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (98*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (6*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (44*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (4*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(89*A - 39*B + 9*C + 30*A*Cos[2*c] - 10*B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (16*A*Cos[2*d*x]*Sin[2*c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(43*A*Sin[(d*x)/2] - 23*B*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(22*A*Sin[(d*x)/2] - 17*B*Sin[(d*x)/2] + 12*C*Sin[(d*x)/2]))/(15*d) - (32*(3*A - B)*Cos[c]*Sin[d*x])/d + (16*A*Cos[2*c]*Sin[2*d*x])/(3*d) - (8*(43*A - 23*B + 9*C)*Tan[c/2])/(3*d) + (8*(22*A - 17*B + 12*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
577,1,1555,313,7.8513822,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{154 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{238 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{98 \sqrt{2} C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{84 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{44 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{52 C \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(27 A \sin \left(\frac{d x}{2}\right)-22 B \sin \left(\frac{d x}{2}\right)+17 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (27 A-22 B+17 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(69 A \sin \left(\frac{d x}{2}\right)-43 B \sin \left(\frac{d x}{2}\right)+23 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (133 \cos (2 c) A+329 A-178 B+78 C-60 B \cos (2 c)+20 C \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{16 (3 A-B) \cos (2 d x) \sin (2 c)}{3 d}+\frac{8 A \cos (3 d x) \sin (3 c)}{5 d}+\frac{8 (133 A-60 B+20 C) \cos (c) \sin (d x)}{5 d}-\frac{16 (3 A-B) \cos (2 c) \sin (2 d x)}{3 d}+\frac{8 A \cos (3 c) \sin (3 d x)}{5 d}+\frac{8 (69 A-43 B+23 C) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^3}","-\frac{(63 A-33 B+13 C) \sin (c+d x)}{10 d \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}+\frac{7 (33 A-17 B+7 C) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(63 A-33 B+13 C) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{(63 A-33 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (33 A-17 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(12 A-7 B+2 C) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"(-154*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (238*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (98*Sqrt[2]*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*E^(I*d*x)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (84*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (44*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (52*C*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(329*A - 178*B + 78*C + 133*A*Cos[2*c] - 60*B*Cos[2*c] + 20*C*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (16*(3*A - B)*Cos[2*d*x]*Sin[2*c])/(3*d) + (8*A*Cos[3*d*x]*Sin[3*c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(27*A*Sin[(d*x)/2] - 22*B*Sin[(d*x)/2] + 17*C*Sin[(d*x)/2]))/(15*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(69*A*Sin[(d*x)/2] - 43*B*Sin[(d*x)/2] + 23*C*Sin[(d*x)/2]))/(3*d) + (8*(133*A - 60*B + 20*C)*Cos[c]*Sin[d*x])/(5*d) - (16*(3*A - B)*Cos[2*c]*Sin[2*d*x])/(3*d) + (8*A*Cos[3*c]*Sin[3*d*x])/(5*d) + (8*(69*A - 43*B + 23*C)*Tan[c/2])/(3*d) - (8*(27*A - 22*B + 17*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
578,1,179,227,2.5057244,"\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((432 A+77 (8 B+7 C)) \cos (c+d x)+4 (48 A+40 B+35 C) \cos (2 (c+d x))+144 A \cos (3 (c+d x))+192 A+120 B \cos (3 (c+d x))+160 B+105 C \cos (3 (c+d x))+332 C)+24 \sqrt{2} (48 A+40 B+35 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3072 d}","\frac{a (48 A+40 B+35 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a (48 A+40 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a (8 B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])]*(24*Sqrt[2]*(48*A + 40*B + 35*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + 4*(192*A + 160*B + 332*C + (432*A + 77*(8*B + 7*C))*Cos[c + d*x] + 4*(48*A + 40*B + 35*C)*Cos[2*(c + d*x)] + 144*A*Cos[3*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 105*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(3072*d)","A",1
579,1,141,179,1.1990713,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+6 B+5 C) \cos (2 (c+d x))+24 A+4 (6 B+5 C) \cos (c+d x)+18 B+31 C)+12 \sqrt{2} (8 A+6 B+5 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{a (8 A+6 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (8 A+6 B+5 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a (6 B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])]*(12*Sqrt[2]*(8*A + 6*B + 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + 4*(24*A + 18*B + 31*C + 4*(6*B + 5*C)*Cos[c + d*x] + 3*(8*A + 6*B + 5*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(192*d)","A",1
580,1,109,131,0.8130723,"\int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (8 A+4 B+3 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 B+2 C \sec (c+d x)+3 C)\right)}{8 d \sqrt{\sec (c+d x)}}","\frac{\sqrt{a} (8 A+4 B+3 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(8*A + 4*B + 3*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*B + 3*C + 2*C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(8*d*Sqrt[Sec[c + d*x]])","A",1
581,1,94,119,0.5868908,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A+C \sec (c+d x))+\sqrt{2} (2 B+C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d \sqrt{\sec (c+d x)}}","\frac{a (2 A-C) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (2 B+C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{d}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(2*B + C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(2*A + C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(2*d*Sqrt[Sec[c + d*x]])","A",1
582,1,94,120,0.7249619,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (A \cos (c+d x)+2 A+3 B)+3 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a (A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{a} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(2*A + 3*B + A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d*Sqrt[Sec[c + d*x]])","A",1
583,1,77,129,0.497478,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (4 A+5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+19 A+20 B+30 C)}{15 d \sqrt{\sec (c+d x)}}","\frac{2 a (7 A+5 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (A+5 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"((19*A + 20*B + 30*C + 2*(4*A + 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d*Sqrt[Sec[c + d*x]])","A",1
584,1,99,178,0.8177461,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((141 A+28 (4 B+5 C)) \cos (c+d x)+6 (6 A+7 B) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+228 A+266 B+280 C)}{210 d \sqrt{\sec (c+d x)}}","\frac{4 a (24 A+28 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((228*A + 266*B + 280*C + (141*A + 28*(4*B + 5*C))*Cos[c + d*x] + 6*(6*A + 7*B)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d*Sqrt[Sec[c + d*x]])","A",1
585,1,121,226,1.3363343,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((752 A+846 B+672 C) \cos (c+d x)+4 (83 A+54 B+63 C) \cos (2 (c+d x))+80 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+1321 A+90 B \cos (3 (c+d x))+1368 B+1596 C)}{1260 d \sqrt{\sec (c+d x)}}","\frac{2 a (16 A+18 B+21 C) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"((1321*A + 1368*B + 1596*C + (752*A + 846*B + 672*C)*Cos[c + d*x] + 4*(83*A + 54*B + 63*C)*Cos[2*(c + d*x)] + 80*A*Cos[3*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d*Sqrt[Sec[c + d*x]])","A",1
586,1,211,283,3.5907942,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (12 (880 A+1070 B+1273 C) \cos (c+d x)+4 (3280 A+3450 B+3059 C) \cos (2 (c+d x))+3520 A \cos (3 (c+d x))+2640 A \cos (4 (c+d x))+10480 A+3000 B \cos (3 (c+d x))+2250 B \cos (4 (c+d x))+11550 B+2660 C \cos (3 (c+d x))+1995 C \cos (4 (c+d x))+13313 C)+240 \sqrt{2} (176 A+150 B+133 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{61440 d}","\frac{a^{3/2} (176 A+150 B+133 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^(9/2)*Sqrt[a*(1 + Sec[c + d*x])]*(240*Sqrt[2]*(176*A + 150*B + 133*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + 4*(10480*A + 11550*B + 13313*C + 12*(880*A + 1070*B + 1273*C)*Cos[c + d*x] + 4*(3280*A + 3450*B + 3059*C)*Cos[2*(c + d*x)] + 3520*A*Cos[3*(c + d*x)] + 3000*B*Cos[3*(c + d*x)] + 2660*C*Cos[3*(c + d*x)] + 2640*A*Cos[4*(c + d*x)] + 2250*B*Cos[4*(c + d*x)] + 1995*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(61440*d)","A",1
587,1,177,233,2.3442049,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((1008 A+1048 B+1155 C) \cos (c+d x)+4 (48 A+88 B+75 C) \cos (2 (c+d x))+336 A \cos (3 (c+d x))+192 A+264 B \cos (3 (c+d x))+352 B+225 C \cos (3 (c+d x))+492 C)+24 \sqrt{2} (112 A+88 B+75 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3072 d}","\frac{a^{3/2} (112 A+88 B+75 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (48 A+56 B+39 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])]*(24*Sqrt[2]*(112*A + 88*B + 75*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + 4*(192*A + 352*B + 492*C + (1008*A + 1048*B + 1155*C)*Cos[c + d*x] + 4*(48*A + 88*B + 75*C)*Cos[2*(c + d*x)] + 336*A*Cos[3*(c + d*x)] + 264*B*Cos[3*(c + d*x)] + 225*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(3072*d)","A",1
588,1,142,181,1.4617211,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+14 B+11 C) \cos (2 (c+d x))+24 A+4 (6 B+11 C) \cos (c+d x)+42 B+49 C)+12 \sqrt{2} (24 A+14 B+11 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{a^{3/2} (24 A+14 B+11 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])]*(12*Sqrt[2]*(24*A + 14*B + 11*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + 4*(24*A + 42*B + 49*C + 4*(6*B + 11*C)*Cos[c + d*x] + 3*(8*A + 14*B + 11*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(192*d)","A",1
589,1,129,183,1.2283922,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (8 A+12 B+7 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) (2 (2 A \cos (2 (c+d x))+2 A+C)+(4 B+7 C) \cos (c+d x))\right)}{8 d \sqrt{\sec (c+d x)}}","\frac{a^{3/2} (8 A+12 B+7 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (8 A-4 B-5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a (4 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(8*A + 12*B + 7*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*((4*B + 7*C)*Cos[c + d*x] + 2*(2*A + C + 2*A*Cos[2*(c + d*x)]))*Sec[c + d*x]^2*Sin[(c + d*x)/2]))/(8*d*Sqrt[Sec[c + d*x]])","A",1
590,1,122,177,1.0695277,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 (5 A+3 B) \cos (c+d x)+A \cos (2 (c+d x))+A+3 C)+3 \sqrt{2} (2 B+3 C) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 d}","\frac{a^{3/2} (2 B+3 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 (8 A+6 B-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{a (2 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(2*B + 3*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(A + 3*C + 2*(5*A + 3*B)*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
591,1,162,172,1.8122258,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{\sec ^3\left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right) (2 (9 A+5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+39 A+50 B+30 C)+15 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 a^{3/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sec[(c + d*x)/2]^3*(a*(1 + Sec[c + d*x]))^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(15*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (39*A + 50*B + 30*C + 2*(9*A + 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))","A",1
592,1,100,181,0.9634159,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((253 A+28 (9 B+5 C)) \cos (c+d x)+6 (13 A+7 B) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+494 A+546 B+700 C)}{210 d \sqrt{\sec (c+d x)}}","\frac{8 a^2 (19 A+21 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (19 A+21 B+35 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d \sqrt{\sec (c+d x)}}+\frac{2 (3 A+7 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*(494*A + 546*B + 700*C + (253*A + 28*(9*B + 5*C))*Cos[c + d*x] + 6*(13*A + 7*B)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d*Sqrt[Sec[c + d*x]])","A",1
593,1,123,232,1.5511666,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (799 A+759 B+756 C) \cos (c+d x)+4 (137 A+117 B+63 C) \cos (2 (c+d x))+170 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+2689 A+90 B \cos (3 (c+d x))+2964 B+3276 C)}{1260 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a*(2689*A + 2964*B + 3276*C + 2*(799*A + 759*B + 756*C)*Cos[c + d*x] + 4*(137*A + 117*B + 63*C)*Cos[2*(c + d*x)] + 170*A*Cos[3*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d*Sqrt[Sec[c + d*x]])","A",1
594,1,158,284,2.2668923,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((34734 A+44 (799 B+759 C)) \cos (c+d x)+8 (1743 A+1507 B+1287 C) \cos (2 (c+d x))+4935 A \cos (3 (c+d x))+1470 A \cos (4 (c+d x))+315 A \cos (5 (c+d x))+55482 A+3740 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+59158 B+1980 C \cos (3 (c+d x))+65208 C)}{27720 d \sqrt{\sec (c+d x)}}","\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a*(55482*A + 59158*B + 65208*C + (34734*A + 44*(799*B + 759*C))*Cos[c + d*x] + 8*(1743*A + 1507*B + 1287*C)*Cos[2*(c + d*x)] + 4935*A*Cos[3*(c + d*x)] + 3740*B*Cos[3*(c + d*x)] + 1980*C*Cos[3*(c + d*x)] + 1470*A*Cos[4*(c + d*x)] + 770*B*Cos[4*(c + d*x)] + 315*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(27720*d*Sqrt[Sec[c + d*x]])","A",1
595,1,245,333,4.5546667,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{11}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((283920 A+303048 B+321370 C) \cos (c+d x)+16 (7480 A+8444 B+8555 C) \cos (2 (c+d x))+127240 A \cos (3 (c+d x))+26080 A \cos (4 (c+d x))+19560 A \cos (5 (c+d x))+93600 A+121124 B \cos (3 (c+d x))+22640 B \cos (4 (c+d x))+16980 B \cos (5 (c+d x))+112464 B+108605 C \cos (3 (c+d x))+20300 C \cos (4 (c+d x))+15225 C \cos (5 (c+d x))+137060 C)+480 \sqrt{2} (1304 A+1132 B+1015 C) \cos ^6(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{491520 d}","\frac{a^{5/2} (1304 A+1132 B+1015 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{512 d}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{768 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{480 d}+\frac{a (12 B+5 C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^(11/2)*Sqrt[a*(1 + Sec[c + d*x])]*(480*Sqrt[2]*(1304*A + 1132*B + 1015*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^6 + 4*(93600*A + 112464*B + 137060*C + (283920*A + 303048*B + 321370*C)*Cos[c + d*x] + 16*(7480*A + 8444*B + 8555*C)*Cos[2*(c + d*x)] + 127240*A*Cos[3*(c + d*x)] + 121124*B*Cos[3*(c + d*x)] + 108605*C*Cos[3*(c + d*x)] + 26080*A*Cos[4*(c + d*x)] + 22640*B*Cos[4*(c + d*x)] + 20300*C*Cos[4*(c + d*x)] + 19560*A*Cos[5*(c + d*x)] + 16980*B*Cos[5*(c + d*x)] + 15225*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(491520*d)","A",1
596,1,213,281,3.4932687,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (12 (1360 A+1950 B+2343 C) \cos (c+d x)+4 (6640 A+6730 B+6509 C) \cos (2 (c+d x))+5440 A \cos (3 (c+d x))+6000 A \cos (4 (c+d x))+20560 A+6520 B \cos (3 (c+d x))+4890 B \cos (4 (c+d x))+22030 B+5660 C \cos (3 (c+d x))+4245 C \cos (4 (c+d x))+24863 C)+240 \sqrt{2} (400 A+326 B+283 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{61440 d}","\frac{a^{5/2} (400 A+326 B+283 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{960 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{240 d}+\frac{a (2 B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^(9/2)*Sqrt[a*(1 + Sec[c + d*x])]*(240*Sqrt[2]*(400*A + 326*B + 283*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + 4*(20560*A + 22030*B + 24863*C + 12*(1360*A + 1950*B + 2343*C)*Cos[c + d*x] + 4*(6640*A + 6730*B + 6509*C)*Cos[2*(c + d*x)] + 5440*A*Cos[3*(c + d*x)] + 6520*B*Cos[3*(c + d*x)] + 5660*C*Cos[3*(c + d*x)] + 6000*A*Cos[4*(c + d*x)] + 4890*B*Cos[4*(c + d*x)] + 4245*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(61440*d)","A",1
597,1,179,233,2.1385436,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((1584 A+2056 B+2203 C) \cos (c+d x)+4 (48 A+136 B+163 C) \cos (2 (c+d x))+528 A \cos (3 (c+d x))+192 A+600 B \cos (3 (c+d x))+544 B+489 C \cos (3 (c+d x))+844 C)+24 \sqrt{2} (304 A+200 B+163 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3072 d}","\frac{a^{5/2} (304 A+200 B+163 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{32 d}+\frac{a (8 B+5 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])]*(24*Sqrt[2]*(304*A + 200*B + 163*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + 4*(192*A + 544*B + 844*C + (1584*A + 2056*B + 2203*C)*Cos[c + d*x] + 4*(48*A + 136*B + 163*C)*Cos[2*(c + d*x)] + 528*A*Cos[3*(c + d*x)] + 600*B*Cos[3*(c + d*x)] + 489*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(3072*d)","A",1
598,1,158,233,1.7336667,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (4 (18 A+6 B+17 C) \cos (c+d x)+3 (8 A+22 B+25 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+24 A+66 B+91 C)+12 \sqrt{2} (40 A+38 B+25 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","\frac{a^{5/2} (40 A+38 B+25 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (24 A-54 B-49 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+42 B+31 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{a (6 B+5 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{12 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])]*(12*Sqrt[2]*(40*A + 38*B + 25*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + 4*(24*A + 66*B + 91*C + 4*(18*A + 6*B + 17*C)*Cos[c + d*x] + 3*(8*A + 22*B + 25*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(192*d)","A",1
599,1,155,233,1.2640809,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (3 (2 A+4 B+11 C) \cos (c+d x)+4 (8 A+3 B) \cos (2 (c+d x))+2 A \cos (3 (c+d x))+32 A+12 B+6 C)+6 \sqrt{2} (8 A+20 B+19 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d}","\frac{a^{5/2} (8 A+20 B+19 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (56 A+12 B-27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{12 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (8 A-12 B-21 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{12 d}-\frac{a (4 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{6 d}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(8*A + 20*B + 19*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 4*(32*A + 12*B + 6*C + 3*(2*A + 4*B + 11*C)*Cos[c + d*x] + 4*(8*A + 3*B)*Cos[2*(c + d*x)] + 2*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d)","A",1
600,1,149,223,1.1254141,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((181 A+160 B+60 C) \cos (c+d x)+2 (14 A+5 B) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+28 A+10 B+30 C)+30 \sqrt{2} (2 B+5 C) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{60 d}","\frac{a^{5/2} (2 B+5 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 (64 A+70 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (16 A+10 B-15 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a (A+B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*(30*Sqrt[2]*(2*B + 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(28*A + 10*B + 30*C + (181*A + 160*B + 60*C)*Cos[c + d*x] + 2*(14*A + 5*B)*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(60*d)","A",1
601,1,194,222,6.3559263,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{\sec ^5\left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-140 (8 A+4 B+C) \sin ^3\left(\frac{1}{2} (c+d x)\right)+210 (4 A+4 B+3 C) \sin \left(\frac{1}{2} (c+d x)\right)+168 (5 A+B) \sin ^5\left(\frac{1}{2} (c+d x)\right)-240 A \sin ^7\left(\frac{1}{2} (c+d x)\right)+105 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{210 d \sec ^{\frac{9}{2}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 a^{5/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+7 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(105*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 210*(4*A + 4*B + 3*C)*Sin[(c + d*x)/2] - 140*(8*A + 4*B + C)*Sin[(c + d*x)/2]^3 + 168*(5*A + B)*Sin[(c + d*x)/2]^5 - 240*A*Sin[(c + d*x)/2]^7))/(210*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","A",1
602,1,124,231,1.9136409,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((3116 A+3030 B+2352 C) \cos (c+d x)+4 (254 A+180 B+63 C) \cos (2 (c+d x))+260 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+5653 A+90 B \cos (3 (c+d x))+6240 B+7476 C)}{1260 d \sqrt{\sec (c+d x)}}","\frac{64 a^3 (13 A+15 B+21 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (13 A+15 B+21 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d \sqrt{\sec (c+d x)}}+\frac{2 a (13 A+15 B+21 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 A+9 B) \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^2*(5653*A + 6240*B + 7476*C + (3116*A + 3030*B + 2352*C)*Cos[c + d*x] + 4*(254*A + 180*B + 63*C)*Cos[2*(c + d*x)] + 260*A*Cos[3*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d*Sqrt[Sec[c + d*x]])","A",1
603,1,157,284,1.5251699,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((69890 A+68552 B+66660 C) \cos (c+d x)+16 (1625 A+1397 B+990 C) \cos (2 (c+d x))+8675 A \cos (3 (c+d x))+2240 A \cos (4 (c+d x))+315 A \cos (5 (c+d x))+114640 A+5720 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+124366 B+1980 C \cos (3 (c+d x))+137280 C)}{27720 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x)}{3465 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (5 A+11 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a^2*(114640*A + 124366*B + 137280*C + (69890*A + 68552*B + 66660*C)*Cos[c + d*x] + 16*(1625*A + 1397*B + 990*C)*Cos[2*(c + d*x)] + 8675*A*Cos[3*(c + d*x)] + 5720*B*Cos[3*(c + d*x)] + 1980*C*Cos[3*(c + d*x)] + 2240*A*Cos[4*(c + d*x)] + 770*B*Cos[4*(c + d*x)] + 315*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(27720*d*Sqrt[Sec[c + d*x]])","A",1
604,1,190,334,1.6116522,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]","\frac{a^2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (4 (453146 A+454285 B+445588 C) \cos (c+d x)+(746519 A+676000 B+581152 C) \cos (2 (c+d x))+287060 A \cos (3 (c+d x))+94010 A \cos (4 (c+d x))+23940 A \cos (5 (c+d x))+3465 A \cos (6 (c+d x))+2798182 A+225550 B \cos (3 (c+d x))+58240 B \cos (4 (c+d x))+8190 B \cos (5 (c+d x))+2980640 B+148720 C \cos (3 (c+d x))+20020 C \cos (4 (c+d x))+3233516 C)}{720720 d \sqrt{\sec (c+d x)}}","\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{15015 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x)}{9009 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{45045 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a (5 A+13 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d \sec ^{\frac{11}{2}}(c+d x)}",1,"(a^2*(2798182*A + 2980640*B + 3233516*C + 4*(453146*A + 454285*B + 445588*C)*Cos[c + d*x] + (746519*A + 676000*B + 581152*C)*Cos[2*(c + d*x)] + 287060*A*Cos[3*(c + d*x)] + 225550*B*Cos[3*(c + d*x)] + 148720*C*Cos[3*(c + d*x)] + 94010*A*Cos[4*(c + d*x)] + 58240*B*Cos[4*(c + d*x)] + 20020*C*Cos[4*(c + d*x)] + 23940*A*Cos[5*(c + d*x)] + 8190*B*Cos[5*(c + d*x)] + 3465*A*Cos[6*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(720720*d*Sqrt[Sec[c + d*x]])","A",1
605,1,198,241,1.4673046,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(48 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \sqrt{2} (8 A-14 B+9 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(3 (8 A-2 B+7 C)+2 (6 B-C) \sec (c+d x)+8 C \sec ^2(c+d x)\right)\right)}{12 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(8 A-2 B+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(8 A-14 B+9 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(6 B-C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(48*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] - 3*Sqrt[2]*(8*A - 14*B + 9*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(3*(8*A - 2*B + 7*C) + 2*(6*B - C)*Sec[c + d*x] + 8*C*Sec[c + d*x]^2)*Sin[(c + d*x)/2]))/(12*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
606,1,174,195,0.8773227,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-8 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (8 A-4 B+7 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 B+2 C \sec (c+d x)-C)\right)}{2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(8 A-4 B+7 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{(4 B-C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-8*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + Sqrt[2]*(8*A - 4*B + 7*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*B - C + 2*C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
607,1,107,141,0.6334808,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(2 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (2 B-C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{d \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(2*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + Sqrt[2]*(2*B - C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*C*Sec[c + d*x]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
608,1,96,138,0.5276526,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-\left((A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 A \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(-((A - B + C)*ArcTanh[Sin[(c + d*x)/2]]) + Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*A*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
609,1,88,143,0.5995275,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(3 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) (A \cos (c+d x)-A+3 B)\right)}{3 d \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(3*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(-A + 3*B + A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
610,1,155,191,0.8784486,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(15 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+20 (A+B) \sin ^3\left(\frac{1}{2} (c+d x)\right)-30 (A+C) \sin \left(\frac{1}{2} (c+d x)\right)-24 A \sin ^5\left(\frac{1}{2} (c+d x)\right)\right)}{15 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 (13 A-5 B+15 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(-4*Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(15*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] - 30*(A + C)*Sin[(c + d*x)/2] + 20*(A + B)*Sin[(c + d*x)/2]^3 - 24*A*Sin[(c + d*x)/2]^5))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
611,1,175,237,1.5774092,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{4 \cos \left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-140 (2 A+B+C) \sin ^3\left(\frac{1}{2} (c+d x)\right)+105 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+168 (2 A+B) \sin ^5\left(\frac{1}{2} (c+d x)\right)-240 A \sin ^7\left(\frac{1}{2} (c+d x)\right)+210 B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{2 (43 A-91 B+35 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (31 A-7 B+35 C) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(4*Cos[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(105*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + 210*B*Sin[(c + d*x)/2] - 140*(2*A + B + C)*Sin[(c + d*x)/2]^3 + 168*(2*A + B)*Sin[(c + d*x)/2]^5 - 240*A*Sin[(c + d*x)/2]^7))/(105*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
612,1,118,152,0.5431677,"\int \frac{\sqrt{\sec (c+d x)} \left(a A+(A b+a B) \sec (c+d x)+b B \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(2 (a-b) (A-B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (2 a B+2 A b-b B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b B \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{d \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (a-b) (A-B) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 a B+2 A b-b B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(2*(a - b)*(A - B)*ArcTanh[Sin[(c + d*x)/2]] + Sqrt[2]*(2*A*b + 2*a*B - b*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*b*B*Sec[c + d*x]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
613,1,239,260,2.3455138,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-2 (5 A-9 B+13 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{\sqrt{2} (8 A-12 B+19 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) ((-2 A+6 B-7 C) \cos (2 (c+d x))-2 A+(8 B-6 C) \cos (c+d x)+6 B-3 C)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{d \sqrt{\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(5 A-9 B+13 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(8 A-12 B+19 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(A-B+2 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(2 A-6 B+7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-2*(5*A - 9*B + 13*C)*ArcTanh[Sin[(c + d*x)/2]] - (Sqrt[2]*(8*A - 12*B + 19*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 + ((-2*A + 6*B - 3*C + (8*B - 6*C)*Cos[c + d*x] + (-2*A + 6*B - 7*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Sin[(c + d*x)/2])/2)/(-1 + Sin[(c + d*x)/2]^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
614,1,177,202,2.2281341,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) (A-B+2 C \sec (c+d x)+3 C)+(A-5 B+9 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sqrt{2} (2 B-3 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A-5 B+9 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(A-B+3 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}",1,"((A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((A - 5*B + 9*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + 2*Sqrt[2]*(2*B - 3*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (A - B + 3*C + 2*C*Sec[c + d*x])*Tan[(c + d*x)/2]))/(a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
615,1,175,149,1.3165007,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{A-B+C}{\sin \left(\frac{1}{2} (c+d x)\right)-1}+\frac{A-B+C}{\sin \left(\frac{1}{2} (c+d x)\right)+1}+2 (3 A+B-5 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(3 A+B-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(3*A + B - 5*C)*ArcTanh[Sin[(c + d*x)/2]] + 8*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (A - B + C)/(-1 + Sin[(c + d*x)/2]) + (A - B + C)/(1 + Sin[(c + d*x)/2])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
616,1,147,161,1.1474059,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 (-7 A+3 B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (4 A \cos (c+d x)+5 A-B+C)\right)}{d \sqrt{\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(7 A-3 B-C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(-7*A + 3*B + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(5*A - B + C + 4*A*Cos[c + d*x])*Sec[(c + d*x)/2]*Tan[(c + d*x)/2]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
617,1,126,213,1.5663473,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (-12 (A-B) \cos (c+d x)+2 A \cos (2 (c+d x))-17 A+15 B-3 C)+6 (11 A-7 B+3 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{12 a d \sqrt{a (\sec (c+d x)+1)}}","\frac{(11 A-7 B+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{(19 A-15 B+3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{(7 A-3 B+3 C) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(6*(11*A - 7*B + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 + 2*(-17*A + 15*B - 3*C - 12*(A - B)*Cos[c + d*x] + 2*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(12*a*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
618,1,148,263,2.0938107,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (3 (39 A+20 (C-B)) \cos (c+d x)+(10 B-6 A) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+141 A-85 B+75 C)-30 (15 A-11 B+7 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{60 a d \sqrt{a (\sec (c+d x)+1)}}","-\frac{(15 A-11 B+7 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B+5 C) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}+\frac{(147 A-95 B+75 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \sec (c+d x)+a}}-\frac{(39 A-35 B+15 C) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(-30*(15*A - 11*B + 7*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 + 2*(141*A - 85*B + 75*C + 3*(39*A + 20*(-B + C))*Cos[c + d*x] + (-6*A + 10*B)*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(60*a*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
619,1,222,254,3.6533036,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left((6 A-86 B+230 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sec ^3\left(\frac{1}{2} (c+d x)\right) (2 (7 A-15 B+55 C) \cos (c+d x)+(3 A-11 B+35 C) \cos (2 (c+d x))+3 A-11 B+67 C)+32 \sqrt{2} (2 B-5 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(3 A-43 B+115 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A-11 B+35 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(A+7 B-15 C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((6*A - 86*B + 230*C)*ArcTanh[Sin[(c + d*x)/2]] + 32*Sqrt[2]*(2*B - 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + ((3*A - 11*B + 67*C + 2*(7*A - 15*B + 55*C)*Cos[c + d*x] + (3*A - 11*B + 35*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Sec[c + d*x]*Tan[(c + d*x)/2])/2))/(4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
620,1,204,201,1.6231756,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 (5 A+3 B-43 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{\sin \left(\frac{1}{2} (c+d x)\right) ((5 A+3 B-11 C) \cos (c+d x)+A+7 B-15 C)+64 \sqrt{2} C \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2}\right)}{4 d (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(5 A+3 B-43 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(5 A+3 B-11 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^5*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(5*A + 3*B - 43*C)*ArcTanh[Sin[(c + d*x)/2]] + (64*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + (A + 7*B - 15*C + (5*A + 3*B - 11*C)*Cos[c + d*x])*Sin[(c + d*x)/2])/(-1 + Sin[(c + d*x)/2]^2)^2))/(4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
621,1,119,163,1.9623045,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \left(8 (19 A+5 B+3 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \sin \left(\frac{1}{2} (c+d x)\right) ((13 A-5 B-3 C) \cos (c+d x)+9 A-B-7 C)\right)}{64 a d (a (\sec (c+d x)+1))^{3/2}}","\frac{(19 A+5 B+3 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(8*(19*A + 5*B + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 - 4*(9*A - B - 7*C + (13*A - 5*B - 3*C)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(64*a*d*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
622,1,128,211,1.7796494,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((85 A-13 B+5 C) \cos (c+d x)+16 A \cos (2 (c+d x))+65 A-9 B+C)-8 (75 A-19 B-5 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{64 a d (a (\sec (c+d x)+1))^{3/2}}","-\frac{(75 A-19 B-5 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A-9 B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(13 A-5 B-3 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(-8*(75*A - 19*B - 5*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(65*A - 9*B + C + (85*A - 13*B + 5*C)*Cos[c + d*x] + 16*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(64*a*d*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
623,1,146,261,2.091019,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((-479 A+255 B-39 C) \cos (c+d x)+(48 B-80 A) \cos (2 (c+d x))+8 A \cos (3 (c+d x))-379 A+195 B-27 C)+24 (163 A-75 B+19 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 a d (a (\sec (c+d x)+1))^{3/2}}","\frac{(163 A-75 B+19 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(299 A-147 B+27 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(95 A-39 B+15 C) \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(17 A-9 B+C) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(24*(163*A - 75*B + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(-379*A + 195*B - 27*C + (-479*A + 255*B - 39*C)*Cos[c + d*x] + (-80*A + 48*B)*Cos[2*(c + d*x)] + 8*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(192*a*d*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
624,1,221,313,3.2127101,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(15 (283 A-163 B+75 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) (5 (887 A-479 B+255 C) \cos (c+d x)+16 (52 A-25 B+15 C) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+12 A \cos (4 (c+d x))+3491 A+40 B \cos (3 (c+d x))-1895 B+975 C)\right)}{60 a d \sqrt{\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(283 A-163 B+75 C) \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A-85 B+45 C) \sin (c+d x)}{80 a^2 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(2671 A-1495 B+735 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{240 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(787 A-475 B+195 C) \sin (c+d x)}{240 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(21 A-13 B+5 C) \sin (c+d x)}{16 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"-1/60*(Sec[(c + d*x)/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(15*(283*A - 163*B + 75*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 - ((3491*A - 1895*B + 975*C + 5*(887*A - 479*B + 255*C)*Cos[c + d*x] + 16*(52*A - 25*B + 15*C)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 40*B*Cos[3*(c + d*x)] + 12*A*Cos[4*(c + d*x)])*Sin[(c + d*x)/2])/2))/(a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
625,1,5449,446,21.2637981,"\int (a+a \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{3 \sqrt{2} A \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 (5 B+2 C) \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{10 d (\sec (c+d x)+1)}-\frac{3^{3/4} (5 B+2 C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} (a \sec (c+d x)+a)^{2/3} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{10 \sqrt[3]{2} d (1-\sec (c+d x)) (\sec (c+d x)+1) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 C \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{5 d}",1,"Result too large to show","B",0
626,1,2931,390,19.8358676,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(1/3),x]","\text{Result too large to show}","\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}-\frac{3^{3/4} (2 B-C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2 \sqrt[3]{2} d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}+\frac{3 C \tan (c+d x)}{2 d \sqrt[3]{a \sec (c+d x)+a}}",1,"(3*C*Cos[c + d*x]^2*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Tan[(c + d*x)/2])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(1/3)) + (2^(2/3)*Cos[c + d*x]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3) + Sec[(c + d*x)/2]^2*(B*(1 + Sec[c + d*x])^(2/3) - (C*(1 + Sec[c + d*x])^(2/3))/2))*Tan[(c + d*x)/2]*(-((2*A - 2*B + C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2) + (27*(2*A + 2*B - C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(1/3)*((Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(-((2*A - 2*B + C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2) + (27*(2*A + 2*B - C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(3*2^(1/3)) + (2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*(-((2*A - 2*B + C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]) - (2*A - 2*B + C)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) - (2*(2*A - 2*B + C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (27*(2*A + 2*B - C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (27*(2*A + 2*B - C)*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (27*(2*A + 2*B - C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/3 + (2*2^(2/3)*Tan[(c + d*x)/2]*(-((2*A - 2*B + C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2) + (27*(2*A + 2*B - C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
627,1,3029,402,19.5873329,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{4/3}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(4/3),x]","\text{Result too large to show}","\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{a d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 (A-B+C) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^{4/3}}+\frac{3^{3/4} (A-B-4 C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[3]{2} a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^2*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-6*Sec[(c + d*x)/2]*(A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2] + C*Sin[(c + d*x)/2]))/5 + (3*Sec[(c + d*x)/2]^3*(A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2] + C*Sin[(c + d*x)/2]))/5))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(4/3)) + (2*2^(2/3)*Cos[c + d*x]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3) + Sec[(c + d*x)/2]^2*(-1/5*(A*(1 + Sec[c + d*x])^(2/3)) + (B*(1 + Sec[c + d*x])^(2/3))/5 + (4*C*(1 + Sec[c + d*x])^(2/3))/5))*Tan[(c + d*x)/2]*((-6*A + B + 4*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(4*A + B + 4*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(4/3)*((2^(2/3)*Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*((-6*A + B + 4*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(4*A + B + 4*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/15 + (2*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*((-6*A + B + 4*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + (-6*A + B + 4*C)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*(-6*A + B + 4*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (27*(4*A + B + 4*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (27*(4*A + B + 4*C)*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (27*(4*A + B + 4*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/15 + (4*2^(2/3)*Tan[(c + d*x)/2]*((-6*A + B + 4*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(4*A + B + 4*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(45*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
628,1,3111,466,19.7215884,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{7/3}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(7/3),x]","\text{Result too large to show}","-\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(-\frac{5}{6};\frac{1}{2},1;\frac{1}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 a^2 d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 (4 A-4 B-7 C) \tan (c+d x)}{55 a^2 d (\sec (c+d x)+1) \sqrt[3]{a \sec (c+d x)+a}}+\frac{3^{3/4} (4 A-4 B-7 C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{55 \sqrt[3]{2} a^2 d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}-\frac{3 (A-B+C) \tan (c+d x)}{11 d (a \sec (c+d x)+a)^{7/3}}",1,"(Cos[c + d*x]^2*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(7/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-6*Sec[(c + d*x)/2]*(20*A*Sin[(c + d*x)/2] - 9*B*Sin[(c + d*x)/2] - 2*C*Sin[(c + d*x)/2]))/55 - (3*Sec[(c + d*x)/2]^5*(A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2] + C*Sin[(c + d*x)/2]))/22 + (3*Sec[(c + d*x)/2]^3*(25*A*Sin[(c + d*x)/2] - 14*B*Sin[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2]))/55))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(7/3)) + (2*2^(2/3)*Cos[c + d*x]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(7/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3) + Sec[(c + d*x)/2]^2*((-3*A*(1 + Sec[c + d*x])^(2/3))/11 + (4*B*(1 + Sec[c + d*x])^(2/3))/55 + (7*C*(1 + Sec[c + d*x])^(2/3))/55))*Tan[(c + d*x)/2]*((-70*A + 4*B + 7*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(40*A + 4*B + 7*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(165*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(7/3)*((2^(2/3)*Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*((-70*A + 4*B + 7*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(40*A + 4*B + 7*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/165 + (2*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*((-70*A + 4*B + 7*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + (-70*A + 4*B + 7*C)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*(-70*A + 4*B + 7*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (27*(40*A + 4*B + 7*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (27*(40*A + 4*B + 7*C)*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (27*(40*A + 4*B + 7*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/165 + (4*2^(2/3)*Tan[(c + d*x)/2]*((-70*A + 4*B + 7*C)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(40*A + 4*B + 7*C)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(495*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
629,1,4995,839,20.384742,"\int (a+a \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{3 C \tan (c+d x) (\sec (c+d x) a+a)^{4/3}}{7 d}+\frac{3 \sqrt{2} a A F_1\left(\frac{11}{6};\frac{1}{2},1;\frac{17}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right) (\sec (c+d x)+1) \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{11 d \sqrt{1-\sec (c+d x)}}+\frac{15 \sqrt[4]{3} a (7 B+4 C) E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{14\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{5\ 3^{3/4} \left(1-\sqrt{3}\right) a (7 B+4 C) F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{28\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 a (7 B+4 C) \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{28 d}-\frac{15 \left(1+\sqrt{3}\right) a (7 B+4 C) \tan (c+d x) \sqrt[3]{\sec (c+d x) a+a}}{28 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}",1,"(Cos[c + d*x]^2*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(a*(1 + Sec[c + d*x]))^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((3*(28*A + 35*B + 20*C)*Sin[c + d*x])/14 + (3*Sec[c + d*x]*(7*B*Sin[c + d*x] + 8*C*Sin[c + d*x]))/14 + (6*C*Sec[c + d*x]*Tan[c + d*x])/7))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(4/3)) + (Cos[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(4*A*(1 + Sec[c + d*x])^(1/3) + (5*B*(1 + Sec[c + d*x])^(1/3))/2 + (10*C*(1 + Sec[c + d*x])^(1/3))/7 + Cos[c + d*x]*(-6*A*(1 + Sec[c + d*x])^(1/3) - (15*B*(1 + Sec[c + d*x])^(1/3))/2 - (30*C*(1 + Sec[c + d*x])^(1/3))/7))*Tan[(c + d*x)/2]*(-(((28*A + 35*B + 20*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (Cos[(c + d*x)/2]^2*(18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + 27*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*(14*A + 35*B + 20*C) + 3*(28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2)))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(21*2^(2/3)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(4/3)*((Sec[(c + d*x)/2]^2*(-(((28*A + 35*B + 20*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (Cos[(c + d*x)/2]^2*(18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + 27*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*(14*A + 35*B + 20*C) + 3*(28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2)))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(42*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (Tan[(c + d*x)/2]*(-(((28*A + 35*B + 20*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - ((28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (2*(28*A + 35*B + 20*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3)) - (Tan[(c + d*x)/2]*(18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + 27*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*(14*A + 35*B + 20*C) + 3*(28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2)))/((-1 + Tan[(c + d*x)/2]^2)^2*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) - (Cos[(c + d*x)/2]*Sin[(c + d*x)/2]*(18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + 27*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*(14*A + 35*B + 20*C) + 3*(28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2)))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) - (Cos[(c + d*x)/2]^2*(18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + 27*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*(14*A + 35*B + 20*C) + 3*(28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2))*(2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2) + (Cos[(c + d*x)/2]^2*(81*(28*A + 35*B + 20*C)*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2] - 18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2]^2 + 18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^3 + 27*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9)*(-4*(14*A + 35*B + 20*C) + 3*(28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2) + 18*(28*A + 35*B + 20*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(21*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) - (2^(1/3)*Tan[(c + d*x)/2]*(-(((28*A + 35*B + 20*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (Cos[(c + d*x)/2]^2*(18*(28*A + 35*B + 20*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + 27*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*(14*A + 35*B + 20*C) + 3*(28*A + 35*B + 20*C)*Tan[(c + d*x)/2]^2)))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(63*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
630,1,4191,786,20.6394722,"\int \sqrt[3]{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{3 \sqrt{2} A \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 d \sqrt{1-\sec (c+d x)}}-\frac{3 \left(1+\sqrt{3}\right) (4 B+C) \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{4 d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}+\frac{3^{3/4} \left(1-\sqrt{3}\right) (4 B+C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{4\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 \sqrt[4]{3} (4 B+C) \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a} E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{2\ 2^{2/3} d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 C \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{4 d}",1,"(Cos[c + d*x]^2*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(a*(1 + Sec[c + d*x]))^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((3*(4*B + C)*Sin[c + d*x])/2 + (3*C*Tan[c + d*x])/2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(1 + Sec[c + d*x])^(1/3)) + (Cos[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*A*(1 + Sec[c + d*x])^(1/3) + 2*B*(1 + Sec[c + d*x])^(1/3) + (C*(1 + Sec[c + d*x])^(1/3))/2 + Cos[c + d*x]*(-6*B*(1 + Sec[c + d*x])^(1/3) - (3*C*(1 + Sec[c + d*x])^(1/3))/2))*Tan[(c + d*x)/2]*(-(((4*B + C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (9*((3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(8*A - 4*B - C + (8*A - 7*(4*B + C))*Cos[c + d*x]))/2 + 2*(4*B + C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(3*2^(2/3)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(1/3)*((Sec[(c + d*x)/2]^2*(-(((4*B + C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (9*((3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(8*A - 4*B - C + (8*A - 7*(4*B + C))*Cos[c + d*x]))/2 + 2*(4*B + C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(6*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (Tan[(c + d*x)/2]*(-(((4*B + C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - ((4*B + C)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (2*(4*B + C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3)) - (9*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*((3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(8*A - 4*B - C + (8*A - 7*(4*B + C))*Cos[c + d*x]))/2 + 2*(4*B + C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/((-1 + Tan[(c + d*x)/2]^2)^2*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) - (9*((3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(8*A - 4*B - C + (8*A - 7*(4*B + C))*Cos[c + d*x]))/2 + 2*(4*B + C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2)*(2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2) + (9*((-3*(8*A - 7*(4*B + C))*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[c + d*x])/2 + 2*(4*B + C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 2*(4*B + C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sin[c + d*x]*Tan[(c + d*x)/2]^2 + (3*(8*A - 4*B - C + (8*A - 7*(4*B + C))*Cos[c + d*x])*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/2 + 2*(4*B + C)*Cos[c + d*x]*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(3*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) - (2^(1/3)*Tan[(c + d*x)/2]*(-(((4*B + C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (9*((3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(8*A - 4*B - C + (8*A - 7*(4*B + C))*Cos[c + d*x]))/2 + 2*(4*B + C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/((-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
631,1,4253,803,22.0256818,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","\frac{3 \sqrt[3]{2} \sqrt[4]{3} (A-B+2 C) E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sqrt[3]{\sec (c+d x) a+a} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x)}{a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3^{3/4} \left(1-\sqrt{3}\right) (A-B+2 C) F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sqrt[3]{\sec (c+d x) a+a} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x)}{2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x)+1)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3 \sqrt{2} A F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right) \sqrt[3]{\sec (c+d x) a+a} \tan (c+d x)}{5 a d \sqrt{1-\sec (c+d x)}}-\frac{3 (A-B+C) \tan (c+d x)}{d (\sec (c+d x) a+a)^{2/3}}-\frac{3 \left(1+\sqrt{3}\right) (A-B+2 C) \sqrt[3]{\sec (c+d x) a+a} \tan (c+d x)}{a d (\sec (c+d x)+1)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}",1,"(Cos[c + d*x]^2*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(1 + Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-6*Sec[(c + d*x)/2]*(A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2] + C*Sin[(c + d*x)/2]) + 6*(A - B + 2*C)*Sin[c + d*x]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(2/3)) - (2*2^(1/3)*Cos[c + d*x]^2*(1 + Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(4*A*(1 + Sec[c + d*x])^(1/3) - 2*B*(1 + Sec[c + d*x])^(1/3) + 4*C*(1 + Sec[c + d*x])^(1/3) + Cos[c + d*x]*(-6*A*(1 + Sec[c + d*x])^(1/3) + 6*B*(1 + Sec[c + d*x])^(1/3) - 12*C*(1 + Sec[c + d*x])^(1/3)))*Tan[(c + d*x)/2]*(((A - B + 2*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (9*(-3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B - 2*C + (-5*A + 7*(B - 2*C))*Cos[c + d*x]) - 4*(A - B + 2*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*(-1/3*(2^(1/3)*Sec[(c + d*x)/2]^2*(((A - B + 2*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (9*(-3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B - 2*C + (-5*A + 7*(B - 2*C))*Cos[c + d*x]) - 4*(A - B + 2*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3) - (2*2^(1/3)*Tan[(c + d*x)/2]*(((A - B + 2*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + ((A - B + 2*C)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) - (2*(A - B + 2*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3)) - (9*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(-3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B - 2*C + (-5*A + 7*(B - 2*C))*Cos[c + d*x]) - 4*(A - B + 2*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)^2*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) - (9*(-3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B - 2*C + (-5*A + 7*(B - 2*C))*Cos[c + d*x]) - 4*(A - B + 2*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2)*(2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2) + (9*(3*(-5*A + 7*(B - 2*C))*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[c + d*x] - 4*(A - B + 2*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 4*(A - B + 2*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sin[c + d*x]*Tan[(c + d*x)/2]^2 - 3*(A + B - 2*C + (-5*A + 7*(B - 2*C))*Cos[c + d*x])*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) - 4*(A - B + 2*C)*Cos[c + d*x]*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(3*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (4*2^(1/3)*Tan[(c + d*x)/2]*(((A - B + 2*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (9*(-3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B - 2*C + (-5*A + 7*(B - 2*C))*Cos[c + d*x]) - 4*(A - B + 2*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
632,1,4383,856,23.1832451,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/3),x]","\text{Result too large to show}","\frac{3 \sqrt[3]{2} \sqrt[4]{3} (2 A-2 B-5 C) E\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x)}{7 a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}+\frac{3^{3/4} \left(1-\sqrt{3}\right) (2 A-2 B-5 C) F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right) \sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \tan (c+d x)}{7\ 2^{2/3} a d (1-\sec (c+d x)) (\sec (c+d x) a+a)^{2/3} \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}}}-\frac{3 \sqrt{2} A F_1\left(-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right) \tan (c+d x)}{a d \sqrt{1-\sec (c+d x)} (\sec (c+d x) a+a)^{2/3}}-\frac{3 (2 A-2 B-5 C) \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3}}-\frac{3 \left(1+\sqrt{3}\right) (2 A-2 B-5 C) \sqrt[3]{\sec (c+d x)+1} \tan (c+d x)}{7 a d (\sec (c+d x) a+a)^{2/3} \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)}-\frac{3 (A-B+C) \tan (c+d x)}{7 d (\sec (c+d x) a+a)^{5/3}}",1,"(Cos[c + d*x]^2*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(1 + Sec[c + d*x])^(5/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-6*Sec[(c + d*x)/2]*(10*A*Sin[(c + d*x)/2] - 3*B*Sin[(c + d*x)/2] - 4*C*Sin[(c + d*x)/2]))/7 + (3*Sec[(c + d*x)/2]^3*(A*Sin[(c + d*x)/2] - B*Sin[(c + d*x)/2] + C*Sin[(c + d*x)/2]))/7 + (6*(9*A - 2*B - 5*C)*Sin[c + d*x])/7))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a*(1 + Sec[c + d*x]))^(5/3)) + (2*2^(1/3)*Cos[c + d*x]^2*(1 + Sec[c + d*x])^(5/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((32*A*(1 + Sec[c + d*x])^(1/3))/7 - (4*B*(1 + Sec[c + d*x])^(1/3))/7 - (10*C*(1 + Sec[c + d*x])^(1/3))/7 + Cos[c + d*x]*((-54*A*(1 + Sec[c + d*x])^(1/3))/7 + (12*B*(1 + Sec[c + d*x])^(1/3))/7 + (30*C*(1 + Sec[c + d*x])^(1/3))/7))*Tan[(c + d*x)/2]*(-(((9*A - 2*B - 5*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(5*A + 2*B + 5*C - 7*(7*A - 2*B - 5*C)*Cos[c + d*x]) + 4*(9*A - 2*B - 5*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(5/3)*((2^(1/3)*Sec[(c + d*x)/2]^2*(-(((9*A - 2*B - 5*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(5*A + 2*B + 5*C - 7*(7*A - 2*B - 5*C)*Cos[c + d*x]) + 4*(9*A - 2*B - 5*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(21*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (2*2^(1/3)*Tan[(c + d*x)/2]*(-(((9*A - 2*B - 5*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - ((9*A - 2*B - 5*C)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (2*(9*A - 2*B - 5*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3)) - (9*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(5*A + 2*B + 5*C - 7*(7*A - 2*B - 5*C)*Cos[c + d*x]) + 4*(9*A - 2*B - 5*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)^2*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(5*A + 2*B + 5*C - 7*(7*A - 2*B - 5*C)*Cos[c + d*x]) + 4*(9*A - 2*B - 5*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2)*(2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2) + (9*(21*(7*A - 2*B - 5*C)*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[c + d*x] + 4*(9*A - 2*B - 5*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 4*(9*A - 2*B - 5*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sin[c + d*x]*Tan[(c + d*x)/2]^2 + 3*(5*A + 2*B + 5*C - 7*(7*A - 2*B - 5*C)*Cos[c + d*x])*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 4*(9*A - 2*B - 5*C)*Cos[c + d*x]*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(21*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) - (4*2^(1/3)*Tan[(c + d*x)/2]*(-(((9*A - 2*B - 5*C)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) + (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(5*A + 2*B + 5*C - 7*(7*A - 2*B - 5*C)*Cos[c + d*x]) + 4*(9*A - 2*B - 5*C)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(63*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
633,0,0,259,4.3173918,"\int \sec ^m(c+d x) (a+a \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^m*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\int \sec ^m(c+d x) (a+a \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","\frac{2^{n+\frac{1}{2}} \tan (c+d x) (A (m+n+1)-B (m+n+1)+C (m-n)) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};1-m,\frac{1}{2}-n;\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (m+n+1)}+\frac{2^{n+\frac{3}{2}} (B (m+n+1)+C n) \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};1-m,-n-\frac{1}{2};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d (m+n+1)}+\frac{C \sin (c+d x) \sec ^{m+1}(c+d x) (a \sec (c+d x)+a)^n}{d (m+n+1)}",1,"Integrate[Sec[c + d*x]^m*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x]","F",-1
634,0,0,258,2.4717497,"\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","\frac{(A n+B (n+1)-C (n+1)) \sin (c+d x) \sec ^{1-n}(c+d x) \left(\frac{\sec (c+d x)+1}{1-\sec (c+d x)}\right)^{\frac{1}{2}-n} (a \sec (c+d x)+a)^n \, _2F_1\left(\frac{1}{2}-n,-n;1-n;-\frac{2 \sec (c+d x)}{1-\sec (c+d x)}\right)}{d n (n+1) (\sec (c+d x)+1)}+\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}+\frac{C 2^{n+\frac{3}{2}} \tan (c+d x) (\sec (c+d x)+1)^{-n-\frac{1}{2}} (a \sec (c+d x)+a)^n F_1\left(\frac{1}{2};n+1,-n-\frac{1}{2};\frac{3}{2};1-\sec (c+d x),\frac{1}{2} (1-\sec (c+d x))\right)}{d}",1,"Integrate[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x]","F",-1
635,1,38,38,0.1570725,"\int \left(\frac{\sec ^{-n}(c+d x) (a+a \sec (c+d x))^n (-a (B+A n+B n)-a C (1+n) \sec (c+d x))}{a (1+n)}+\sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)\right) \, dx","Integrate[((a + a*Sec[c + d*x])^n*(-(a*(B + A*n + B*n)) - a*C*(1 + n)*Sec[c + d*x]))/(a*(1 + n)*Sec[c + d*x]^n) + Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a (\sec (c+d x)+1))^n}{d (n+1)}","\frac{A \sin (c+d x) \sec ^{-n}(c+d x) (a \sec (c+d x)+a)^n}{d (n+1)}",1,"(A*(a*(1 + Sec[c + d*x]))^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n)","A",1
636,1,1817,171,14.6566199,"\int (a+a \sec (c+d x))^m \left(B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + a*Sec[c + d*x])^m*(B - C + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\frac{2^{m+1} B \cos (c+d x) \, _2F_1\left(\frac{1}{2},m+1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^m \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^m (a (\sec (c+d x)+1))^{m+1} (B-C+C \sec (c+d x)) \tan \left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^{-m-1}}{d (-\cos (c+d x) C+C+B \cos (c+d x))}+\frac{C \left(4 m \cos (c+d x) \, _2F_1\left(\frac{1}{2},m+2;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^m+\cos (c+d x)+1\right) (a (\sec (c+d x)+1))^{m+1} (B-C+C \sec (c+d x)) \tan \left(\frac{1}{2} (c+d x)\right)}{d (2 m+1) (-\cos (c+d x) C+C+B \cos (c+d x)) (\sec (c+d x)+1)}+\frac{30 B F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) (a (\sec (c+d x)+1))^{m+1} (B-C+C \sec (c+d x)) \sin (c+d x) \left(3 F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d (-\cos (c+d x) C+C+B \cos (c+d x)) \left(45 F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2 (-2 \cos (c+d x) m+2 m+\cos (2 (c+d x))+1) \cos ^2\left(\frac{1}{2} (c+d x)\right)+40 \left(F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2 \cos (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sin ^2\left(\frac{1}{2} (c+d x)\right) \left(-48 \left(2 F_1\left(\frac{5}{2};m,3;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 m F_1\left(\frac{5}{2};m+1,2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+m (m+1) F_1\left(\frac{5}{2};m+2,1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \cot (c+d x) \csc (c+d x) \sin ^4\left(\frac{1}{2} (c+d x)\right)-5 F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 m-2 (m+2) \cos (c+d x)+\cos (2 (c+d x))+1)+5 m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 m-2 (m+2) \cos (c+d x)+\cos (2 (c+d x))+1)\right)\right) (\sec (c+d x)+1)}-\frac{30 C F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \cos ^2\left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) (a (\sec (c+d x)+1))^{m+1} (B-C+C \sec (c+d x)) \sin (c+d x) \left(3 F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 \left(F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)\right)}{d (-\cos (c+d x) C+C+B \cos (c+d x)) \left(45 F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right){}^2 (-2 \cos (c+d x) m+2 m+\cos (2 (c+d x))+1) \cos ^2\left(\frac{1}{2} (c+d x)\right)+40 \left(F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right){}^2 \cos (c+d x) \sin ^2\left(\frac{1}{2} (c+d x)\right) \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 F_1\left(\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) \sin ^2\left(\frac{1}{2} (c+d x)\right) \left(-48 \left(2 F_1\left(\frac{5}{2};m,3;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)-2 m F_1\left(\frac{5}{2};m+1,2;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)+m (m+1) F_1\left(\frac{5}{2};m+2,1;\frac{7}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right)\right) \cot (c+d x) \csc (c+d x) \sin ^4\left(\frac{1}{2} (c+d x)\right)-5 F_1\left(\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 m-2 (m+2) \cos (c+d x)+\cos (2 (c+d x))+1)+5 m F_1\left(\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left(\frac{1}{2} (c+d x)\right),-\tan ^2\left(\frac{1}{2} (c+d x)\right)\right) (2 m-2 (m+2) \cos (c+d x)+\cos (2 (c+d x))+1)\right)\right) (\sec (c+d x)+1)}}{a}","\frac{\sqrt{2} (B-C) \tan (c+d x) (\sec (c+d x)+1) (a \sec (c+d x)+a)^m F_1\left(m+\frac{3}{2};\frac{1}{2},1;m+\frac{5}{2};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d (2 m+3) \sqrt{1-\sec (c+d x)}}+\frac{C 2^{m+\frac{3}{2}} \tan (c+d x) (\sec (c+d x)+1)^{-m-\frac{1}{2}} (a \sec (c+d x)+a)^m \, _2F_1\left(\frac{1}{2},-m-\frac{1}{2};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)}{d}",1,"((C*(1 + Cos[c + d*x] + 4*m*Cos[c + d*x]*Hypergeometric2F1[1/2, 2 + m, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m)*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Tan[(c + d*x)/2])/(d*(1 + 2*m)*(C + B*Cos[c + d*x] - C*Cos[c + d*x])*(1 + Sec[c + d*x])) + (2^(1 + m)*B*Cos[c + d*x]*Hypergeometric2F1[1/2, 1 + m, 3/2, Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^m*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^m*(1 + Sec[c + d*x])^(-1 - m)*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Tan[(c + d*x)/2])/(d*(C + B*Cos[c + d*x] - C*Cos[c + d*x])) + (30*B*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*Cos[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Sin[c + d*x]*(3*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(d*(C + B*Cos[c + d*x] - C*Cos[c + d*x])*(1 + Sec[c + d*x])*(45*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 + 2*m - 2*m*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(-5*AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) - 48*(2*AppellF1[5/2, m, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*m*AppellF1[5/2, 1 + m, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + m*(1 + m)*AppellF1[5/2, 2 + m, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cot[c + d*x]*Csc[c + d*x]*Sin[(c + d*x)/2]^4) + 40*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)) - (30*C*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*Cos[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(1 + m)*(B - C + C*Sec[c + d*x])*Sin[c + d*x]*(3*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))/(d*(C + B*Cos[c + d*x] - C*Cos[c + d*x])*(1 + Sec[c + d*x])*(45*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]^2*Cos[(c + d*x)/2]^2*(1 + 2*m - 2*m*Cos[c + d*x] + Cos[2*(c + d*x)]) + 6*AppellF1[1/2, m, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[(c + d*x)/2]^2*(-5*AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) + 5*m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(1 + 2*m - 2*(2 + m)*Cos[c + d*x] + Cos[2*(c + d*x)]) - 48*(2*AppellF1[5/2, m, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - 2*m*AppellF1[5/2, 1 + m, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + m*(1 + m)*AppellF1[5/2, 2 + m, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cot[c + d*x]*Csc[c + d*x]*Sin[(c + d*x)/2]^4) + 40*(AppellF1[3/2, m, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - m*AppellF1[3/2, 1 + m, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])^2*Cos[c + d*x]*Sin[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2)))/a","B",0
637,1,96,140,0.9226835,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(15 a (4 A+3 C) \sec (c+d x)+30 a C \sec ^3(c+d x)+8 b \left(5 (A+2 C) \tan ^2(c+d x)+15 (A+C)+3 C \tan ^4(c+d x)\right)\right)+15 a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{120 d}","\frac{a (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{b (5 A+4 C) \tan ^3(c+d x)}{15 d}+\frac{b (5 A+4 C) \tan (c+d x)}{5 d}+\frac{b C \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(15*a*(4*A + 3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*a*(4*A + 3*C)*Sec[c + d*x] + 30*a*C*Sec[c + d*x]^3 + 8*b*(15*(A + C) + 5*(A + 2*C)*Tan[c + d*x]^2 + 3*C*Tan[c + d*x]^4)))/(120*d)","A",1
638,1,80,117,0.5088411,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(8 a \left(3 (A+C)+C \tan ^2(c+d x)\right)+3 b (4 A+3 C) \sec (c+d x)+6 b C \sec ^3(c+d x)\right)+3 b (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{24 d}","\frac{a (3 A+2 C) \tan (c+d x)}{3 d}+\frac{a C \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{b (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (4 A+3 C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*b*(4*A + 3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*b*(4*A + 3*C)*Sec[c + d*x] + 6*b*C*Sec[c + d*x]^3 + 8*a*(3*(A + C) + C*Tan[c + d*x]^2)))/(24*d)","A",1
639,1,59,86,0.335291,"\int \sec (c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(3 a C \sec (c+d x)+6 b (A+C)+2 b C \tan ^2(c+d x)\right)+3 a (2 A+C) \tanh ^{-1}(\sin (c+d x))}{6 d}","\frac{a (2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a C \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b (3 A+2 C) \tan (c+d x)}{3 d}+\frac{b C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*a*(2*A + C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*b*(A + C) + 3*a*C*Sec[c + d*x] + 2*b*C*Tan[c + d*x]^2))/(6*d)","A",1
640,1,67,58,0.0196802,"\int (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","a A x+\frac{a C \tan (c+d x)}{d}+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d}","a A x+\frac{a C \tan (c+d x)}{d}+\frac{b (2 A+C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d}",1,"a*A*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (b*C*ArcTanh[Sin[c + d*x]])/(2*d) + (a*C*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
641,1,54,42,0.0202271,"\int \cos (c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+A b x+\frac{b C \tan (c+d x)}{d}","\frac{a A \sin (c+d x)}{d}+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+A b x+\frac{b C \tan (c+d x)}{d}",1,"A*b*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Cos[d*x]*Sin[c])/d + (a*A*Cos[c]*Sin[d*x])/d + (b*C*Tan[c + d*x])/d","A",1
642,1,73,58,0.1734486,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{a A (c+d x)}{2 d}+\frac{a A \sin (2 (c+d x))}{4 d}+a C x+\frac{A b \sin (c) \cos (d x)}{d}+\frac{A b \cos (c) \sin (d x)}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} a x (A+2 C)+\frac{A b \sin (c+d x)}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*C*x + (a*A*(c + d*x))/(2*d) + (b*C*ArcTanh[Sin[c + d*x]])/d + (A*b*Cos[d*x]*Sin[c])/d + (A*b*Cos[c]*Sin[d*x])/d + (a*A*Sin[2*(c + d*x)])/(4*d)","A",1
643,1,64,77,0.1322165,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{3 a (3 A+4 C) \sin (c+d x)+a A \sin (3 (c+d x))+3 A b \sin (2 (c+d x))+6 A b c+6 A b d x+12 b C d x}{12 d}","\frac{a (2 A+3 C) \sin (c+d x)}{3 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{A b \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} b x (A+2 C)",1,"(6*A*b*c + 6*A*b*d*x + 12*b*C*d*x + 3*a*(3*A + 4*C)*Sin[c + d*x] + 3*A*b*Sin[2*(c + d*x)] + a*A*Sin[3*(c + d*x)])/(12*d)","A",1
644,1,84,95,0.2367858,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{24 a (A+C) \sin (2 (c+d x))+3 a A \sin (4 (c+d x))+36 a A c+36 a A d x+48 a c C+48 a C d x+24 b (3 A+4 C) \sin (c+d x)+8 A b \sin (3 (c+d x))}{96 d}","\frac{a (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} a x (3 A+4 C)+\frac{b (A+C) \sin (c+d x)}{d}-\frac{A b \sin ^3(c+d x)}{3 d}",1,"(36*a*A*c + 48*a*c*C + 36*a*A*d*x + 48*a*C*d*x + 24*b*(3*A + 4*C)*Sin[c + d*x] + 24*a*(A + C)*Sin[2*(c + d*x)] + 8*A*b*Sin[3*(c + d*x)] + 3*a*A*Sin[4*(c + d*x)])/(96*d)","A",1
645,1,89,131,0.3542834,"\int \cos ^5(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\frac{-160 a (2 A+C) \sin ^3(c+d x)+480 a (A+C) \sin (c+d x)+96 a A \sin ^5(c+d x)+15 b (4 (3 A+4 C) (c+d x)+8 (A+C) \sin (2 (c+d x))+A \sin (4 (c+d x)))}{480 d}","-\frac{a (4 A+5 C) \sin ^3(c+d x)}{15 d}+\frac{a (4 A+5 C) \sin (c+d x)}{5 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{b (3 A+4 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{A b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} b x (3 A+4 C)",1,"(480*a*(A + C)*Sin[c + d*x] - 160*a*(2*A + C)*Sin[c + d*x]^3 + 96*a*A*Sin[c + d*x]^5 + 15*b*(4*(3*A + 4*C)*(c + d*x) + 8*(A + C)*Sin[2*(c + d*x)] + A*Sin[4*(c + d*x)]))/(480*d)","A",1
646,1,275,226,2.5218258,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","-\frac{\sec ^5(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(60 a b (4 A+3 C) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sin (c+d x) \left(24 \left(5 a^2 (A+C)+b^2 (5 A+4 C)\right) \cos (2 (c+d x))+30 a^2 A \cos (4 (c+d x))+90 a^2 A+20 a^2 C \cos (4 (c+d x))+100 a^2 C+15 a b (12 A+17 C) \cos (c+d x)+60 a A b \cos (3 (c+d x))+45 a b C \cos (3 (c+d x))+20 A b^2 \cos (4 (c+d x))+100 A b^2+16 b^2 C \cos (4 (c+d x))+128 b^2 C\right)\right)}{120 d (A \cos (2 (c+d x))+A+2 C)}","\frac{\left(a^2 C+2 b^2 (5 A+4 C)\right) \tan (c+d x) (a+b \sec (c+d x))^2}{30 b^2 d}+\frac{a \left(2 a^2 C+20 A b^2+13 b^2 C\right) \tan (c+d x) \sec (c+d x)}{60 b d}+\frac{\left(a^4 C+2 a^2 b^2 (5 A+3 C)+2 b^4 (5 A+4 C)\right) \tan (c+d x)}{15 b^2 d}+\frac{a b (4 A+3 C) \tanh ^{-1}(\sin (c+d x))}{4 d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^3}{10 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^3}{5 b d}",1,"-1/120*((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(60*a*b*(4*A + 3*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - (90*a^2*A + 100*A*b^2 + 100*a^2*C + 128*b^2*C + 15*a*b*(12*A + 17*C)*Cos[c + d*x] + 24*(5*a^2*(A + C) + b^2*(5*A + 4*C))*Cos[2*(c + d*x)] + 60*a*A*b*Cos[3*(c + d*x)] + 45*a*b*C*Cos[3*(c + d*x)] + 30*a^2*A*Cos[4*(c + d*x)] + 20*A*b^2*Cos[4*(c + d*x)] + 20*a^2*C*Cos[4*(c + d*x)] + 16*b^2*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
647,1,1123,170,6.3388673,"\int \sec (c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\left(-8 A a^2-4 C a^2-4 A b^2-3 b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \cos ^4(c+d x)}{4 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C)}+\frac{\left(8 A a^2+4 C a^2+4 A b^2+3 b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \cos ^4(c+d x)}{4 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C)}+\frac{2 a b C (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{4 (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \left(3 a A b \sin \left(\frac{1}{2} (c+d x)\right)+2 a b C \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \left(3 a A b \sin \left(\frac{1}{2} (c+d x)\right)+2 a b C \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\left(12 C a^2+8 b C a+12 A b^2+9 b^2 C\right) (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \cos ^4(c+d x)}{24 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(-12 C a^2-8 b C a-12 A b^2-9 b^2 C\right) (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \cos ^4(c+d x)}{24 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 a b C (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^4(c+d x)}{3 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{b^2 C (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \cos ^4(c+d x)}{8 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{b^2 C (a+b \sec (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \cos ^4(c+d x)}{8 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}","\frac{a \left(a^2 (-C)+12 A b^2+8 b^2 C\right) \tan (c+d x)}{6 b d}+\frac{\left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left(2 a^2 C-3 b^2 (4 A+3 C)\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d}",1,"((-8*a^2*A - 4*A*b^2 - 4*a^2*C - 3*b^2*C)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((8*a^2*A + 4*A*b^2 + 4*a^2*C + 3*b^2*C)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + (b^2*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((12*A*b^2 + 12*a^2*C + 8*a*b*C + 9*b^2*C)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(24*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*a*b*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (b^2*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (2*a*b*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((-12*A*b^2 - 12*a^2*C - 8*a*b*C - 9*b^2*C)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/(24*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(3*a*A*b*Sin[(c + d*x)/2] + 2*a*b*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(3*a*A*b*Sin[(c + d*x)/2] + 2*a*b*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
648,1,242,103,1.2382927,"\int (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^3(c+d x) \left(2 \sin (c+d x) \left(\left(3 a^2 C+3 A b^2+2 b^2 C\right) \cos (2 (c+d x))+3 a^2 C+6 a b C \cos (c+d x)+3 A b^2+4 b^2 C\right)+9 a \cos (c+d x) \left(a A (c+d x)-b (2 A+C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b (2 A+C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+3 a \cos (3 (c+d x)) \left(a A (c+d x)-b (2 A+C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b (2 A+C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{12 d}","\frac{\left(2 C \left(a^2+b^2\right)+3 A b^2\right) \tan (c+d x)}{3 d}+a^2 A x+\frac{a b (2 A+C) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a b C \tan (c+d x) \sec (c+d x)}{3 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(Sec[c + d*x]^3*(9*a*Cos[c + d*x]*(a*A*(c + d*x) - b*(2*A + C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*(2*A + C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*a*Cos[3*(c + d*x)]*(a*A*(c + d*x) - b*(2*A + C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*(2*A + C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(3*A*b^2 + 3*a^2*C + 4*b^2*C + 6*a*b*C*Cos[c + d*x] + (3*A*b^2 + 3*a^2*C + 2*b^2*C)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(12*d)","B",1
649,1,352,109,0.8829999,"\int \cos (c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^2(c+d x) \left(\left(a^2 A+2 b^2 C\right) \sin (c+d x)+\cos (2 (c+d x)) \left(-\left(C \left(2 a^2+b^2\right)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2 C+2 A b^2+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 a A b (c+d x)\right)+a^2 A \sin (3 (c+d x))-2 a^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 a A b c+4 a A b d x+4 a b C \sin (2 (c+d x))-2 A b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 A b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-b^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d}","\frac{\left(C \left(2 a^2+b^2\right)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{2 a b (A-C) \tan (c+d x)}{d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^2}{d}+2 a A b x-\frac{b^2 (2 A-C) \tan (c+d x) \sec (c+d x)}{2 d}",1,"(Sec[c + d*x]^2*(4*a*A*b*c + 4*a*A*b*d*x - 2*A*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*a^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - b^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*A*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*a^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Cos[2*(c + d*x)]*(4*a*A*b*(c + d*x) - (2*A*b^2 + (2*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (2*A*b^2 + 2*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (a^2*A + 2*b^2*C)*Sin[c + d*x] + 4*a*b*C*Sin[2*(c + d*x)] + a^2*A*Sin[3*(c + d*x)]))/(4*d)","B",1
650,1,130,103,0.7731539,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{2 (c+d x) \left(a^2 (A+2 C)+2 A b^2\right)+\tan (c+d x) \left(a^2 A \cos (2 (c+d x))+a^2 A+4 b^2 C\right)+8 a A b \sin (c+d x)-8 a b C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 a b C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{1}{2} x \left(a^2 (A+2 C)+2 A b^2\right)+\frac{a A b \sin (c+d x)}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{2 a b C \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d}",1,"(2*(2*A*b^2 + a^2*(A + 2*C))*(c + d*x) - 8*a*b*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*a*b*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*a*A*b*Sin[c + d*x] + (a^2*A + 4*b^2*C + a^2*A*Cos[2*(c + d*x)])*Tan[c + d*x])/(4*d)","A",1
651,1,144,112,0.2539435,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{3 \left(a^2 (3 A+4 C)+4 A b^2\right) \sin (c+d x)+a^2 A \sin (3 (c+d x))+6 a A b \sin (2 (c+d x))+12 a A b c+12 a A b d x+24 a b c C+24 a b C d x-12 b^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 d}","\frac{\left(a^2 (2 A+3 C)+2 A b^2\right) \sin (c+d x)}{3 d}+\frac{a A b \sin (c+d x) \cos (c+d x)}{3 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+a b x (A+2 C)+\frac{b^2 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(12*a*A*b*c + 24*a*b*c*C + 12*a*A*b*d*x + 24*a*b*C*d*x - 12*b^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] + 6*a*A*b*Sin[2*(c + d*x)] + a^2*A*Sin[3*(c + d*x)])/(12*d)","A",1
652,1,104,145,0.4124728,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{12 (c+d x) \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right)+24 \left(a^2 (A+C)+A b^2\right) \sin (2 (c+d x))+3 a^2 A \sin (4 (c+d x))+48 a b (3 A+4 C) \sin (c+d x)+16 a A b \sin (3 (c+d x))}{96 d}","\frac{\left(a^2 (3 A+4 C)+2 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right)+\frac{2 a b (2 A+3 C) \sin (c+d x)}{3 d}+\frac{a A b \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"(12*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*(c + d*x) + 48*a*b*(3*A + 4*C)*Sin[c + d*x] + 24*(A*b^2 + a^2*(A + C))*Sin[2*(c + d*x)] + 16*a*A*b*Sin[3*(c + d*x)] + 3*a^2*A*Sin[4*(c + d*x)])/(96*d)","A",1
653,1,126,161,0.4483233,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{30 \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \sin (c+d x)+5 \left(a^2 (5 A+4 C)+4 A b^2\right) \sin (3 (c+d x))+3 a^2 A \sin (5 (c+d x))+60 a b (3 A+4 C) (c+d x)+120 a b (A+C) \sin (2 (c+d x))+15 a A b \sin (4 (c+d x))}{240 d}","-\frac{\left(a^2 (4 A+5 C)+2 A b^2\right) \sin ^3(c+d x)}{15 d}+\frac{\left(a^2+b^2\right) (4 A+5 C) \sin (c+d x)}{5 d}+\frac{a b (3 A+4 C) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{a A b \sin (c+d x) \cos ^3(c+d x)}{10 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}+\frac{1}{4} a b x (3 A+4 C)",1,"(60*a*b*(3*A + 4*C)*(c + d*x) + 30*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sin[c + d*x] + 120*a*b*(A + C)*Sin[2*(c + d*x)] + 5*(4*A*b^2 + a^2*(5*A + 4*C))*Sin[3*(c + d*x)] + 15*a*A*b*Sin[4*(c + d*x)] + 3*a^2*A*Sin[5*(c + d*x)])/(240*d)","A",1
654,1,407,306,3.6529085,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^6(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(2 \sin (c+d x) \left(600 a^3 A \cos (3 (c+d x))+120 a^3 A \cos (5 (c+d x))+560 a^3 C \cos (3 (c+d x))+80 a^3 C \cos (5 (c+d x))+16 a \left(a^2 (75 A+80 C)+24 b^2 (10 A+11 C)\right) \cos (c+d x)+20 b \left(18 a^2 (4 A+5 C)+5 b^2 (6 A+5 C)\right) \cos (2 (c+d x))+360 a^2 A b \cos (4 (c+d x))+1080 a^2 A b+270 a^2 b C \cos (4 (c+d x))+1530 a^2 b C+1680 a A b^2 \cos (3 (c+d x))+240 a A b^2 \cos (5 (c+d x))+1344 a b^2 C \cos (3 (c+d x))+192 a b^2 C \cos (5 (c+d x))+90 A b^3 \cos (4 (c+d x))+510 A b^3+75 b^3 C \cos (4 (c+d x))+745 b^3 C\right)-240 b \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{1920 d (A \cos (2 (c+d x))+A+2 C)}","\frac{b \left(6 a^2 (4 A+3 C)+b^2 (6 A+5 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(2 a^2 C+5 b^2 (6 A+5 C)\right) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b^2 d}+\frac{a \left(2 a^2 C+30 A b^2+21 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b^2 d}+\frac{a \left(2 a^4 C+a^2 b^2 (30 A+17 C)+24 b^4 (5 A+4 C)\right) \tan (c+d x)}{60 b^2 d}+\frac{\left(4 a^4 C+12 a^2 b^2 (5 A+3 C)+15 b^4 (6 A+5 C)\right) \tan (c+d x) \sec (c+d x)}{240 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^4}{15 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}",1,"((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^6*(-240*b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(1080*a^2*A*b + 510*A*b^3 + 1530*a^2*b*C + 745*b^3*C + 16*a*(24*b^2*(10*A + 11*C) + a^2*(75*A + 80*C))*Cos[c + d*x] + 20*b*(18*a^2*(4*A + 5*C) + 5*b^2*(6*A + 5*C))*Cos[2*(c + d*x)] + 600*a^3*A*Cos[3*(c + d*x)] + 1680*a*A*b^2*Cos[3*(c + d*x)] + 560*a^3*C*Cos[3*(c + d*x)] + 1344*a*b^2*C*Cos[3*(c + d*x)] + 360*a^2*A*b*Cos[4*(c + d*x)] + 90*A*b^3*Cos[4*(c + d*x)] + 270*a^2*b*C*Cos[4*(c + d*x)] + 75*b^3*C*Cos[4*(c + d*x)] + 120*a^3*A*Cos[5*(c + d*x)] + 240*a*A*b^2*Cos[5*(c + d*x)] + 80*a^3*C*Cos[5*(c + d*x)] + 192*a*b^2*C*Cos[5*(c + d*x)])*Sin[c + d*x]))/(1920*d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
655,1,324,234,1.8663675,"\int \sec (c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\frac{\sec ^5(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(120 a \left(4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2 \sin (c+d x) \left(60 a^3 C \cos (3 (c+d x))+45 a \left(4 a^2 C+12 A b^2+17 b^2 C\right) \cos (c+d x)+48 b \left(15 a^2 (A+C)+b^2 (5 A+4 C)\right) \cos (2 (c+d x))+180 a^2 A b \cos (4 (c+d x))+540 a^2 A b+120 a^2 b C \cos (4 (c+d x))+600 a^2 b C+180 a A b^2 \cos (3 (c+d x))+135 a b^2 C \cos (3 (c+d x))+40 A b^3 \cos (4 (c+d x))+200 A b^3+32 b^3 C \cos (4 (c+d x))+256 b^3 C\right)\right)}{480 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a \left(4 a^2 (2 A+C)+3 b^2 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{\left(3 a^2 C-4 b^2 (5 A+4 C)\right) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}+\frac{a \left(-6 a^2 C+100 A b^2+71 b^2 C\right) \tan (c+d x) \sec (c+d x)}{120 d}-\frac{\left(3 a^4 C-4 a^2 b^2 (20 A+13 C)-4 b^4 (5 A+4 C)\right) \tan (c+d x)}{30 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}",1,"-1/480*((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(120*a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*C))*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*(540*a^2*A*b + 200*A*b^3 + 600*a^2*b*C + 256*b^3*C + 45*a*(12*A*b^2 + 4*a^2*C + 17*b^2*C)*Cos[c + d*x] + 48*b*(15*a^2*(A + C) + b^2*(5*A + 4*C))*Cos[2*(c + d*x)] + 180*a*A*b^2*Cos[3*(c + d*x)] + 60*a^3*C*Cos[3*(c + d*x)] + 135*a*b^2*C*Cos[3*(c + d*x)] + 180*a^2*A*b*Cos[4*(c + d*x)] + 40*A*b^3*Cos[4*(c + d*x)] + 120*a^2*b*C*Cos[4*(c + d*x)] + 32*b^3*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
656,1,1241,167,6.4249319,"\int (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\left(-4 A b^3-3 C b^3-24 a^2 A b-12 a^2 C b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \cos ^5(c+d x)}{4 d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C)}+\frac{\left(4 A b^3+3 C b^3+24 a^2 A b+12 a^2 C b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \cos ^5(c+d x)}{4 d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C)}+\frac{2 a^3 A (c+d x) (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C)}+\frac{a b^2 C (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \left(C \sin \left(\frac{1}{2} (c+d x)\right) a^3+3 A b^2 \sin \left(\frac{1}{2} (c+d x)\right) a+2 b^2 C \sin \left(\frac{1}{2} (c+d x)\right) a\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \left(C \sin \left(\frac{1}{2} (c+d x)\right) a^3+3 A b^2 \sin \left(\frac{1}{2} (c+d x)\right) a+2 b^2 C \sin \left(\frac{1}{2} (c+d x)\right) a\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\left(4 A b^3+3 C b^3+4 a C b^2+12 a^2 C b\right) (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(-4 A b^3-3 C b^3-4 a C b^2-12 a^2 C b\right) (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{a b^2 C (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{b^3 C (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{b^3 C (a+b \sec (c+d x))^3 \left(C \sec ^2(c+d x)+A\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}","a^3 A x+\frac{a \left(C \left(a^2+4 b^2\right)+6 A b^2\right) \tan (c+d x)}{2 d}+\frac{b \left(12 a^2 (2 A+C)+b^2 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b \left(2 a^2 C+b^2 (4 A+3 C)\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a C \tan (c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(2*a^3*A*(c + d*x)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((-24*a^2*A*b - 4*A*b^3 - 12*a^2*b*C - 3*b^3*C)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((24*a^2*A*b + 4*A*b^3 + 12*a^2*b*C + 3*b^3*C)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])) + (b^3*C*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((4*A*b^3 + 12*a^2*b*C + 4*a*b^2*C + 3*b^3*C)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (a*b^2*C*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (b^3*C*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (a*b^2*C*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((-4*A*b^3 - 12*a^2*b*C - 4*a*b^2*C - 3*b^3*C)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(3*a*A*b^2*Sin[(c + d*x)/2] + a^3*C*Sin[(c + d*x)/2] + 2*a*b^2*C*Sin[(c + d*x)/2]))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(3*a*A*b^2*Sin[(c + d*x)/2] + a^3*C*Sin[(c + d*x)/2] + 2*a*b^2*C*Sin[(c + d*x)/2]))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
657,1,325,167,1.6273786,"\int \cos (c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^3(c+d x) \left(9 a \cos (c+d x) \left(-\left(2 a^2 C+6 A b^2+3 b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2 C+6 A b^2+3 b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 a A b (c+d x)\right)+3 a \cos (3 (c+d x)) \left(-\left(2 a^2 C+6 A b^2+3 b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2 C+6 A b^2+3 b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 a A b (c+d x)\right)+2 \sin (c+d x) \left(3 a^3 A \cos (3 (c+d x))+2 \left(9 a^2 b C+3 A b^3+2 b^3 C\right) \cos (2 (c+d x))+9 a \left(a^2 A+2 b^2 C\right) \cos (c+d x)+18 a^2 b C+6 A b^3+8 b^3 C\right)\right)}{24 d}","-\frac{b \left(a^2 (6 A-8 C)-b^2 (3 A+2 C)\right) \tan (c+d x)}{3 d}+\frac{a \left(2 a^2 C+6 A b^2+3 b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+3 a^2 A b x-\frac{a b^2 (6 A-5 C) \tan (c+d x) \sec (c+d x)}{6 d}-\frac{b (3 A-C) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^3}{d}",1,"(Sec[c + d*x]^3*(9*a*Cos[c + d*x]*(6*a*A*b*(c + d*x) - (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*a*Cos[3*(c + d*x)]*(6*a*A*b*(c + d*x) - (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (6*A*b^2 + 2*a^2*C + 3*b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(6*A*b^3 + 18*a^2*b*C + 8*b^3*C + 9*a*(a^2*A + 2*b^2*C)*Cos[c + d*x] + 2*(3*A*b^3 + 9*a^2*b*C + 2*b^3*C)*Cos[2*(c + d*x)] + 3*a^3*A*Cos[3*(c + d*x)])*Sin[c + d*x]))/(24*d)","A",1
658,1,287,168,1.90599,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 A \sin (2 (c+d x))+2 a (c+d x) \left(a^2 (A+2 C)+6 A b^2\right)-2 b \left(C \left(6 a^2+b^2\right)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \left(C \left(6 a^2+b^2\right)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 a^2 A b \sin (c+d x)+\frac{12 a b^2 C \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 a b^2 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{b^3 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^3 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 d}","\frac{b \left(C \left(6 a^2+b^2\right)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} a x \left(a^2 (A+2 C)+6 A b^2\right)-\frac{3 a b^2 (3 A-2 C) \tan (c+d x)}{2 d}+\frac{3 A b \sin (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^3}{2 d}-\frac{b^3 (4 A-C) \tan (c+d x) \sec (c+d x)}{2 d}",1,"(2*a*(6*A*b^2 + a^2*(A + 2*C))*(c + d*x) - 2*b*(2*A*b^2 + (6*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*b*(2*A*b^2 + (6*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^3*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*a*b^2*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^3*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (12*a*b^2*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*a^2*A*b*Sin[c + d*x] + a^3*A*Sin[2*(c + d*x)])/(4*d)","A",1
659,1,184,163,1.0103025,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 A \sin (3 (c+d x))+3 a \left(a^2 (3 A+4 C)+12 A b^2\right) \sin (c+d x)+9 a^2 A b \sin (2 (c+d x))+18 a^2 A b c+18 a^2 A b d x+36 a^2 b c C+36 a^2 b C d x-36 a b^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+36 a b^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+12 A b^3 c+12 A b^3 d x+12 b^3 C \tan (c+d x)}{12 d}","\frac{a \left(a^2 (2 A+3 C)+3 A b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} b x \left(3 a^2 (A+2 C)+2 A b^2\right)+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac{A b \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{3 a b^2 C \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^3 (5 A-6 C) \tan (c+d x)}{6 d}",1,"(18*a^2*A*b*c + 12*A*b^3*c + 36*a^2*b*c*C + 18*a^2*A*b*d*x + 12*A*b^3*d*x + 36*a^2*b*C*d*x - 36*a*b^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 36*a*b^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*a*(12*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] + 9*a^2*A*b*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)] + 12*b^3*C*Tan[c + d*x])/(12*d)","A",1
660,1,177,182,0.6420551,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 A \sin (4 (c+d x))+4 a (c+d x) \left(a^2 (3 A+4 C)+12 b^2 (A+2 C)\right)+8 a \left(a^2 (A+C)+3 A b^2\right) \sin (2 (c+d x))+8 b \left(3 a^2 (3 A+4 C)+4 A b^2\right) \sin (c+d x)+8 a^2 A b \sin (3 (c+d x))-32 b^3 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 b^3 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{32 d}","\frac{b \left(a^2 (4 A+6 C)+A b^2\right) \sin (c+d x)}{2 d}+\frac{a \left(a^2 (3 A+4 C)+2 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x \left(a^2 (3 A+4 C)+12 b^2 (A+2 C)\right)+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{4 d}+\frac{b^3 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(4*a*(12*b^2*(A + 2*C) + a^2*(3*A + 4*C))*(c + d*x) - 32*b^3*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*b^3*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*b*(4*A*b^2 + 3*a^2*(3*A + 4*C))*Sin[c + d*x] + 8*a*(3*A*b^2 + a^2*(A + C))*Sin[2*(c + d*x)] + 8*a^2*A*b*Sin[3*(c + d*x)] + a^3*A*Sin[4*(c + d*x)])/(32*d)","A",1
661,1,155,218,0.8118546,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{6 a^3 A \sin (5 (c+d x))+60 b (c+d x) \left(3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right)+60 a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \sin (c+d x)+10 a \left(a^2 (5 A+4 C)+12 A b^2\right) \sin (3 (c+d x))+120 b \left(3 a^2 (A+C)+A b^2\right) \sin (2 (c+d x))+45 a^2 A b \sin (4 (c+d x))}{480 d}","\frac{a \left(2 a^2 (4 A+5 C)+15 b^2 (2 A+3 C)\right) \sin (c+d x)}{15 d}+\frac{a \left(2 a^2 (4 A+5 C)+3 A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{30 d}+\frac{3 b \left(5 a^2 (3 A+4 C)+2 A b^2\right) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{1}{8} b x \left(3 a^2 (3 A+4 C)+4 b^2 (A+2 C)\right)+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}+\frac{3 A b \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{20 d}",1,"(60*b*(4*b^2*(A + 2*C) + 3*a^2*(3*A + 4*C))*(c + d*x) + 60*a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sin[c + d*x] + 120*b*(A*b^2 + 3*a^2*(A + C))*Sin[2*(c + d*x)] + 10*a*(12*A*b^2 + a^2*(5*A + 4*C))*Sin[3*(c + d*x)] + 45*a^2*A*b*Sin[4*(c + d*x)] + 6*a^3*A*Sin[5*(c + d*x)])/(480*d)","A",1
662,1,253,257,1.0860004,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{45 a^3 A \sin (4 (c+d x))+5 a^3 A \sin (6 (c+d x))+300 a^3 A c+300 a^3 A d x+30 a^3 C \sin (4 (c+d x))+360 a^3 c C+360 a^3 C d x+15 a \left(a^2 (15 A+16 C)+48 b^2 (A+C)\right) \sin (2 (c+d x))+120 b \left(3 a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \sin (c+d x)+300 a^2 A b \sin (3 (c+d x))+36 a^2 A b \sin (5 (c+d x))+240 a^2 b C \sin (3 (c+d x))+90 a A b^2 \sin (4 (c+d x))+1080 a A b^2 c+1080 a A b^2 d x+1440 a b^2 c C+1440 a b^2 C d x+80 A b^3 \sin (3 (c+d x))}{960 d}","-\frac{b \left(3 a^2 (4 A+5 C)+A b^2\right) \sin ^3(c+d x)}{15 d}+\frac{b \left(9 a^2 (4 A+5 C)+b^2 (11 A+15 C)\right) \sin (c+d x)}{15 d}+\frac{a \left(5 a^2 (5 A+6 C)+6 A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac{a \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} a x \left(a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}+\frac{A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}",1,"(300*a^3*A*c + 1080*a*A*b^2*c + 360*a^3*c*C + 1440*a*b^2*c*C + 300*a^3*A*d*x + 1080*a*A*b^2*d*x + 360*a^3*C*d*x + 1440*a*b^2*C*d*x + 120*b*(2*b^2*(3*A + 4*C) + 3*a^2*(5*A + 6*C))*Sin[c + d*x] + 15*a*(48*b^2*(A + C) + a^2*(15*A + 16*C))*Sin[2*(c + d*x)] + 300*a^2*A*b*Sin[3*(c + d*x)] + 80*A*b^3*Sin[3*(c + d*x)] + 240*a^2*b*C*Sin[3*(c + d*x)] + 45*a^3*A*Sin[4*(c + d*x)] + 90*a*A*b^2*Sin[4*(c + d*x)] + 30*a^3*C*Sin[4*(c + d*x)] + 36*a^2*A*b*Sin[5*(c + d*x)] + 5*a^3*A*Sin[6*(c + d*x)])/(960*d)","A",1
663,1,371,381,2.7038517,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\frac{\sec ^6(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(-2 b^2 \left(3 \left(6 C \left(7 a^2+b^2\right)+7 A b^2\right) \sin (2 (c+d x))+140 a b C \sin (c+d x)+30 b^2 C \tan (c+d x)\right)-105 a b \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \sin (c+d x) \cos ^4(c+d x)-70 a b \left(6 a^2 C+6 A b^2+5 b^2 C\right) \sin (c+d x) \cos ^2(c+d x)+105 a b \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-4 \left(35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right) \sin (c+d x) \cos ^5(c+d x)-4 \left(35 a^4 C+42 a^2 b^2 (5 A+4 C)+4 b^4 (7 A+6 C)\right) \sin (c+d x) \cos ^3(c+d x)\right)}{210 d (A \cos (2 (c+d x))+A+2 C)}","\frac{a b \left(a^2 (8 A+6 C)+b^2 (6 A+5 C)\right) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac{\left(a^2 C+3 b^2 (7 A+6 C)\right) \tan (c+d x) (a+b \sec (c+d x))^4}{105 b^2 d}+\frac{a \left(2 a^2 C+42 A b^2+31 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^3}{210 b^2 d}+\frac{\left(2 a^4 C+3 a^2 b^2 (14 A+9 C)+8 b^4 (7 A+6 C)\right) \tan (c+d x) (a+b \sec (c+d x))^2}{210 b^2 d}+\frac{a \left(4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right) \tan (c+d x) \sec (c+d x)}{420 b d}+\frac{\left(2 a^6 C+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)+8 b^6 (7 A+6 C)\right) \tan (c+d x)}{105 b^2 d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^5}{21 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d}",1,"-1/210*((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^6*(105*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 70*a*b*(6*A*b^2 + 6*a^2*C + 5*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x] - 4*(35*a^4*C + 42*a^2*b^2*(5*A + 4*C) + 4*b^4*(7*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x] - 105*a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*Cos[c + d*x]^4*Sin[c + d*x] - 4*(35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Cos[c + d*x]^5*Sin[c + d*x] - 2*b^2*(140*a*b*C*Sin[c + d*x] + 3*(7*A*b^2 + 6*(7*a^2 + b^2)*C)*Sin[2*(c + d*x)] + 30*b^2*C*Tan[c + d*x])))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
664,1,460,310,2.7414368,"\int \sec (c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\frac{\sec ^6(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(240 \left(8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2 \sin (c+d x) \left(120 a^4 C \cos (4 (c+d x))+360 a^4 C+2400 a^3 A b \cos (3 (c+d x))+480 a^3 A b \cos (5 (c+d x))+2240 a^3 b C \cos (3 (c+d x))+320 a^3 b C \cos (5 (c+d x))+64 a b \left(a^2 (75 A+80 C)+8 b^2 (10 A+11 C)\right) \cos (c+d x)+720 a^2 A b^2 \cos (4 (c+d x))+2160 a^2 A b^2+540 a^2 b^2 C \cos (4 (c+d x))+3060 a^2 b^2 C+20 \left(24 a^4 C+36 a^2 b^2 (4 A+5 C)+5 b^4 (6 A+5 C)\right) \cos (2 (c+d x))+2240 a A b^3 \cos (3 (c+d x))+320 a A b^3 \cos (5 (c+d x))+1792 a b^3 C \cos (3 (c+d x))+256 a b^3 C \cos (5 (c+d x))+90 A b^4 \cos (4 (c+d x))+510 A b^4+75 b^4 C \cos (4 (c+d x))+745 b^4 C\right)\right)}{1920 d (A \cos (2 (c+d x))+A+2 C)}","-\frac{\left(4 a^2 C-5 b^2 (6 A+5 C)\right) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}+\frac{a \left(-4 a^2 C+70 A b^2+53 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}-\frac{a \left(4 a^4 C-a^2 b^2 (190 A+121 C)-32 b^4 (5 A+4 C)\right) \tan (c+d x)}{60 b d}+\frac{\left(8 a^4 (2 A+C)+12 a^2 b^2 (4 A+3 C)+b^4 (6 A+5 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\left(8 a^4 C-2 a^2 b^2 (130 A+89 C)-15 b^4 (6 A+5 C)\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}-\frac{a C \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}",1,"-1/1920*((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^6*(240*(8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*(2160*a^2*A*b^2 + 510*A*b^4 + 360*a^4*C + 3060*a^2*b^2*C + 745*b^4*C + 64*a*b*(8*b^2*(10*A + 11*C) + a^2*(75*A + 80*C))*Cos[c + d*x] + 20*(24*a^4*C + 36*a^2*b^2*(4*A + 5*C) + 5*b^4*(6*A + 5*C))*Cos[2*(c + d*x)] + 2400*a^3*A*b*Cos[3*(c + d*x)] + 2240*a*A*b^3*Cos[3*(c + d*x)] + 2240*a^3*b*C*Cos[3*(c + d*x)] + 1792*a*b^3*C*Cos[3*(c + d*x)] + 720*a^2*A*b^2*Cos[4*(c + d*x)] + 90*A*b^4*Cos[4*(c + d*x)] + 120*a^4*C*Cos[4*(c + d*x)] + 540*a^2*b^2*C*Cos[4*(c + d*x)] + 75*b^4*C*Cos[4*(c + d*x)] + 480*a^3*A*b*Cos[5*(c + d*x)] + 320*a*A*b^3*Cos[5*(c + d*x)] + 320*a^3*b*C*Cos[5*(c + d*x)] + 256*a*b^3*C*Cos[5*(c + d*x)])*Sin[c + d*x]))/(d*(A + 2*C + A*Cos[2*(c + d*x)]))","A",1
665,1,503,227,2.5328883,"\int (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^5(c+d x) \left(A \cos ^2(c+d x)+C\right) \left(150 a^4 A (c+d x) \cos (c+d x)+75 a^4 A (c+d x) \cos (3 (c+d x))+15 a^4 A c \cos (5 (c+d x))+15 a^4 A d x \cos (5 (c+d x))+30 a^4 C \sin (c+d x)+45 a^4 C \sin (3 (c+d x))+15 a^4 C \sin (5 (c+d x))+120 a^3 b C \sin (2 (c+d x))+60 a^3 b C \sin (4 (c+d x))-120 a b \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+180 a^2 A b^2 \sin (c+d x)+270 a^2 A b^2 \sin (3 (c+d x))+90 a^2 A b^2 \sin (5 (c+d x))+240 a^2 b^2 C \sin (c+d x)+300 a^2 b^2 C \sin (3 (c+d x))+60 a^2 b^2 C \sin (5 (c+d x))+120 a A b^3 \sin (2 (c+d x))+60 a A b^3 \sin (4 (c+d x))+210 a b^3 C \sin (2 (c+d x))+45 a b^3 C \sin (4 (c+d x))+40 A b^4 \sin (c+d x)+50 A b^4 \sin (3 (c+d x))+10 A b^4 \sin (5 (c+d x))+80 b^4 C \sin (c+d x)+40 b^4 C \sin (3 (c+d x))+8 b^4 C \sin (5 (c+d x))\right)}{120 d (A \cos (2 (c+d x))+A+2 C)}","a^4 A x+\frac{a b \left(4 a^2 (2 A+C)+b^2 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a b \left(6 a^2 C+40 A b^2+29 b^2 C\right) \tan (c+d x) \sec (c+d x)}{30 d}+\frac{\left(3 a^2 C+b^2 (5 A+4 C)\right) \tan (c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac{\left(6 a^4 C+a^2 b^2 (85 A+56 C)+2 b^4 (5 A+4 C)\right) \tan (c+d x)}{15 d}+\frac{a C \tan (c+d x) (a+b \sec (c+d x))^3}{5 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}",1,"((C + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(150*a^4*A*(c + d*x)*Cos[c + d*x] + 75*a^4*A*(c + d*x)*Cos[3*(c + d*x)] + 15*a^4*A*c*Cos[5*(c + d*x)] + 15*a^4*A*d*x*Cos[5*(c + d*x)] - 120*a*b*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 180*a^2*A*b^2*Sin[c + d*x] + 40*A*b^4*Sin[c + d*x] + 30*a^4*C*Sin[c + d*x] + 240*a^2*b^2*C*Sin[c + d*x] + 80*b^4*C*Sin[c + d*x] + 120*a*A*b^3*Sin[2*(c + d*x)] + 120*a^3*b*C*Sin[2*(c + d*x)] + 210*a*b^3*C*Sin[2*(c + d*x)] + 270*a^2*A*b^2*Sin[3*(c + d*x)] + 50*A*b^4*Sin[3*(c + d*x)] + 45*a^4*C*Sin[3*(c + d*x)] + 300*a^2*b^2*C*Sin[3*(c + d*x)] + 40*b^4*C*Sin[3*(c + d*x)] + 60*a*A*b^3*Sin[4*(c + d*x)] + 60*a^3*b*C*Sin[4*(c + d*x)] + 45*a*b^3*C*Sin[4*(c + d*x)] + 90*a^2*A*b^2*Sin[5*(c + d*x)] + 10*A*b^4*Sin[5*(c + d*x)] + 15*a^4*C*Sin[5*(c + d*x)] + 60*a^2*b^2*C*Sin[5*(c + d*x)] + 8*b^4*C*Sin[5*(c + d*x)]))/(120*d*(A + 2*C + A*Cos[2*(c + d*x)]))","B",1
666,1,1357,229,6.5561295,"\int \cos (c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{\left(-8 C a^4-48 A b^2 a^2-24 b^2 C a^2-4 A b^4-3 b^4 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \cos ^6(c+d x)}{4 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C)}+\frac{\left(8 C a^4+48 A b^2 a^2+24 b^2 C a^2+4 A b^4+3 b^4 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \cos ^6(c+d x)}{4 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C)}+\frac{8 a^3 A b (c+d x) (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \cos ^6(c+d x)}{d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C)}+\frac{4 a b^3 C (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^6(c+d x)}{3 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{8 (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \left(3 b C \sin \left(\frac{1}{2} (c+d x)\right) a^3+3 A b^3 \sin \left(\frac{1}{2} (c+d x)\right) a+2 b^3 C \sin \left(\frac{1}{2} (c+d x)\right) a\right) \cos ^6(c+d x)}{3 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{8 (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \left(3 b C \sin \left(\frac{1}{2} (c+d x)\right) a^3+3 A b^3 \sin \left(\frac{1}{2} (c+d x)\right) a+2 b^3 C \sin \left(\frac{1}{2} (c+d x)\right) a\right) \cos ^6(c+d x)}{3 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 a^4 A (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \sin (c+d x) \cos ^6(c+d x)}{d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C)}+\frac{\left(12 A b^4+9 C b^4+16 a C b^3+72 a^2 C b^2\right) (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \cos ^6(c+d x)}{24 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(-12 A b^4-9 C b^4-16 a C b^3-72 a^2 C b^2\right) (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \cos ^6(c+d x)}{24 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 a b^3 C (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^6(c+d x)}{3 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{b^4 C (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \cos ^6(c+d x)}{8 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}-\frac{b^4 C (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+A\right) \cos ^6(c+d x)}{8 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}","4 a^3 A b x-\frac{a b \left(a^2 (12 A-19 C)-8 b^2 (3 A+2 C)\right) \tan (c+d x)}{6 d}-\frac{b^2 \left(a^2 (24 A-26 C)-3 b^2 (4 A+3 C)\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{\left(8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}-\frac{a b (12 A-7 C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^4}{d}",1,"(8*a^3*A*b*(c + d*x)*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((-48*a^2*A*b^2 - 4*A*b^4 - 8*a^4*C - 24*a^2*b^2*C - 3*b^4*C)*Cos[c + d*x]^6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((48*a^2*A*b^2 + 4*A*b^4 + 8*a^4*C + 24*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]^6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])) + (b^4*C*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((12*A*b^4 + 72*a^2*b^2*C + 16*a*b^3*C + 9*b^4*C)*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(24*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*a*b^3*C*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) - (b^4*C*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) + (4*a*b^3*C*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((-12*A*b^4 - 72*a^2*b^2*C - 16*a*b^3*C - 9*b^4*C)*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2))/(24*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (8*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(3*a*A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2] + 2*a*b^3*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (8*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*(3*a*A*b^3*Sin[(c + d*x)/2] + 3*a^3*b*C*Sin[(c + d*x)/2] + 2*a*b^3*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (2*a^4*A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + A*Cos[2*c + 2*d*x]))","B",1
667,1,864,219,6.3208354,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\frac{2 \left(2 b C a^3+2 A b^3 a+b^3 C a\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4}{d (b+a \cos (c+d x))^4}+\frac{2 \left(2 b C a^3+2 A b^3 a+b^3 C a\right) \cos ^4(c+d x) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4}{d (b+a \cos (c+d x))^4}+\frac{b^4 C \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \sec (c+d x))^4}{6 d (b+a \cos (c+d x))^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{\cos ^4(c+d x) \left(3 A \sin \left(\frac{1}{2} (c+d x)\right) b^4+2 C \sin \left(\frac{1}{2} (c+d x)\right) b^4+18 a^2 C \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) (a+b \sec (c+d x))^4}{3 d (b+a \cos (c+d x))^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\cos ^4(c+d x) \left(3 A \sin \left(\frac{1}{2} (c+d x)\right) b^4+2 C \sin \left(\frac{1}{2} (c+d x)\right) b^4+18 a^2 C \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) (a+b \sec (c+d x))^4}{3 d (b+a \cos (c+d x))^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 a^3 A b \cos ^4(c+d x) \sin (c+d x) (a+b \sec (c+d x))^4}{d (b+a \cos (c+d x))^4}+\frac{a^4 A \cos ^4(c+d x) \sin (2 (c+d x)) (a+b \sec (c+d x))^4}{4 d (b+a \cos (c+d x))^4}+\frac{\left(C b^4+12 a C b^3\right) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{12 d (b+a \cos (c+d x))^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(-C b^4-12 a C b^3\right) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{12 d (b+a \cos (c+d x))^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^4 C \cos ^4(c+d x) \sin \left(\frac{1}{2} (c+d x)\right) (a+b \sec (c+d x))^4}{6 d (b+a \cos (c+d x))^4 \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{a^2 \left(A a^2+2 C a^2+12 A b^2\right) (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{2 d (b+a \cos (c+d x))^4}","-\frac{b^2 \left(a^2 (39 A-34 C)-2 b^2 (3 A+2 C)\right) \tan (c+d x)}{6 d}+\frac{2 a b \left(C \left(2 a^2+b^2\right)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{1}{2} a^2 x \left(a^2 (A+2 C)+12 A b^2\right)-\frac{a b^3 (9 A-4 C) \tan (c+d x) \sec (c+d x)}{3 d}-\frac{b^2 (15 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^2}{6 d}+\frac{2 A b \sin (c+d x) (a+b \sec (c+d x))^3}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^4}{2 d}",1,"(a^2*(a^2*A + 12*A*b^2 + 2*a^2*C)*(c + d*x)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4)/(2*d*(b + a*Cos[c + d*x])^4) - (2*(2*a*A*b^3 + 2*a^3*b*C + a*b^3*C)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4)/(d*(b + a*Cos[c + d*x])^4) + (2*(2*a*A*b^3 + 2*a^3*b*C + a*b^3*C)*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4)/(d*(b + a*Cos[c + d*x])^4) + ((12*a*b^3*C + b^4*C)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4)/(12*d*(b + a*Cos[c + d*x])^4*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (b^4*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[(c + d*x)/2])/(6*d*(b + a*Cos[c + d*x])^4*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (b^4*C*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[(c + d*x)/2])/(6*d*(b + a*Cos[c + d*x])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((-12*a*b^3*C - b^4*C)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4)/(12*d*(b + a*Cos[c + d*x])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*(3*A*b^4*Sin[(c + d*x)/2] + 18*a^2*b^2*C*Sin[(c + d*x)/2] + 2*b^4*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*(3*A*b^4*Sin[(c + d*x)/2] + 18*a^2*b^2*C*Sin[(c + d*x)/2] + 2*b^4*C*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (4*a^3*A*b*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])^4) + (a^4*A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[2*(c + d*x)])/(4*d*(b + a*Cos[c + d*x])^4)","B",1
668,1,324,251,3.0859154,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 A \sin (3 (c+d x))+12 a^3 A b \sin (2 (c+d x))+24 a b (c+d x) \left(a^2 (A+2 C)+2 A b^2\right)+3 a^2 \left(a^2 (3 A+4 C)+24 A b^2\right) \sin (c+d x)-6 b^2 \left(C \left(12 a^2+b^2\right)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 b^2 \left(C \left(12 a^2+b^2\right)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{48 a b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{48 a b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{3 b^4 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{3 b^4 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{12 d}","-\frac{2 a b \left(a^2 (2 A+3 C)+b^2 (11 A-6 C)\right) \tan (c+d x)}{3 d}+\frac{b^2 \left(C \left(12 a^2+b^2\right)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\left(a^2 (2 A+3 C)+6 A b^2\right) \sin (c+d x) (a+b \sec (c+d x))^2}{3 d}-\frac{b^2 \left(a^2 (4 A+6 C)+3 b^2 (6 A-C)\right) \tan (c+d x) \sec (c+d x)}{6 d}+2 a b x \left(a^2 (A+2 C)+2 A b^2\right)+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^4}{3 d}+\frac{2 A b \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^3}{3 d}",1,"(24*a*b*(2*A*b^2 + a^2*(A + 2*C))*(c + d*x) - 6*b^2*(2*A*b^2 + (12*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*b^2*(2*A*b^2 + (12*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*b^4*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (48*a*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*b^4*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (48*a*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*a^2*(24*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] + 12*a^3*A*b*Sin[2*(c + d*x)] + a^4*A*Sin[3*(c + d*x)])/(12*d)","A",1
669,1,270,246,1.6957772,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{3 a^4 A \sin (4 (c+d x))+32 a^3 A b \sin (3 (c+d x))+24 a^2 \left(a^2 (A+C)+6 A b^2\right) \sin (2 (c+d x))+96 a b \left(a^2 (3 A+4 C)+4 A b^2\right) \sin (c+d x)+12 (c+d x) \left(a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right)-384 a b^3 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+384 a b^3 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{96 b^4 C \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{96 b^4 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{96 d}","\frac{a b \left(a^2 (23 A+36 C)+12 A b^2\right) \sin (c+d x)}{12 d}-\frac{b^2 \left(3 a^2 (3 A+4 C)+2 b^2 (13 A-12 C)\right) \tan (c+d x)}{24 d}+\frac{\left(a^2 (3 A+4 C)+4 A b^2\right) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{8 d}+\frac{1}{8} x \left(a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+8 A b^4\right)+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac{4 a b^3 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(12*(8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*(c + d*x) - 384*a*b^3*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 384*a*b^3*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (96*b^4*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (96*b^4*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 96*a*b*(4*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] + 24*a^2*(6*A*b^2 + a^2*(A + C))*Sin[2*(c + d*x)] + 32*a^3*A*b*Sin[3*(c + d*x)] + 3*a^4*A*Sin[4*(c + d*x)])/(96*d)","A",1
670,1,223,250,0.8541895,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{3 a^4 A \sin (5 (c+d x))+30 a^3 A b \sin (4 (c+d x))+120 a b (c+d x) \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right)+5 a^2 \left(a^2 (5 A+4 C)+24 A b^2\right) \sin (3 (c+d x))+240 a b \left(a^2 (A+C)+A b^2\right) \sin (2 (c+d x))+30 \left(a^4 (5 A+6 C)+12 a^2 b^2 (3 A+4 C)+8 A b^4\right) \sin (c+d x)-240 b^4 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+240 b^4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{240 d}","\frac{a b \left(a^2 (29 A+40 C)+6 A b^2\right) \sin (c+d x) \cos (c+d x)}{30 d}+\frac{\left(a^2 (4 A+5 C)+3 A b^2\right) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{15 d}+\frac{1}{2} a b x \left(a^2 (3 A+4 C)+4 b^2 (A+2 C)\right)+\frac{\left(2 a^4 (4 A+5 C)+a^2 b^2 (56 A+85 C)+6 A b^4\right) \sin (c+d x)}{15 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac{A b \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{5 d}+\frac{b^4 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(120*a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*(c + d*x) - 240*b^4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 240*b^4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 30*(8*A*b^4 + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*Sin[c + d*x] + 240*a*b*(A*b^2 + a^2*(A + C))*Sin[2*(c + d*x)] + 5*a^2*(24*A*b^2 + a^2*(5*A + 4*C))*Sin[3*(c + d*x)] + 30*a^3*A*b*Sin[4*(c + d*x)] + 3*a^4*A*Sin[5*(c + d*x)])/(240*d)","A",1
671,1,302,298,0.8830639,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{45 a^4 A \sin (4 (c+d x))+5 a^4 A \sin (6 (c+d x))+300 a^4 A c+300 a^4 A d x+30 a^4 C \sin (4 (c+d x))+360 a^4 c C+360 a^4 C d x+400 a^3 A b \sin (3 (c+d x))+48 a^3 A b \sin (5 (c+d x))+320 a^3 b C \sin (3 (c+d x))+480 a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \sin (c+d x)+180 a^2 A b^2 \sin (4 (c+d x))+2160 a^2 A b^2 c+2160 a^2 A b^2 d x+2880 a^2 b^2 c C+2880 a^2 b^2 C d x+15 \left(a^4 (15 A+16 C)+96 a^2 b^2 (A+C)+16 A b^4\right) \sin (2 (c+d x))+320 a A b^3 \sin (3 (c+d x))+480 A b^4 c+480 A b^4 d x+960 b^4 c C+960 b^4 C d x}{960 d}","\frac{4 a b \left(2 a^2 (4 A+5 C)+5 b^2 (2 A+3 C)\right) \sin (c+d x)}{15 d}+\frac{a b \left(a^2 (39 A+50 C)+4 A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac{\left(5 a^2 (5 A+6 C)+12 A b^2\right) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{120 d}+\frac{\left(15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)+24 A b^4\right) \sin (c+d x) \cos (c+d x)}{240 d}+\frac{1}{16} x \left(a^4 (5 A+6 C)+12 a^2 b^2 (3 A+4 C)+8 b^4 (A+2 C)\right)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}+\frac{2 A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{15 d}",1,"(300*a^4*A*c + 2160*a^2*A*b^2*c + 480*A*b^4*c + 360*a^4*c*C + 2880*a^2*b^2*c*C + 960*b^4*c*C + 300*a^4*A*d*x + 2160*a^2*A*b^2*d*x + 480*A*b^4*d*x + 360*a^4*C*d*x + 2880*a^2*b^2*C*d*x + 960*b^4*C*d*x + 480*a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sin[c + d*x] + 15*(16*A*b^4 + 96*a^2*b^2*(A + C) + a^4*(15*A + 16*C))*Sin[2*(c + d*x)] + 400*a^3*A*b*Sin[3*(c + d*x)] + 320*a*A*b^3*Sin[3*(c + d*x)] + 320*a^3*b*C*Sin[3*(c + d*x)] + 45*a^4*A*Sin[4*(c + d*x)] + 180*a^2*A*b^2*Sin[4*(c + d*x)] + 30*a^4*C*Sin[4*(c + d*x)] + 48*a^3*A*b*Sin[5*(c + d*x)] + 5*a^4*A*Sin[6*(c + d*x)])/(960*d)","A",1
672,1,351,339,0.8580583,"\int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{735 a^4 A \sin (3 (c+d x))+147 a^4 A \sin (5 (c+d x))+15 a^4 A \sin (7 (c+d x))+700 a^4 C \sin (3 (c+d x))+84 a^4 C \sin (5 (c+d x))+1260 a^3 A b \sin (4 (c+d x))+140 a^3 A b \sin (6 (c+d x))+8400 a^3 A b c+8400 a^3 A b d x+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+420 a b \left(a^2 (15 A+16 C)+16 b^2 (A+C)\right) \sin (2 (c+d x))+4200 a^2 A b^2 \sin (3 (c+d x))+504 a^2 A b^2 \sin (5 (c+d x))+3360 a^2 b^2 C \sin (3 (c+d x))+105 \left(5 a^4 (7 A+8 C)+48 a^2 b^2 (5 A+6 C)+16 b^4 (3 A+4 C)\right) \sin (c+d x)+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+13440 a b^3 c C+13440 a b^3 C d x+560 A b^4 \sin (3 (c+d x))}{6720 d}","\frac{a b \left(a^2 (103 A+126 C)+6 A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{210 d}+\frac{a b \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right) \sin (c+d x) \cos (c+d x)}{4 d}+\frac{\left(a^2 (6 A+7 C)+2 A b^2\right) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{35 d}+\frac{1}{4} a b x \left(a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right)-\frac{\left(4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right) \sin ^3(c+d x)}{105 d}+\frac{\left(12 a^4 (6 A+7 C)+3 a^2 b^2 (162 A+203 C)+b^4 (74 A+105 C)\right) \sin (c+d x)}{105 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}+\frac{2 A b \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{21 d}",1,"(8400*a^3*A*b*c + 10080*a*A*b^3*c + 10080*a^3*b*c*C + 13440*a*b^3*c*C + 8400*a^3*A*b*d*x + 10080*a*A*b^3*d*x + 10080*a^3*b*C*d*x + 13440*a*b^3*C*d*x + 105*(16*b^4*(3*A + 4*C) + 48*a^2*b^2*(5*A + 6*C) + 5*a^4*(7*A + 8*C))*Sin[c + d*x] + 420*a*b*(16*b^2*(A + C) + a^2*(15*A + 16*C))*Sin[2*(c + d*x)] + 735*a^4*A*Sin[3*(c + d*x)] + 4200*a^2*A*b^2*Sin[3*(c + d*x)] + 560*A*b^4*Sin[3*(c + d*x)] + 700*a^4*C*Sin[3*(c + d*x)] + 3360*a^2*b^2*C*Sin[3*(c + d*x)] + 1260*a^3*A*b*Sin[4*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 147*a^4*A*Sin[5*(c + d*x)] + 504*a^2*A*b^2*Sin[5*(c + d*x)] + 84*a^4*C*Sin[5*(c + d*x)] + 140*a^3*A*b*Sin[6*(c + d*x)] + 15*a^4*A*Sin[7*(c + d*x)])/(6720*d)","A",1
673,1,1299,158,6.4135992,"\int (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2),x]","\frac{\left(3 b^5+8 a^2 b^3-24 a^4 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \cos ^5(c+d x)}{4 d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right)}+\frac{\left(-3 b^5-8 a^2 b^3+24 a^4 b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \cos ^5(c+d x)}{4 d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right)}+\frac{2 a^5 (c+d x) (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right)}-\frac{a b^4 (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{4 (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \left(a b^4 \sin \left(\frac{1}{2} (c+d x)\right)-a^3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{4 (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \left(a b^4 \sin \left(\frac{1}{2} (c+d x)\right)-a^3 b^2 \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{\left(-3 b^5-4 a b^4-8 a^2 b^3\right) (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(3 b^5+4 a b^4+8 a^2 b^3\right) (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{a b^4 (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}-\frac{b^5 (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{b^5 (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \cos ^5(c+d x)}{8 d (b+a \cos (c+d x))^3 \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^4}","a^5 x+\frac{a b^2 \left(5 a^2-4 b^2\right) \tan (c+d x)}{2 d}+\frac{b^3 \left(2 a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b \left(24 a^4-8 a^2 b^2-3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{a b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{4 d}-\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(2*a^5*(c + d*x)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2))/(d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])) + ((-24*a^4*b + 8*a^2*b^3 + 3*b^5)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])) + ((24*a^4*b - 8*a^2*b^3 - 3*b^5)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])) - (b^5*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((-8*a^2*b^3 - 4*a*b^4 - 3*b^5)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - (a*b^4*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (b^5*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4) - (a*b^4*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/(d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((8*a^2*b^3 + 4*a*b^4 + 3*b^5)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2))/(8*d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) - (4*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2)*(-(a^3*b^2*Sin[(c + d*x)/2]) + a*b^4*Sin[(c + d*x)/2]))/(d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (4*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2)*(-(a^3*b^2*Sin[(c + d*x)/2]) + a*b^4*Sin[(c + d*x)/2]))/(d*(b + a*Cos[c + d*x])^3*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
674,1,86,106,0.2468997,"\int (a+b \sec (c+d x))^2 \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^2*(a^2 - b^2*Sec[c + d*x]^2),x]","a^4 x+\frac{2 a^3 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^3 \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^3 \tan (c+d x) \sec (c+d x)}{d}-\frac{b^4 \left(\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right)}{d}","a^4 x+\frac{b^2 \left(a^2-2 b^2\right) \tan (c+d x)}{3 d}+\frac{a b \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^3 \tan (c+d x) \sec (c+d x)}{3 d}-\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"a^4*x + (2*a^3*b*ArcTanh[Sin[c + d*x]])/d - (a*b^3*ArcTanh[Sin[c + d*x]])/d - (a*b^3*Sec[c + d*x]*Tan[c + d*x])/d - (b^4*(Tan[c + d*x] + Tan[c + d*x]^3/3))/d","A",1
675,1,75,75,0.0208787,"\int (a+b \sec (c+d x)) \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])*(a^2 - b^2*Sec[c + d*x]^2),x]","a^3 x+\frac{a^2 b \tanh ^{-1}(\sin (c+d x))}{d}-\frac{a b^2 \tan (c+d x)}{d}-\frac{b^3 \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^3 \tan (c+d x) \sec (c+d x)}{2 d}","a^3 x+\frac{b \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{a b^2 \tan (c+d x)}{2 d}-\frac{b^2 \tan (c+d x) (a+b \sec (c+d x))}{2 d}",1,"a^3*x + (a^2*b*ArcTanh[Sin[c + d*x]])/d - (b^3*ArcTanh[Sin[c + d*x]])/(2*d) - (a*b^2*Tan[c + d*x])/d - (b^3*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
676,1,657,186,3.9659502,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\cos (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 b \sin \left(\frac{d x}{2}\right) \left(3 a^2 C+3 A b^2+2 b^2 C\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 b \sin \left(\frac{d x}{2}\right) \left(3 a^2 C+3 A b^2+2 b^2 C\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+6 a \left(C \left(2 a^2+b^2\right)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a \left(C \left(2 a^2+b^2\right)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{24 i a^2 (\cos (c)-i \sin (c)) \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{b^2 C \left((3 a+b) \sin \left(\frac{c}{2}\right)+(b-3 a) \cos \left(\frac{c}{2}\right)\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^2 C \left((3 a+b) \sin \left(\frac{c}{2}\right)+(3 a-b) \cos \left(\frac{c}{2}\right)\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 b^3 C \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 b^3 C \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)}{6 b^4 d (a+b \sec (c+d x)) (A \cos (2 (c+d x))+A+2 C)}","-\frac{a \left(C \left(2 a^2+b^2\right)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{2 a^2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(3 a^2 C+b^2 (3 A+2 C)\right) \tan (c+d x)}{3 b^3 d}-\frac{a C \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 b d}",1,"(Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*(6*a*(2*A*b^2 + (2*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*a*(2*A*b^2 + (2*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((24*I)*a^2*(A*b^2 + a^2*C)*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(Cos[c] - I*Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) + (2*b^3*C*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (b^2*C*((-3*a + b)*Cos[c/2] + (3*a + b)*Sin[c/2]))/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*b*(3*A*b^2 + 3*a^2*C + 2*b^2*C)*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*b^3*C*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (b^2*C*((3*a - b)*Cos[c/2] + (3*a + b)*Sin[c/2]))/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*b*(3*A*b^2 + 3*a^2*C + 2*b^2*C)*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(6*b^4*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x]))","C",1
677,1,428,137,2.2850153,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\cos (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(-2 \left(C \left(2 a^2+b^2\right)+2 A b^2\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(C \left(2 a^2+b^2\right)+2 A b^2\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 a (\sin (c)+i \cos (c)) \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}-\frac{4 a b C \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{4 a b C \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{b^2 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{2 b^3 d (a+b \sec (c+d x)) (A \cos (2 (c+d x))+A+2 C)}","\frac{\left(2 a^2 C+b^2 (2 A+C)\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{2 a \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C \tan (c+d x)}{b^2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 b d}",1,"(Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*(-2*(2*A*b^2 + (2*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + (2*a^2 + b^2)*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (8*a*(A*b^2 + a^2*C)*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(I*Cos[c] + Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b^2*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - (4*a*b*C*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (b^2*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (4*a*b*C*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(2*b^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x]))","C",1
678,1,331,95,2.3433358,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{2 \cos (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(-\frac{2 i (\cos (c)-i \sin (c)) \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}+a C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-a C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b C \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{b C \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{b^2 d (a+b \sec (c+d x)) (A \cos (2 (c+d x))+A+2 C)}","\frac{2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a C \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{C \tan (c+d x)}{b d}",1,"(2*Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*(a*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - a*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((2*I)*(A*b^2 + a^2*C)*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(Cos[c] - I*Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b*C*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (b*C*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(b^2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x]))","C",1
679,1,239,88,0.4273969,"\int \frac{A+C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","\frac{2 \left(A \cos ^2(c+d x)+C\right) \left(\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2} \left(-a C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+a C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+A b d x\right)+2 (\sin (c)+i \cos (c)) \left(a^2 C+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)\right)}{a b d \sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2} (A \cos (2 (c+d x))+A+2 C)}","-\frac{2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A x}{a}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b d}",1,"(2*(C + A*Cos[c + d*x]^2)*(Sqrt[a^2 - b^2]*(A*b*d*x - a*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + a*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[(Cos[c] - I*Sin[c])^2] + 2*(A*b^2 + a^2*C)*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(I*Cos[c] + Sin[c])))/(a*b*Sqrt[a^2 - b^2]*d*(A + 2*C + A*Cos[2*(c + d*x)])*Sqrt[(Cos[c] - I*Sin[c])^2])","C",1
680,1,82,86,0.2365279,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a A \sin (c+d x)-A b (c+d x)}{a^2 d}","\frac{2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{A b x}{a^2}+\frac{A \sin (c+d x)}{a d}",1,"(-(A*b*(c + d*x)) - (2*(A*b^2 + a^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*A*Sin[c + d*x])/(a^2*d)","A",1
681,1,115,128,0.3757735,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{2 (c+d x) \left(a^2 (A+2 C)+2 A b^2\right)+\frac{8 b \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a^2 A \sin (2 (c+d x))-4 a A b \sin (c+d x)}{4 a^3 d}","-\frac{A b \sin (c+d x)}{a^2 d}-\frac{2 b \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{x \left(a^2 (A+2 C)+2 A b^2\right)}{2 a^3}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d}",1,"(2*(2*A*b^2 + a^2*(A + 2*C))*(c + d*x) + (8*b*(A*b^2 + a^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 4*a*A*b*Sin[c + d*x] + a^2*A*Sin[2*(c + d*x)])/(4*a^3*d)","A",1
682,1,149,175,0.5061225,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{a^3 A \sin (3 (c+d x))-6 b (c+d x) \left(a^2 (A+2 C)+2 A b^2\right)+3 a \left(a^2 (3 A+4 C)+4 A b^2\right) \sin (c+d x)-\frac{24 b^2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-3 a^2 A b \sin (2 (c+d x))}{12 a^4 d}","-\frac{A b \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{2 b^2 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{b x \left(a^2 (A+2 C)+2 A b^2\right)}{2 a^4}+\frac{\left(a^2 (2 A+3 C)+3 A b^2\right) \sin (c+d x)}{3 a^3 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"(-6*b*(2*A*b^2 + a^2*(A + 2*C))*(c + d*x) - (24*b^2*(A*b^2 + a^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 3*a*(4*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] - 3*a^2*A*b*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^4*d)","A",1
683,1,191,232,0.6615509,"\int \frac{\cos ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{3 a^4 A \sin (4 (c+d x))-8 a^3 A b \sin (3 (c+d x))+24 a^2 \left(a^2 (A+C)+A b^2\right) \sin (2 (c+d x))-24 a b \left(a^2 (3 A+4 C)+4 A b^2\right) \sin (c+d x)+\frac{192 b^3 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+12 (c+d x) \left(a^4 (3 A+4 C)+4 a^2 b^2 (A+2 C)+8 A b^4\right)}{96 a^5 d}","-\frac{A b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}-\frac{2 b^3 \left(a^2 C+A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{b \left(a^2 (2 A+3 C)+3 A b^2\right) \sin (c+d x)}{3 a^4 d}+\frac{\left(a^2 (3 A+4 C)+4 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{x \left(a^4 (3 A+4 C)+4 a^2 b^2 (A+2 C)+8 A b^4\right)}{8 a^5}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"(12*(8*A*b^4 + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*(c + d*x) + (192*b^3*(A*b^2 + a^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 24*a*b*(4*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] + 24*a^2*(A*b^2 + a^2*(A + C))*Sin[2*(c + d*x)] - 8*a^3*A*b*Sin[3*(c + d*x)] + 3*a^4*A*Sin[4*(c + d*x)])/(96*a^5*d)","A",1
684,1,461,271,4.3595203,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{(a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 a^2 b \left(a^2 C+A b^2\right) \sin (c+d x)}{(b-a) (a+b)}-2 \left(C \left(6 a^2+b^2\right)+2 A b^2\right) (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(C \left(6 a^2+b^2\right)+2 A b^2\right) (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 a \left(3 a^4 C+a^2 b^2 (A-4 C)-2 A b^4\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b^2 C (a \cos (c+d x)+b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 C (a \cos (c+d x)+b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{8 a b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{8 a b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{2 b^4 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 C)}","-\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(3 a^2 C+2 A b^2-b^2 C\right) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left(a^2-b^2\right)}+\frac{\left(C \left(6 a^2+b^2\right)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{a \left(3 a^2 C+A b^2-2 b^2 C\right) \tan (c+d x)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(3 a^4 C+a^2 A b^2-4 a^2 b^2 C-2 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*((8*a*(-2*A*b^4 + a^2*b^2*(A - 4*C) + 3*a^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) - 2*(2*A*b^2 + (6*a^2 + b^2)*C)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + (6*a^2 + b^2)*C)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*C*(b + a*Cos[c + d*x]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - (8*a*b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^2*C*(b + a*Cos[c + d*x]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (8*a*b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (4*a^2*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((-a + b)*(a + b))))/(2*b^4*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","A",0
685,1,336,153,2.7517353,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{a b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b)}+\frac{2 \left(-2 a^4 C+3 a^2 b^2 C+A b^4\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+2 a C (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a C (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{b^3 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 C)}","\frac{a \left(a^2 C+A b^2\right) \tan (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 \left(-2 a^4 C+3 a^2 b^2 C+A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{2 a C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}",1,"(2*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*((2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) + 2*a*C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*a*C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b))))/(b^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","B",1
686,1,331,135,2.3915309,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{b \left(a^2 C+A b^2\right) (b \sin (c)-a \sin (d x))}{a (a-b) (a+b) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}+\frac{2 a (\sin (c)+i \cos (c)) \left(C \left(a^2-2 b^2\right)-A b^2\right) (a \cos (c+d x)+b) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{3/2} \sqrt{(\cos (c)-i \sin (c))^2}}-C (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+C (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{b^2 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 C)}","\frac{2 a \left(a^2 (-C)+A b^2+2 b^2 C\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"(2*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*(-(C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*a*(-(A*b^2) + (a^2 - 2*b^2)*C)*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(3/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b*(A*b^2 + a^2*C)*(b*Sin[c] - a*Sin[d*x]))/(a*(a - b)*(a + b)*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(b^2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","C",0
687,1,270,125,2.0263998,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","\frac{2 (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{\left(a^2 C+A b^2\right) (a \sin (d x)-b \sin (c))}{d (a-b) (a+b) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}+\frac{2 b (\sin (c)+i \cos (c)) \left(a^2 (2 A+C)-A b^2\right) (a \cos (c+d x)+b) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{d \left(a^2-b^2\right)^{3/2} \sqrt{(\cos (c)-i \sin (c))^2}}+A x (a \cos (c+d x)+b)\right)}{a^2 (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 C)}","-\frac{2 b \left(2 a^2 A+a^2 C-A b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{A x}{a^2}",1,"(2*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*(A*x*(b + a*Cos[c + d*x]) + (2*b*(-(A*b^2) + a^2*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(3/2)*d*Sqrt[(Cos[c] - I*Sin[c])^2]) + ((A*b^2 + a^2*C)*(-(b*Sin[c]) + a*Sin[d*x]))/((a - b)*(a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(a^2*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","C",0
688,1,137,171,1.1002041,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{-\frac{a b \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-\frac{2 \left(a^4 C+3 a^2 A b^2-2 A b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+a A \sin (c+d x)-2 A b (c+d x)}{a^3 d}","-\frac{2 A b x}{a^3}-\frac{\left(2 A b^2-a^2 (A-C)\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{2 \left(a^4 C+3 a^2 A b^2-2 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*A*b*(c + d*x) - (2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + a*A*Sin[c + d*x] - (a*b*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])))/(a^3*d)","A",1
689,1,176,256,1.0900886,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (c+d x) \left(a^2 (A+2 C)+6 A b^2\right)+\frac{4 a b^2 \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}+a^2 A \sin (2 (c+d x))-\frac{8 b \left(-2 a^4 C+a^2 b^2 (C-4 A)+3 A b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-8 a A b \sin (c+d x)}{4 a^4 d}","-\frac{\left(3 A b^2-a^2 (A-2 C)\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x \left(a^2 (A+2 C)+6 A b^2\right)}{2 a^4}-\frac{2 b \left(2 a^4 C+4 a^2 A b^2-a^2 b^2 C-3 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{b \left(3 A b^2-a^2 (2 A-C)\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}",1,"(2*(6*A*b^2 + a^2*(A + 2*C))*(c + d*x) - (8*b*(3*A*b^4 - 2*a^4*C + a^2*b^2*(-4*A + C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - 8*a*A*b*Sin[c + d*x] + (4*a*b^2*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*A*Sin[2*(c + d*x)])/(4*a^4*d)","A",1
690,1,212,326,1.2855128,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{a^3 A \sin (3 (c+d x))-12 b (c+d x) \left(a^2 (A+2 C)+4 A b^2\right)+3 a \left(a^2 (3 A+4 C)+12 A b^2\right) \sin (c+d x)-\frac{12 a b^3 \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-6 a^2 A b \sin (2 (c+d x))+\frac{24 b^2 \left(-3 a^4 C+a^2 b^2 (2 C-5 A)+4 A b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}}{12 a^5 d}","-\frac{\left(4 A b^2-a^2 (A-3 C)\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{b x \left(a^2 (A+2 C)+4 A b^2\right)}{a^5}-\frac{\left(-\left(a^4 (2 A+3 C)\right)-a^2 b^2 (7 A-6 C)+12 A b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}+\frac{b \left(2 A b^2-a^2 (A-C)\right) \sin (c+d x) \cos (c+d x)}{a^3 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(3 a^4 C+5 a^2 A b^2-2 a^2 b^2 C-4 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-12*b*(4*A*b^2 + a^2*(A + 2*C))*(c + d*x) + (24*b^2*(4*A*b^4 - 3*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 3*a*(12*A*b^2 + a^2*(3*A + 4*C))*Sin[c + d*x] - (12*a*b^3*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) - 6*a^2*A*b*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^5*d)","A",1
691,1,559,381,4.8190473,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{2 a^2 b^2 \left(a^2 C+A b^2\right) \sin (c+d x)}{(b-a) (a+b)}-2 \left(C \left(12 a^2+b^2\right)+2 A b^2\right) (a \cos (c+d x)+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(C \left(12 a^2+b^2\right)+2 A b^2\right) (a \cos (c+d x)+b)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 a^2 b \left(-6 a^4 C+a^2 b^2 (9 C-2 A)+5 A b^4\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}+\frac{4 a \left(12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{b^2 C (a \cos (c+d x)+b)^2}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 C (a \cos (c+d x)+b)^2}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{12 a b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}-\frac{12 a b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{2 b^5 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 C)}","-\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(C \left(12 a^2+b^2\right)+2 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac{a \left(12 a^4 C+a^2 b^2 (2 A-21 C)-b^4 (5 A-6 C)\right) \tan (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-4 a^4 C+7 a^2 b^2 C+3 A b^4\right) \tan (c+d x) \sec ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(6 a^4 C+a^2 b^2 (A-10 C)-b^4 (4 A-C)\right) \tan (c+d x) \sec (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}-\frac{a \left(12 a^6 C+a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+6 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((4*a*(6*A*b^6 + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) - 2*(2*A*b^2 + (12*a^2 + b^2)*C)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 + (12*a^2 + b^2)*C)*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*C*(b + a*Cos[c + d*x])^2)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - (12*a*b*C*(b + a*Cos[c + d*x])^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^2*C*(b + a*Cos[c + d*x])^2)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (12*a*b*C*(b + a*Cos[c + d*x])^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (2*a^2*b^2*(A*b^2 + a^2*C)*Sin[c + d*x])/((-a + b)*(a + b)) + (2*a^2*b*(5*A*b^4 - 6*a^4*C + a^2*b^2*(-2*A + 9*C))*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2)))/(2*b^5*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","A",0
692,1,421,271,2.6637702,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{a b^2 \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b)}+\frac{a b \left(4 a^4 C-7 a^2 b^2 C-3 A b^4\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{2 \left(6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)+2 A b^6\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{2 b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{2 b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)^2}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+6 a C (a \cos (c+d x)+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 a C (a \cos (c+d x)+b)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{b^4 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 C)}","-\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(3 a^2 C+A b^2-2 b^2 C\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \left(-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(6 a^6 C-15 a^4 b^2 C+a^2 b^4 (A+12 C)+2 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{3 a C \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((-2*(2*A*b^6 + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) + 6*a*C*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*a*C*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*C*(b + a*Cos[c + d*x])^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*b*C*(b + a*Cos[c + d*x])^2*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (a*b^2*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)) + (a*b*(-3*A*b^4 + 4*a^4*C - 7*a^2*b^2*C)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2)))/(b^4*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","A",0
693,1,445,212,5.6038139,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 a (\sin (c)+i \cos (c)) \left(C \left(2 a^4-5 a^2 b^2+6 b^4\right)+3 A b^4\right) (a \cos (c+d x)+b)^2 \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{5/2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{b \left(\left(a^2+2 b^2\right) \tan (c) \left(2 a^4 C-a^2 b^2 (2 A+5 C)-A b^4\right)+a \sec (c) \left(\sin (d x) \left(-7 a^4 b C+a^2 b^3 (5 A+16 C)+4 A b^5\right)+a \left(a b \left(C \left(a^2-4 b^2\right)-3 A b^2\right) \sin (2 c+d x)+\left(-2 a^4 C+a^2 b^2 (2 A+5 C)+A b^4\right) \sin (c+2 d x)\right)\right)\right)}{a \left(a^2-b^2\right)^2}-4 C (a \cos (c+d x)+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 C (a \cos (c+d x)+b)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 b^3 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 C)}","\frac{a \left(a^2 C+A b^2\right) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(-3 a^4 C+a^2 b^2 (A+6 C)+2 A b^4\right) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{a \left(C \left(2 a^4-5 a^2 b^2+6 b^4\right)+3 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(-4*C*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*C*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b*(a*Sec[c]*((4*A*b^5 - 7*a^4*b*C + a^2*b^3*(5*A + 16*C))*Sin[d*x] + a*(a*b*(-3*A*b^2 + (a^2 - 4*b^2)*C)*Sin[2*c + d*x] + (A*b^4 - 2*a^4*C + a^2*b^2*(2*A + 5*C))*Sin[c + 2*d*x])) + (a^2 + 2*b^2)*(-(A*b^4) + 2*a^4*C - a^2*b^2*(2*A + 5*C))*Tan[c]))/(a*(a^2 - b^2)^2)))/(2*b^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","C",0
694,1,342,177,3.6202885,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{b \left(a^2+2 b^2\right) \tan (c) \left(a^2 (4 A+3 C)-A b^2\right)+a \sec (c) \left(a b \left(A b^2-a^2 (4 A+3 C)\right) \sin (c+2 d x)+\left(a^4 C+a^2 b^2 (5 A+2 C)-2 A b^4\right) \sin (2 c+d x)+\sin (d x) \left(a^4 C-a^2 b^2 (11 A+10 C)+2 A b^4\right)\right)}{\left(a^3-a b^2\right)^2}-\frac{4 i (\cos (c)-i \sin (c)) \left(a^2 (2 A+C)+b^2 (A+2 C)\right) (a \cos (c+d x)+b)^2 \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{5/2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{2 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 C)}","\frac{\left(a^2 (2 A+C)+b^2 (A+2 C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{a \left(a^2 (-C)+3 A b^2+4 b^2 C\right) \tan (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(((-4*I)*(a^2*(2*A + C) + b^2*(A + 2*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (a*Sec[c]*((2*A*b^4 + a^4*C - a^2*b^2*(11*A + 10*C))*Sin[d*x] + (-2*A*b^4 + a^4*C + a^2*b^2*(5*A + 2*C))*Sin[2*c + d*x] + a*b*(A*b^2 - a^2*(4*A + 3*C))*Sin[c + 2*d*x]) + b*(a^2 + 2*b^2)*(-(A*b^2) + a^2*(4*A + 3*C))*Tan[c])/(a^3 - a*b^2)^2))/(2*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","C",0
695,1,642,202,5.3167292,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{\sec (c) \left(a^6 A d x \cos (c+2 d x)+a^6 A d x \cos (3 c+2 d x)+2 a^6 C \sin (c+2 d x)-2 a^6 C \sin (c)+4 a^5 A b d x \cos (2 c+d x)-3 a^5 b C \sin (2 c+d x)+5 a^5 b C \sin (d x)+6 a^4 A b^2 \sin (c+2 d x)-2 a^4 A b^2 d x \cos (c+2 d x)-2 a^4 A b^2 d x \cos (3 c+2 d x)-6 a^4 A b^2 \sin (c)+a^4 b^2 C \sin (c+2 d x)-5 a^4 b^2 C \sin (c)-7 a^3 A b^3 \sin (2 c+d x)-8 a^3 A b^3 d x \cos (2 c+d x)+17 a^3 A b^3 \sin (d x)+4 a^3 b^3 C \sin (d x)-3 a^2 A b^4 \sin (c+2 d x)+a^2 A b^4 d x \cos (c+2 d x)+a^2 A b^4 d x \cos (3 c+2 d x)-9 a^2 A b^4 \sin (c)+2 A d x \left(a^2-b^2\right)^2 \left(a^2+2 b^2\right) \cos (c)+4 a A b d x \left(a^2-b^2\right)^2 \cos (d x)-2 a^2 b^4 C \sin (c)+4 a A b^5 \sin (2 c+d x)+4 a A b^5 d x \cos (2 c+d x)-8 a A b^5 \sin (d x)+6 A b^6 \sin (c)\right)}{\left(a^2-b^2\right)^2}+\frac{4 b (\sin (c)+i \cos (c)) \left(3 a^4 (2 A+C)-5 a^2 A b^2+2 A b^4\right) (a \cos (c+d x)+b)^2 \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{5/2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{2 a^3 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 C)}","\frac{A x}{a^3}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\left(a^4 (-C)-a^2 b^2 (5 A+2 C)+2 A b^4\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b \left(-3 a^4 (2 A+C)+5 a^2 A b^2-2 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((4*b*(-5*a^2*A*b^2 + 2*A*b^4 + 3*a^4*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (Sec[c]*(2*A*(a^2 - b^2)^2*(a^2 + 2*b^2)*d*x*Cos[c] + 4*a*A*b*(a^2 - b^2)^2*d*x*Cos[d*x] + 4*a^5*A*b*d*x*Cos[2*c + d*x] - 8*a^3*A*b^3*d*x*Cos[2*c + d*x] + 4*a*A*b^5*d*x*Cos[2*c + d*x] + a^6*A*d*x*Cos[c + 2*d*x] - 2*a^4*A*b^2*d*x*Cos[c + 2*d*x] + a^2*A*b^4*d*x*Cos[c + 2*d*x] + a^6*A*d*x*Cos[3*c + 2*d*x] - 2*a^4*A*b^2*d*x*Cos[3*c + 2*d*x] + a^2*A*b^4*d*x*Cos[3*c + 2*d*x] - 6*a^4*A*b^2*Sin[c] - 9*a^2*A*b^4*Sin[c] + 6*A*b^6*Sin[c] - 2*a^6*C*Sin[c] - 5*a^4*b^2*C*Sin[c] - 2*a^2*b^4*C*Sin[c] + 17*a^3*A*b^3*Sin[d*x] - 8*a*A*b^5*Sin[d*x] + 5*a^5*b*C*Sin[d*x] + 4*a^3*b^3*C*Sin[d*x] - 7*a^3*A*b^3*Sin[2*c + d*x] + 4*a*A*b^5*Sin[2*c + d*x] - 3*a^5*b*C*Sin[2*c + d*x] + 6*a^4*A*b^2*Sin[c + 2*d*x] - 3*a^2*A*b^4*Sin[c + 2*d*x] + 2*a^6*C*Sin[c + 2*d*x] + a^4*b^2*C*Sin[c + 2*d*x]))/(a^2 - b^2)^2))/(2*a^3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","C",0
696,1,1186,266,6.8531617,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\left(2 C a^6+12 A b^2 a^4+b^2 C a^4-15 A b^4 a^2+6 A b^6\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{2 i \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{2 \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^3}{\left(b^2-a^2\right)^2 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^3}+\frac{\sec (c) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(A \sin (d x) a^7+A \sin (2 c+d x) a^7+A \sin (2 c+3 d x) a^7+A \sin (4 c+3 d x) a^7-12 A b d x \cos (c) a^6-6 A b d x \cos (c+2 d x) a^6-6 A b d x \cos (3 c+2 d x) a^6+8 b C \sin (c) a^6+4 A b \sin (c+2 d x) a^6-8 b C \sin (c+2 d x) a^6+4 A b \sin (3 c+2 d x) a^6-24 A b^2 d x \cos (d x) a^5-24 A b^2 d x \cos (2 c+d x) a^5+2 A b^2 \sin (d x) a^5-22 b^2 C \sin (d x) a^5+2 A b^2 \sin (2 c+d x) a^5+10 b^2 C \sin (2 c+d x) a^5-2 A b^2 \sin (2 c+3 d x) a^5-2 A b^2 \sin (4 c+3 d x) a^5+12 A b^3 d x \cos (c+2 d x) a^4+12 A b^3 d x \cos (3 c+2 d x) a^4+16 A b^3 \sin (c) a^4+14 b^3 C \sin (c) a^4-24 A b^3 \sin (c+2 d x) a^4+2 b^3 C \sin (c+2 d x) a^4-8 A b^3 \sin (3 c+2 d x) a^4+48 A b^4 d x \cos (d x) a^3+48 A b^4 d x \cos (2 c+d x) a^3-53 A b^4 \sin (d x) a^3+4 b^4 C \sin (d x) a^3+11 A b^4 \sin (2 c+d x) a^3-4 b^4 C \sin (2 c+d x) a^3+A b^4 \sin (2 c+3 d x) a^3+A b^4 \sin (4 c+3 d x) a^3+36 A b^5 d x \cos (c) a^2-6 A b^5 d x \cos (c+2 d x) a^2-6 A b^5 d x \cos (3 c+2 d x) a^2+22 A b^5 \sin (c) a^2-4 b^5 C \sin (c) a^2+14 A b^5 \sin (c+2 d x) a^2+4 A b^5 \sin (3 c+2 d x) a^2-24 A b^6 d x \cos (d x) a-24 A b^6 d x \cos (2 c+d x) a+32 A b^6 \sin (d x) a-8 A b^6 \sin (2 c+d x) a-24 A b^7 d x \cos (c)-20 A b^7 \sin (c)\right) (b+a \cos (c+d x))}{4 a^4 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^3}","-\frac{3 A b x}{a^4}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\left(-2 a^4 C-a^2 b^2 (6 A+C)+3 A b^4\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(-2 a^6 C-a^4 b^2 (12 A+C)+15 a^2 A b^4-6 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-\left(a^4 (2 A-3 C)\right)+11 a^2 A b^2-6 A b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}",1,"((12*a^4*A*b^2 - 15*a^2*A*b^4 + 6*A*b^6 + 2*a^6*C + a^4*b^2*C)*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*(-12*a^6*A*b*d*x*Cos[c] + 36*a^2*A*b^5*d*x*Cos[c] - 24*A*b^7*d*x*Cos[c] - 24*a^5*A*b^2*d*x*Cos[d*x] + 48*a^3*A*b^4*d*x*Cos[d*x] - 24*a*A*b^6*d*x*Cos[d*x] - 24*a^5*A*b^2*d*x*Cos[2*c + d*x] + 48*a^3*A*b^4*d*x*Cos[2*c + d*x] - 24*a*A*b^6*d*x*Cos[2*c + d*x] - 6*a^6*A*b*d*x*Cos[c + 2*d*x] + 12*a^4*A*b^3*d*x*Cos[c + 2*d*x] - 6*a^2*A*b^5*d*x*Cos[c + 2*d*x] - 6*a^6*A*b*d*x*Cos[3*c + 2*d*x] + 12*a^4*A*b^3*d*x*Cos[3*c + 2*d*x] - 6*a^2*A*b^5*d*x*Cos[3*c + 2*d*x] + 16*a^4*A*b^3*Sin[c] + 22*a^2*A*b^5*Sin[c] - 20*A*b^7*Sin[c] + 8*a^6*b*C*Sin[c] + 14*a^4*b^3*C*Sin[c] - 4*a^2*b^5*C*Sin[c] + a^7*A*Sin[d*x] + 2*a^5*A*b^2*Sin[d*x] - 53*a^3*A*b^4*Sin[d*x] + 32*a*A*b^6*Sin[d*x] - 22*a^5*b^2*C*Sin[d*x] + 4*a^3*b^4*C*Sin[d*x] + a^7*A*Sin[2*c + d*x] + 2*a^5*A*b^2*Sin[2*c + d*x] + 11*a^3*A*b^4*Sin[2*c + d*x] - 8*a*A*b^6*Sin[2*c + d*x] + 10*a^5*b^2*C*Sin[2*c + d*x] - 4*a^3*b^4*C*Sin[2*c + d*x] + 4*a^6*A*b*Sin[c + 2*d*x] - 24*a^4*A*b^3*Sin[c + 2*d*x] + 14*a^2*A*b^5*Sin[c + 2*d*x] - 8*a^6*b*C*Sin[c + 2*d*x] + 2*a^4*b^3*C*Sin[c + 2*d*x] + 4*a^6*A*b*Sin[3*c + 2*d*x] - 8*a^4*A*b^3*Sin[3*c + 2*d*x] + 4*a^2*A*b^5*Sin[3*c + 2*d*x] + a^7*A*Sin[2*c + 3*d*x] - 2*a^5*A*b^2*Sin[2*c + 3*d*x] + a^3*A*b^4*Sin[2*c + 3*d*x] + a^7*A*Sin[4*c + 3*d*x] - 2*a^5*A*b^2*Sin[4*c + 3*d*x] + a^3*A*b^4*Sin[4*c + 3*d*x]))/(4*a^4*(a^2 - b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","C",0
697,1,256,369,2.6039961,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{2 (c+d x) \left(a^2 (A+2 C)+12 A b^2\right)-\frac{2 a b^3 \left(a^2 C+A b^2\right) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^2}+a^2 A \sin (2 (c+d x))+\frac{2 a b^2 \left(6 a^4 C+a^2 b^2 (10 A-3 C)-7 A b^4\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{4 b \left(6 a^6 C+5 a^4 b^2 (4 A-C)+a^2 b^4 (2 C-29 A)+12 A b^6\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-12 a A b \sin (c+d x)}{4 a^5 d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x \left(a^2 (A+2 C)+12 A b^2\right)}{2 a^5}-\frac{b \left(a^4 (6 A-5 C)-a^2 b^2 (21 A-2 C)+12 A b^4\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^4 C+7 a^2 A b^2-4 A b^4\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(a^4 (A-4 C)-a^2 b^2 (10 A-C)+6 A b^4\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(6 a^6 C+5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+12 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*(12*A*b^2 + a^2*(A + 2*C))*(c + d*x) + (4*b*(12*A*b^6 + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 12*a*A*b*Sin[c + d*x] - (2*a*b^3*(A*b^2 + a^2*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + (2*a*b^2*(-7*A*b^4 + a^2*b^2*(10*A - 3*C) + 6*a^4*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])) + a^2*A*Sin[2*(c + d*x)])/(4*a^5*d)","A",1
698,1,564,378,4.5407677,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(-\frac{48 \left(8 a^8 C-28 a^6 b^2 C+35 a^4 b^4 C-a^2 b^6 (3 A+20 C)-2 A b^8\right) \cos (c+d x) (a \cos (c+d x)+b)^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}-\frac{2 b \sin (c+d x) \left(24 a^9 C \cos (3 (c+d x))+120 a^8 b C-68 a^7 b^2 C \cos (3 (c+d x))-318 a^6 b^3 C+4 a^5 A b^4 \cos (3 (c+d x))+65 a^5 b^4 C \cos (3 (c+d x))+6 a^4 A b^5+246 a^4 b^5 C+11 a^3 A b^6 \cos (3 (c+d x))-6 a^3 b^6 C \cos (3 (c+d x))+54 a^2 A b^7+36 a^2 b^7 C+6 a^2 b \left(20 a^6 C-57 a^4 b^2 C+a^2 b^4 (A+53 C)+3 b^6 (3 A-2 C)\right) \cos (2 (c+d x))+a \left(72 a^8 C-28 a^6 b^2 C+5 a^4 b^4 (4 A-61 C)+a^2 b^6 (13 A+438 C)+72 b^8 (A-C)\right) \cos (c+d x)-24 b^9 C\right)}{\left(b^2-a^2\right)^3}+192 a C \cos (c+d x) (a \cos (c+d x)+b)^3 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-192 a C \cos (c+d x) (a \cos (c+d x)+b)^3 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 b^5 d (a+b \sec (c+d x))^4 (A \cos (2 (c+d x))+A+2 C)}","-\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^3(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\left(5 A b^4-C \left(12 a^4-23 a^2 b^2+6 b^4\right)\right) \tan (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{\left(-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right) \tan (c+d x) \sec ^2(c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{a \left(4 a^6 C-11 a^4 b^2 C+3 a^2 b^4 (A+4 C)+2 A b^6\right) \tan (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\left(-8 a^8 C+28 a^6 b^2 C-35 a^4 b^4 C+a^2 b^6 (3 A+20 C)+2 A b^8\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2)*((-48*(-2*A*b^8 + 8*a^8*C - 28*a^6*b^2*C + 35*a^4*b^4*C - a^2*b^6*(3*A + 20*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Cos[c + d*x]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + 192*a*C*Cos[c + d*x]*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 192*a*C*Cos[c + d*x]*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*b*(6*a^4*A*b^5 + 54*a^2*A*b^7 + 120*a^8*b*C - 318*a^6*b^3*C + 246*a^4*b^5*C + 36*a^2*b^7*C - 24*b^9*C + a*(5*a^4*b^4*(4*A - 61*C) + 72*b^8*(A - C) + 72*a^8*C - 28*a^6*b^2*C + a^2*b^6*(13*A + 438*C))*Cos[c + d*x] + 6*a^2*b*(3*b^6*(3*A - 2*C) + 20*a^6*C - 57*a^4*b^2*C + a^2*b^4*(A + 53*C))*Cos[2*(c + d*x)] + 4*a^5*A*b^4*Cos[3*(c + d*x)] + 11*a^3*A*b^6*Cos[3*(c + d*x)] + 24*a^9*C*Cos[3*(c + d*x)] - 68*a^7*b^2*C*Cos[3*(c + d*x)] + 65*a^5*b^4*C*Cos[3*(c + d*x)] - 6*a^3*b^6*C*Cos[3*(c + d*x)])*Sin[c + d*x])/(-a^2 + b^2)^3))/(24*b^5*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^4)","A",0
699,1,1092,313,7.2636198,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\frac{2 C \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) (b+a \cos (c+d x))^4}{b^4 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{2 C \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) (b+a \cos (c+d x))^4}{b^4 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\left(-2 C a^6+7 b^2 C a^4+A b^4 a^2-8 b^4 C a^2+4 A b^6+8 b^6 C\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 i a \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{b^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}+\frac{2 a \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{b^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^4}{\left(b^2-a^2\right)^3 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(6 C \sin (d x) a^6-3 b C \sin (c) a^5-17 b^2 C \sin (d x) a^4-3 A b^3 \sin (c) a^3+6 b^3 C \sin (c) a^3+13 A b^4 \sin (d x) a^2+26 b^4 C \sin (d x) a^2-12 A b^5 \sin (c) a-18 b^5 C \sin (c) a+2 A b^6 \sin (d x)\right) (b+a \cos (c+d x))^3}{3 b^3 \left(b^2-a^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-3 C \sin (d x) a^4+b C \sin (c) a^3+3 A b^2 \sin (d x) a^2+8 b^2 C \sin (d x) a^2-5 A b^3 \sin (c) a-6 b^3 C \sin (c) a+2 A b^4 \sin (d x)\right) (b+a \cos (c+d x))^2}{3 b^2 \left(b^2-a^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}-\frac{2 \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-C \sin (d x) a^3+b C \sin (c) a^2-A b^2 \sin (d x) a+A b^3 \sin (c)\right) (b+a \cos (c+d x))}{3 a b \left(b^2-a^2\right) d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}","-\frac{\left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{a \left(-3 a^4 C+a^2 b^2 (3 A+8 C)+2 A b^4\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{a \left(-2 a^6 C+7 a^4 b^2 C+a^2 b^4 (A-8 C)+4 b^6 (A+2 C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(9 a^6 C-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+4 A b^6\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(-2*C*(b + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*C*(b + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((a^2*A*b^4 + 4*A*b^6 - 2*a^6*C + 7*a^4*b^2*C - 8*a^2*b^4*C + 8*b^6*C)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(((2*I)*a*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(b^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) + (2*a*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(b^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(A*b^3*Sin[c] + a^2*b*C*Sin[c] - a*A*b^2*Sin[d*x] - a^3*C*Sin[d*x]))/(3*a*b*(-a^2 + b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(-5*a*A*b^3*Sin[c] + a^3*b*C*Sin[c] - 6*a*b^3*C*Sin[c] + 3*a^2*A*b^2*Sin[d*x] + 2*A*b^4*Sin[d*x] - 3*a^4*C*Sin[d*x] + 8*a^2*b^2*C*Sin[d*x]))/(3*b^2*(-a^2 + b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(-3*a^3*A*b^3*Sin[c] - 12*a*A*b^5*Sin[c] - 3*a^5*b*C*Sin[c] + 6*a^3*b^3*C*Sin[c] - 18*a*b^5*C*Sin[c] + 13*a^2*A*b^4*Sin[d*x] + 2*A*b^6*Sin[d*x] + 6*a^6*C*Sin[d*x] - 17*a^4*b^2*C*Sin[d*x] + 26*a^2*b^4*C*Sin[d*x]))/(3*b^3*(-a^2 + b^2)^3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","C",0
700,1,221,261,1.4172248,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\frac{\frac{24 b \left(a^2 (4 A+3 C)+b^2 (A+2 C)\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{2 \sin (c+d x) \left(6 b \left(a^4 (2 A+C)+9 a^2 b^2 (A+C)-A b^4\right) \cos (c+d x)+a \left(a^4 (6 A+8 C)+a^2 b^2 (14 A+C)+\left(a^4 (6 A+4 C)+a^2 b^2 (10 A+11 C)-A b^4\right) \cos (2 (c+d x))+b^4 (25 A+36 C)\right)\right)}{(a \cos (c+d x)+b)^3}}{24 d \left(b^2-a^2\right)^3}","-\frac{b \left(a^2 (4 A+3 C)+b^2 (A+2 C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(a^2 C+A b^2\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a \left(2 a^4 C+a^2 b^2 (2 A-5 C)+b^4 (13 A+18 C)\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\left(-4 a^4 C+a^2 b^2 (2 A+9 C)+3 A b^4\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}",1,"-1/24*((24*b*(b^2*(A + 2*C) + a^2*(4*A + 3*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*(6*b*(-(A*b^4) + 9*a^2*b^2*(A + C) + a^4*(2*A + C))*Cos[c + d*x] + a*(a^2*b^2*(14*A + C) + a^4*(6*A + 8*C) + b^4*(25*A + 36*C) + (-(A*b^4) + a^4*(6*A + 4*C) + a^2*b^2*(10*A + 11*C))*Cos[2*(c + d*x)]))*Sin[c + d*x])/(b + a*Cos[c + d*x])^3)/((-a^2 + b^2)^3*d)","A",1
701,1,438,252,6.9387935,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(-\frac{6 i a (\cos (c)-i \sin (c)) \left(a^2 (2 A+C)+b^2 (3 A+4 C)\right) (a \cos (c+d x)+b)^3 \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{7/2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{2 b \sec (c) \left(a^2 C+A b^2\right) (b \sin (c)-a \sin (d x))}{a^5-a^3 b^2}+\frac{\sec (c) (a \cos (c+d x)+b) \left(\sin (c) \left(-5 a^4 b C-11 a^2 A b^3+6 A b^5\right)+a \sin (d x) \left(3 a^4 C+a^2 b^2 (9 A+2 C)-4 A b^4\right)\right)}{a^3 \left(a^2-b^2\right)^2}+\frac{\sec (c) (a \cos (c+d x)+b)^2 \left(3 \sin (c) \left(a^6 C+a^4 b^2 (9 A+4 C)-6 a^2 A b^4+2 A b^6\right)-a b \sin (d x) \left(a^4 (18 A+13 C)+a^2 b^2 (2 C-5 A)+2 A b^4\right)\right)}{\left(a^3-a b^2\right)^3}\right)}{3 d (a+b \sec (c+d x))^4 (A \cos (2 (c+d x))+A+2 C)}","\frac{a \left(a^2 (2 A+C)+b^2 (3 A+4 C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{a \left(a^2 (-C)+5 A b^2+6 b^2 C\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{\left(a^4 C-a^2 b^2 (11 A+10 C)-2 b^4 (2 A+3 C)\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(((-6*I)*a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^3*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^(7/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (2*b*(A*b^2 + a^2*C)*Sec[c]*(b*Sin[c] - a*Sin[d*x]))/(a^5 - a^3*b^2) + ((b + a*Cos[c + d*x])*Sec[c]*((-11*a^2*A*b^3 + 6*A*b^5 - 5*a^4*b*C)*Sin[c] + a*(-4*A*b^4 + 3*a^4*C + a^2*b^2*(9*A + 2*C))*Sin[d*x]))/(a^3*(a^2 - b^2)^2) + ((b + a*Cos[c + d*x])^2*Sec[c]*(3*(-6*a^2*A*b^4 + 2*A*b^6 + a^6*C + a^4*b^2*(9*A + 4*C))*Sin[c] - a*b*(2*A*b^4 + a^2*b^2*(-5*A + 2*C) + a^4*(18*A + 13*C))*Sin[d*x]))/(a^3 - a*b^2)^3))/(3*d*(A + 2*C + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^4)","C",0
702,1,995,292,7.2842556,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4,x]","\frac{2 A x \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) (b+a \cos (c+d x))^4}{a^4 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\left(-8 A a^6-4 C a^6+8 A b^2 a^4-b^2 C a^4-7 A b^4 a^2+2 A b^6\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 i b \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}+\frac{2 b \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^4}{\left(b^2-a^2\right)^3 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(6 C \sin (d x) a^7-12 b C \sin (c) a^6+36 A b^2 \sin (d x) a^5+10 b^2 C \sin (d x) a^5-48 A b^3 \sin (c) a^4-3 b^3 C \sin (c) a^4-32 A b^4 \sin (d x) a^3-b^4 C \sin (d x) a^3+51 A b^5 \sin (c) a^2+11 A b^6 \sin (d x) a-18 A b^7 \sin (c)\right) (b+a \cos (c+d x))^3}{3 a^4 \left(a^2-b^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-9 A \sin (c) b^6+7 a A \sin (d x) b^5+14 a^2 A \sin (c) b^4-3 a^2 C \sin (c) b^4-12 a^3 A \sin (d x) b^3+a^3 C \sin (d x) b^3+8 a^4 C \sin (c) b^2-6 a^5 C \sin (d x) b\right) (b+a \cos (c+d x))^2}{3 a^4 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}-\frac{2 \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(A \sin (c) b^5-a A \sin (d x) b^4+a^2 C \sin (c) b^3-a^3 C \sin (d x) b^2\right) (b+a \cos (c+d x))}{3 a^4 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}","\frac{A x}{a^4}+\frac{\left(a^2 C+A b^2\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\left(-2 a^4 C-a^2 b^2 (8 A+3 C)+3 A b^4\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{b \left(4 a^6 (2 A+C)-a^4 b^2 (8 A-C)+7 a^2 A b^4-2 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-2 a^6 C-13 a^4 b^2 (2 A+C)+17 a^2 A b^4-6 A b^6\right) \tan (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"(2*A*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a^4*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-8*a^6*A + 8*a^4*A*b^2 - 7*a^2*A*b^4 + 2*A*b^6 - 4*a^6*C - a^4*b^2*C)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(((2*I)*b*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) + (2*b*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(A*b^5*Sin[c] + a^2*b^3*C*Sin[c] - a*A*b^4*Sin[d*x] - a^3*b^2*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(14*a^2*A*b^4*Sin[c] - 9*A*b^6*Sin[c] + 8*a^4*b^2*C*Sin[c] - 3*a^2*b^4*C*Sin[c] - 12*a^3*A*b^3*Sin[d*x] + 7*a*A*b^5*Sin[d*x] - 6*a^5*b*C*Sin[d*x] + a^3*b^3*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(-48*a^4*A*b^3*Sin[c] + 51*a^2*A*b^5*Sin[c] - 18*A*b^7*Sin[c] - 12*a^6*b*C*Sin[c] - 3*a^4*b^3*C*Sin[c] + 36*a^5*A*b^2*Sin[d*x] - 32*a^3*A*b^4*Sin[d*x] + 11*a*A*b^6*Sin[d*x] + 6*a^7*C*Sin[d*x] + 10*a^5*b^2*C*Sin[d*x] - a^3*b^4*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)^3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","C",0
703,1,1089,367,7.6567861,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\frac{8 A b x \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) (b+a \cos (c+d x))^4}{a^5 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\left(-2 C a^8-20 A b^2 a^6-3 b^2 C a^6+35 A b^4 a^4-28 A b^6 a^2+8 A b^8\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{2 i \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{a^5 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{2 \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{a^5 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^4}{\left(b^2-a^2\right)^3 (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{2 A \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \tan (c+d x) (b+a \cos (c+d x))^4}{a^4 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(36 A \sin (c) b^8-26 a A \sin (d x) b^7-96 a^2 A \sin (c) b^6+6 a^2 C \sin (c) b^6+71 a^3 A \sin (d x) b^5-2 a^3 C \sin (d x) b^5+75 a^4 A \sin (c) b^4-18 a^4 C \sin (c) b^4-60 a^5 A \sin (d x) b^3+5 a^5 C \sin (d x) b^3+27 a^6 C \sin (c) b^2-18 a^7 C \sin (d x) b\right) (b+a \cos (c+d x))^3}{3 a^5 \left(a^2-b^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(12 A \sin (c) b^7-10 a A \sin (d x) b^6-17 a^2 A \sin (c) b^5+6 a^2 C \sin (c) b^5+15 a^3 A \sin (d x) b^4-4 a^3 C \sin (d x) b^4-11 a^4 C \sin (c) b^3+9 a^5 C \sin (d x) b^2\right) (b+a \cos (c+d x))^2}{3 a^5 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}+\frac{2 \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(A \sin (c) b^6-a A \sin (d x) b^5+a^2 C \sin (c) b^4-a^3 C \sin (d x) b^3\right) (b+a \cos (c+d x))}{3 a^5 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^4}","-\frac{4 A b x}{a^5}+\frac{\left(a^2 C+A b^2\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\left(-3 a^4 C-a^2 b^2 (9 A+2 C)+4 A b^4\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\left(a^6 (6 A-11 C)-a^4 b^2 (65 A+4 C)+68 a^2 A b^4-24 A b^6\right) \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-2 a^6 C-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-4 A b^6\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\left(-2 a^8 C-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6+8 A b^8\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"(-8*A*b*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a^5*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-20*a^6*A*b^2 + 35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - 3*a^6*b^2*C)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(A*b^6*Sin[c] + a^2*b^4*C*Sin[c] - a*A*b^5*Sin[d*x] - a^3*b^3*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(-17*a^2*A*b^5*Sin[c] + 12*A*b^7*Sin[c] - 11*a^4*b^3*C*Sin[c] + 6*a^2*b^5*C*Sin[c] + 15*a^3*A*b^4*Sin[d*x] - 10*a*A*b^6*Sin[d*x] + 9*a^5*b^2*C*Sin[d*x] - 4*a^3*b^4*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*(75*a^4*A*b^4*Sin[c] - 96*a^2*A*b^6*Sin[c] + 36*A*b^8*Sin[c] + 27*a^6*b^2*C*Sin[c] - 18*a^4*b^4*C*Sin[c] + 6*a^2*b^6*C*Sin[c] - 60*a^5*A*b^3*Sin[d*x] + 71*a^3*A*b^5*Sin[d*x] - 26*a*A*b^7*Sin[d*x] - 18*a^7*b*C*Sin[d*x] + 5*a^5*b^3*C*Sin[d*x] - 2*a^3*b^5*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*A*(b + a*Cos[c + d*x])^4*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Tan[c + d*x])/(a^4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","C",0
704,1,1314,513,5.8677451,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{12 A c \cos (3 (c+d x)) a^{11}+24 c C \cos (3 (c+d x)) a^{11}+12 A d x \cos (3 (c+d x)) a^{11}+24 C d x \cos (3 (c+d x)) a^{11}+6 A \sin (c+d x) a^{11}+9 A \sin (3 (c+d x)) a^{11}+3 A \sin (5 (c+d x)) a^{11}+72 A b c a^{10}+144 b c C a^{10}+72 A b d x a^{10}+144 b C d x a^{10}-60 A b \sin (2 (c+d x)) a^{10}-30 A b \sin (4 (c+d x)) a^{10}+204 A b^2 c \cos (3 (c+d x)) a^9-72 b^2 c C \cos (3 (c+d x)) a^9+204 A b^2 d x \cos (3 (c+d x)) a^9-72 b^2 C d x \cos (3 (c+d x)) a^9-270 A b^2 \sin (c+d x) a^9+144 b^2 C \sin (c+d x) a^9-279 A b^2 \sin (3 (c+d x)) a^9+144 b^2 C \sin (3 (c+d x)) a^9-9 A b^2 \sin (5 (c+d x)) a^9+1272 A b^3 c a^8-336 b^3 c C a^8+1272 A b^3 d x a^8-336 b^3 C d x a^8-372 A b^3 \sin (2 (c+d x)) a^8+480 b^3 C \sin (2 (c+d x)) a^8+90 A b^3 \sin (4 (c+d x)) a^8-684 A b^4 c \cos (3 (c+d x)) a^7+72 b^4 c C \cos (3 (c+d x)) a^7-684 A b^4 d x \cos (3 (c+d x)) a^7+72 b^4 C d x \cos (3 (c+d x)) a^7+750 A b^4 \sin (c+d x) a^7+288 b^4 C \sin (c+d x) a^7+1143 A b^4 \sin (3 (c+d x)) a^7-128 b^4 C \sin (3 (c+d x)) a^7+9 A b^4 \sin (5 (c+d x)) a^7-3288 A b^5 c a^6+144 b^5 c C a^6-3288 A b^5 d x a^6+144 b^5 C d x a^6+2772 A b^5 \sin (2 (c+d x)) a^6-360 b^5 C \sin (2 (c+d x)) a^6-90 A b^5 \sin (4 (c+d x)) a^6+708 A b^6 c \cos (3 (c+d x)) a^5-24 b^6 c C \cos (3 (c+d x)) a^5+708 A b^6 d x \cos (3 (c+d x)) a^5-24 b^6 C d x \cos (3 (c+d x)) a^5+1086 A b^6 \sin (c+d x) a^5-228 b^6 C \sin (c+d x) a^5-1253 A b^6 \sin (3 (c+d x)) a^5+44 b^6 C \sin (3 (c+d x)) a^5-3 A b^6 \sin (5 (c+d x)) a^5+1512 A b^7 c a^4+144 b^7 c C a^4+1512 A b^7 d x a^4+144 b^7 C d x a^4-3300 A b^7 \sin (2 (c+d x)) a^4+120 b^7 C \sin (2 (c+d x)) a^4+30 A b^7 \sin (4 (c+d x)) a^4-240 A b^8 c \cos (3 (c+d x)) a^3-240 A b^8 d x \cos (3 (c+d x)) a^3-2232 A b^8 \sin (c+d x) a^3+96 b^8 C \sin (c+d x) a^3+440 A b^8 \sin (3 (c+d x)) a^3+1392 A b^9 c a^2-96 b^9 c C a^2+1392 A b^9 d x a^2-96 b^9 C d x a^2+72 b \left(a^2-b^2\right)^3 \left((A+2 C) a^2+20 A b^2\right) (c+d x) \cos (2 (c+d x)) a^2+1200 A b^9 \sin (2 (c+d x)) a^2+36 \left(a^2-b^2\right)^3 \left(a^2+4 b^2\right) \left((A+2 C) a^2+20 A b^2\right) (c+d x) \cos (c+d x) a+960 A b^{10} \sin (c+d x) a-960 A b^{11} c-960 A b^{11} d x}{\left(a^2-b^2\right)^3 (b+a \cos (c+d x))^3}-\frac{96 b \left(-8 C a^8+8 b^2 (C-5 A) a^6+7 b^4 (12 A-C) a^4+b^6 (2 C-69 A) a^2+20 A b^8\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}}{96 a^6 d}","\frac{\left(a^2 C+A b^2\right) \sin (c+d x) \cos (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{x \left(a^2 (A+2 C)+20 A b^2\right)}{2 a^6}-\frac{\left(-4 a^4 C-a^2 b^2 (10 A+C)+5 A b^4\right) \sin (c+d x) \cos (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{\left(-\left(a^6 (A-6 C)\right)+a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+10 A b^6\right) \sin (c+d x) \cos (c+d x)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\left(-8 a^8 b C-8 a^6 b^3 (5 A-C)+7 a^4 b^5 (12 A-C)-a^2 b^7 (69 A-2 C)+20 A b^9\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}+\frac{b \left(-\left(a^6 (24 A-26 C)\right)+a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+60 A b^6\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{\left(12 a^6 C+a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 A b^6\right) \sin (c+d x) \cos (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((-96*b*(20*A*b^8 + 7*a^4*b^4*(12*A - C) - 8*a^8*C + 8*a^6*b^2*(-5*A + C) + a^2*b^6*(-69*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (72*a^10*A*b*c + 1272*a^8*A*b^3*c - 3288*a^6*A*b^5*c + 1512*a^4*A*b^7*c + 1392*a^2*A*b^9*c - 960*A*b^11*c + 144*a^10*b*c*C - 336*a^8*b^3*c*C + 144*a^6*b^5*c*C + 144*a^4*b^7*c*C - 96*a^2*b^9*c*C + 72*a^10*A*b*d*x + 1272*a^8*A*b^3*d*x - 3288*a^6*A*b^5*d*x + 1512*a^4*A*b^7*d*x + 1392*a^2*A*b^9*d*x - 960*A*b^11*d*x + 144*a^10*b*C*d*x - 336*a^8*b^3*C*d*x + 144*a^6*b^5*C*d*x + 144*a^4*b^7*C*d*x - 96*a^2*b^9*C*d*x + 36*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(20*A*b^2 + a^2*(A + 2*C))*(c + d*x)*Cos[c + d*x] + 72*a^2*b*(a^2 - b^2)^3*(20*A*b^2 + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] + 12*a^11*A*c*Cos[3*(c + d*x)] + 204*a^9*A*b^2*c*Cos[3*(c + d*x)] - 684*a^7*A*b^4*c*Cos[3*(c + d*x)] + 708*a^5*A*b^6*c*Cos[3*(c + d*x)] - 240*a^3*A*b^8*c*Cos[3*(c + d*x)] + 24*a^11*c*C*Cos[3*(c + d*x)] - 72*a^9*b^2*c*C*Cos[3*(c + d*x)] + 72*a^7*b^4*c*C*Cos[3*(c + d*x)] - 24*a^5*b^6*c*C*Cos[3*(c + d*x)] + 12*a^11*A*d*x*Cos[3*(c + d*x)] + 204*a^9*A*b^2*d*x*Cos[3*(c + d*x)] - 684*a^7*A*b^4*d*x*Cos[3*(c + d*x)] + 708*a^5*A*b^6*d*x*Cos[3*(c + d*x)] - 240*a^3*A*b^8*d*x*Cos[3*(c + d*x)] + 24*a^11*C*d*x*Cos[3*(c + d*x)] - 72*a^9*b^2*C*d*x*Cos[3*(c + d*x)] + 72*a^7*b^4*C*d*x*Cos[3*(c + d*x)] - 24*a^5*b^6*C*d*x*Cos[3*(c + d*x)] + 6*a^11*A*Sin[c + d*x] - 270*a^9*A*b^2*Sin[c + d*x] + 750*a^7*A*b^4*Sin[c + d*x] + 1086*a^5*A*b^6*Sin[c + d*x] - 2232*a^3*A*b^8*Sin[c + d*x] + 960*a*A*b^10*Sin[c + d*x] + 144*a^9*b^2*C*Sin[c + d*x] + 288*a^7*b^4*C*Sin[c + d*x] - 228*a^5*b^6*C*Sin[c + d*x] + 96*a^3*b^8*C*Sin[c + d*x] - 60*a^10*A*b*Sin[2*(c + d*x)] - 372*a^8*A*b^3*Sin[2*(c + d*x)] + 2772*a^6*A*b^5*Sin[2*(c + d*x)] - 3300*a^4*A*b^7*Sin[2*(c + d*x)] + 1200*a^2*A*b^9*Sin[2*(c + d*x)] + 480*a^8*b^3*C*Sin[2*(c + d*x)] - 360*a^6*b^5*C*Sin[2*(c + d*x)] + 120*a^4*b^7*C*Sin[2*(c + d*x)] + 9*a^11*A*Sin[3*(c + d*x)] - 279*a^9*A*b^2*Sin[3*(c + d*x)] + 1143*a^7*A*b^4*Sin[3*(c + d*x)] - 1253*a^5*A*b^6*Sin[3*(c + d*x)] + 440*a^3*A*b^8*Sin[3*(c + d*x)] + 144*a^9*b^2*C*Sin[3*(c + d*x)] - 128*a^7*b^4*C*Sin[3*(c + d*x)] + 44*a^5*b^6*C*Sin[3*(c + d*x)] - 30*a^10*A*b*Sin[4*(c + d*x)] + 90*a^8*A*b^3*Sin[4*(c + d*x)] - 90*a^6*A*b^5*Sin[4*(c + d*x)] + 30*a^4*A*b^7*Sin[4*(c + d*x)] + 3*a^11*A*Sin[5*(c + d*x)] - 9*a^9*A*b^2*Sin[5*(c + d*x)] + 9*a^7*A*b^4*Sin[5*(c + d*x)] - 3*a^5*A*b^6*Sin[5*(c + d*x)])/((a^2 - b^2)^3*(b + a*Cos[c + d*x])^3))/(96*a^6*d)","B",1
705,1,17,17,0.0068804,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","a x-\frac{b \tanh ^{-1}(\sin (c+d x))}{d}","a x-\frac{b \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*x - (b*ArcTanh[Sin[c + d*x]])/d","A",1
706,1,56,52,0.1068849,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","\frac{4 b \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{d \sqrt{a^2-b^2}}+\frac{c}{d}+x","x-\frac{4 b \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d \sqrt{a-b} \sqrt{a+b}}",1,"c/d + x + (4*b*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(Sqrt[a^2 - b^2]*d)","A",1
707,1,139,107,0.4882378,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{b \left(\left(a^2-b^2\right) (c+d x)+2 a b \sin (c+d x)\right)+a \left(a^2-b^2\right) (c+d x) \cos (c+d x)}{a \cos (c+d x)+b}-\frac{2 b \left(b^2-3 a^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{a d (a-b) (a+b)}","-\frac{2 b \left(3 a^2-b^2\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d (a-b)^{3/2} (a+b)^{3/2}}+\frac{2 b^2 \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x}{a}",1,"((-2*b*(-3*a^2 + b^2)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (a*(a^2 - b^2)*(c + d*x)*Cos[c + d*x] + b*((a^2 - b^2)*(c + d*x) + 2*a*b*Sin[c + d*x]))/(b + a*Cos[c + d*x]))/(a*(a - b)*(a + b)*d)","A",1
708,1,223,162,0.7937673,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) (a-b \sec (c+d x)) \left(\frac{a b^2 \left(5 a^2-2 b^2\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}+\frac{2 b \left(4 a^4-2 a^2 b^2+b^4\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^3 \sin (c+d x)}{(b-a) (a+b)}+(c+d x) (a \cos (c+d x)+b)^2\right)}{a^2 d (a \cos (c+d x)-b) (a+b \sec (c+d x))^3}","\frac{b^2 \left(4 a^2-b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{b^2 \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x}{a^2}-\frac{2 b \left(4 a^4-2 a^2 b^2+b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(a - b*Sec[c + d*x])*((c + d*x)*(b + a*Cos[c + d*x])^2 + (2*b*(4*a^4 - 2*a^2*b^2 + b^4)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) + (a*b^3*Sin[c + d*x])/((-a + b)*(a + b)) + (a*b^2*(5*a^2 - 2*b^2)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2)))/(a^2*d*(-b + a*Cos[c + d*x])*(a + b*Sec[c + d*x])^3)","A",1
709,1,3518,467,23.925692,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \left(6 a^2 C-7 b^2 (9 A+7 C)\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}+\frac{2 a \left(8 a^2 C+21 A b^2+13 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(16 a^4 C+6 a^2 b^2 (7 A+4 C)-21 b^4 (9 A+7 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^5 d}+\frac{2 (a-b) \sqrt{a+b} \left(16 a^3 C+12 a^2 b C+6 a b^2 (7 A+6 C)+21 b^3 (9 A+7 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}+\frac{2 a C \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d}",1,"(Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((4*(-42*a^2*A*b^2 + 189*A*b^4 - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^4) + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] - 6*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b^2) + (4*Sec[c + d*x]*(21*a*A*b^2*Sin[c + d*x] + 8*a^3*C*Sin[c + d*x] + 13*a*b^2*C*Sin[c + d*x]))/(315*b^3) + (4*a*C*Sec[c + d*x]^2*Tan[c + d*x])/(63*b) + (4*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (4*((4*a^2*A)/(15*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*A*b)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^4*C)/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^2*C)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*A*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*A*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a*C*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (32*a^5*C*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x]]) + (8*a^3*C*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (6*a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (14*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (32*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x]]) + (16*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^4*d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(5/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^4*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - a*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*(a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a + b)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*((b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])) + (a + b)*((16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-16*a^3*C + 12*a^2*b*C - 6*a*b^2*(7*A + 6*C) + 21*b^3*(9*A + 7*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(315*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))","B",0
710,1,560,375,19.6520024,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 \sec (c+d x) \left(-4 a^2 C \sin (c+d x)+35 A b^2 \sin (c+d x)+25 b^2 C \sin (c+d x)\right)}{105 b^2}+\frac{4 a \left(8 a^2 C+35 A b^2+19 b^2 C\right) \sin (c+d x)}{105 b^3}+\frac{4 a C \tan (c+d x) \sec (c+d x)}{35 b}+\frac{4}{7} C \tan (c+d x) \sec ^2(c+d x)\right)}{d (A \cos (2 c+2 d x)+A+2 C)}+\frac{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(-a \left(8 a^2 C+35 A b^2+19 b^2 C\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 b (a+b) \left(C \left(8 a^2-6 a b+25 b^2\right)+35 A b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a (a+b) \left(8 a^2 C+35 A b^2+19 b^2 C\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{105 b^3 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 \left(8 a^2 C+5 b^2 (7 A+5 C)\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b^2 d}-\frac{2 a (a-b) \sqrt{a+b} \left(8 a^2 C+35 A b^2+19 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(C \left(8 a^2+6 a b+25 b^2\right)+35 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}-\frac{8 a C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{3/2}}{7 b d}",1,"(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(-2*a*(a + b)*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(35*A*b^2 + (8*a^2 - 6*a*b + 25*b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^3*d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((4*a*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*Sin[c + d*x])/(105*b^3) + (4*Sec[c + d*x]*(35*A*b^2*Sin[c + d*x] - 4*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b^2) + (4*a*C*Sec[c + d*x]*Tan[c + d*x])/(35*b) + (4*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))","A",0
711,1,507,308,18.3828731,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 \left(-2 a^2 C+15 A b^2+9 b^2 C\right) \sin (c+d x)}{15 b^2}+\frac{4 a C \tan (c+d x)}{15 b}+\frac{4}{5} C \tan (c+d x) \sec (c+d x)\right)}{d (A \cos (2 c+2 d x)+A+2 C)}+\frac{4 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left((a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left(\left(2 a^2 C-15 A b^2-9 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+b (-2 a C+15 A b+9 b C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)-\left(-2 a^2 C+15 A b^2+9 b^2 C\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)}{15 b^2 d \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 (a-b) \sqrt{a+b} \left(2 a^2 C-3 b^2 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}+\frac{2 (a-b) \sqrt{a+b} (2 a C+15 A b+9 b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 b d}-\frac{4 a C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b d}",1,"(4*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((a + b)*((-15*A*b^2 + 2*a^2*C - 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(15*A*b - 2*a*C + 9*b*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] - (15*A*b^2 - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(15*b^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((4*(15*A*b^2 - 2*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*b^2) + (4*a*C*Tan[c + d*x])/(15*b) + (4*C*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))","A",0
712,1,570,355,11.4071652,"\int \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 a C \sin (c+d x)}{3 b}+\frac{4}{3} C \tan (c+d x)\right)}{d (A \cos (2 c+2 d x)+A+2 C)}+\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(2 i b (a-b) (3 A+C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-12 i a A b \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-a C \sqrt{\frac{b-a}{a+b}} \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 i a C (a-b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{3 b d \sqrt{\frac{b-a}{a+b}} (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 \sqrt{a+b} (3 A b-C (a-b)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 a C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(4*Cos[(c + d*x)/2]^2*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((2*I)*a*(a - b)*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (2*I)*(a - b)*b*(3*A + C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (12*I)*a*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - a*Sqrt[(-a + b)/(a + b)]*C*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*((4*a*C*Sin[c + d*x])/(3*b) + (4*C*Tan[c + d*x])/3))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
713,1,723,352,18.5005159,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{a+b \sec (c+d x)} \left(-2 (A b-C (a+b)) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+(a+b) (A-2 C) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 A b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 A b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+a A \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A \tan ^3\left(\frac{1}{2} (c+d x)\right)+a A \tan \left(\frac{1}{2} (c+d x)\right)-2 a C \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a C \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a C \tan \left(\frac{1}{2} (c+d x)\right)-A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+A b \tan \left(\frac{1}{2} (c+d x)\right)+2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 b C \tan \left(\frac{1}{2} (c+d x)\right)\right)}{d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+b} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{2 C \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}","\frac{\sqrt{a+b} (2 C (a-b)+A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{(a-b) \sqrt{a+b} (A-2 C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}-\frac{A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(2*C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*A*Tan[(c + d*x)/2] + A*b*Tan[(c + d*x)/2] - 2*a*C*Tan[(c + d*x)/2] - 2*b*C*Tan[(c + d*x)/2] - 2*a*A*Tan[(c + d*x)/2]^3 + 4*a*C*Tan[(c + d*x)/2]^3 + a*A*Tan[(c + d*x)/2]^5 - A*b*Tan[(c + d*x)/2]^5 - 2*a*C*Tan[(c + d*x)/2]^5 + 2*b*C*Tan[(c + d*x)/2]^5 + 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(A - 2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(A*b - (a + b)*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
714,1,1417,411,19.5465519,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{A \sqrt{a+b \sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{\sqrt{a+b \sec (c+d x)} \left(A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i A (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i (a-b) (A b+2 a (A+2 C)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a \sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \left(A b^2-4 a^2 (A+2 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{\sqrt{a+b} (2 a (A+4 C)+A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{A b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}+\frac{A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}",1,"(A*Sqrt[a + b*Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (Sqrt[a + b*Sec[c + d*x]]*(-(a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]) - A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 2*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*A*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(a - b)*(A*b + 2*a*(A + 2*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a*Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
715,1,1334,502,19.1518201,"\int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{1}{12} A \sin (c+d x)+\frac{A b \sin (2 (c+d x))}{24 a}+\frac{1}{12} A \sin (3 (c+d x))\right)}{d}-\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-48 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+24 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+24 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(8 a^2 (2 A+3 C)-3 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a b (14 a A-b A+24 a C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 a^2 d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{\left(3 A b^2-8 a^2 (2 A+3 C)\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 a^2 d}+\frac{\sqrt{a+b} \left(8 a^2 (2 A+3 C)+2 a A b-3 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a^2 d}-\frac{(a-b) \sqrt{a+b} \left(3 A b^2-8 a^2 (2 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a^2 b d}-\frac{b \sqrt{a+b} \left(4 a^2 (A+2 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a^3 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{A b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((A*Sin[c + d*x])/12 + (A*b*Sin[2*(c + d*x)])/(24*a) + (A*Sin[3*(c + d*x)])/12))/d - (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] - 3*a*A*b^2*Tan[(c + d*x)/2] - 3*A*b^3*Tan[(c + d*x)/2] + 24*a^3*C*Tan[(c + d*x)/2] + 24*a^2*b*C*Tan[(c + d*x)/2] - 32*a^3*A*Tan[(c + d*x)/2]^3 + 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 48*a^3*C*Tan[(c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 - 3*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*A*b^3*Tan[(c + d*x)/2]^5 + 24*a^3*C*Tan[(c + d*x)/2]^5 - 24*a^2*b*C*Tan[(c + d*x)/2]^5 + 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-3*A*b^2 + 8*a^2*(2*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*b*(14*a*A - A*b + 24*a*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a^2*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
716,1,1825,587,16.4514294,"\int \cos ^4(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{A b \sin (c+d x)}{96 a}+\frac{\left(48 A a^2+48 C a^2-5 A b^2\right) \sin (2 (c+d x))}{192 a^2}+\frac{A b \sin (3 (c+d x))}{96 a}+\frac{1}{32} A \sin (4 (c+d x))\right)}{d}+\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-15 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-28 a^2 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+28 a^3 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 a^2 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+48 a^3 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a A b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)-56 a^3 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-96 a^3 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-30 A b^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-48 a^2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+288 a^4 A \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+384 a^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-96 a^2 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+15 A b^4 \tan \left(\frac{1}{2} (c+d x)\right)+15 a A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+28 a^2 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+28 a^3 A b \tan \left(\frac{1}{2} (c+d x)\right)+48 a^2 b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+48 a^3 b C \tan \left(\frac{1}{2} (c+d x)\right)+b (a+b) \left(4 (7 A+12 C) a^2+15 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a \left(24 (3 A+4 C) a^3-12 b (3 A+4 C) a^2+2 A b^2 a+5 A b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 A b^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-48 a^2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+288 a^4 A \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+384 a^4 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-96 a^2 b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{192 a^3 d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{\left(5 A b^2-12 a^2 (3 A+4 C)\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 a^2 d}+\frac{\sqrt{a+b} \left(-16 a^4 (3 A+4 C)+8 a^2 b^2 (A+2 C)+5 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^4 d}+\frac{b \left(4 a^2 (7 A+12 C)+15 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a^3 d}+\frac{(a-b) \sqrt{a+b} \left(4 a^2 (7 A+12 C)+15 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a^3 d}-\frac{\sqrt{a+b} \left(-24 a^3 (3 A+4 C)-4 a^2 b (7 A+12 C)+10 a A b^2-15 A b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a^3 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 a d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((A*b*Sin[c + d*x])/(96*a) + ((48*a^2*A - 5*A*b^2 + 48*a^2*C)*Sin[2*(c + d*x)])/(192*a^2) + (A*b*Sin[3*(c + d*x)])/(96*a) + (A*Sin[4*(c + d*x)])/32))/d + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(28*a^3*A*b*Tan[(c + d*x)/2] + 28*a^2*A*b^2*Tan[(c + d*x)/2] + 15*a*A*b^3*Tan[(c + d*x)/2] + 15*A*b^4*Tan[(c + d*x)/2] + 48*a^3*b*C*Tan[(c + d*x)/2] + 48*a^2*b^2*C*Tan[(c + d*x)/2] - 56*a^3*A*b*Tan[(c + d*x)/2]^3 - 30*a*A*b^3*Tan[(c + d*x)/2]^3 - 96*a^3*b*C*Tan[(c + d*x)/2]^3 + 28*a^3*A*b*Tan[(c + d*x)/2]^5 - 28*a^2*A*b^2*Tan[(c + d*x)/2]^5 + 15*a*A*b^3*Tan[(c + d*x)/2]^5 - 15*A*b^4*Tan[(c + d*x)/2]^5 + 48*a^3*b*C*Tan[(c + d*x)/2]^5 - 48*a^2*b^2*C*Tan[(c + d*x)/2]^5 + 288*a^4*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 384*a^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 96*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 288*a^4*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 48*a^2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 384*a^4*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 96*a^2*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(15*A*b^2 + 4*a^2*(7*A + 12*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(2*a*A*b^2 + 5*A*b^3 + 24*a^3*(3*A + 4*C) - 12*a^2*b*(3*A + 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(192*a^3*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
717,1,3988,550,29.0855038,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \left(a^2 C+3 b^2 (11 A+9 C)\right) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{231 b d}+\frac{4 a \left(-3 a^2 C+132 A b^2+101 b^2 C\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{1155 b^2 d}+\frac{4 a (a-b) \sqrt{a+b} \left(8 a^4 C+3 a^2 b^2 (11 A+6 C)-b^4 (451 A+348 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{1155 b^5 d}+\frac{2 \left(8 a^4 C+a^2 b^2 (33 A+19 C)+25 b^4 (11 A+9 C)\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{1155 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(16 a^4 C+12 a^3 b C+6 a^2 b^2 (11 A+8 C)+3 a b^3 (209 A+157 C)-25 b^4 (11 A+9 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{1155 b^4 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}+\frac{2 a C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{33 d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((-8*a*(33*a^2*A*b^2 - 451*A*b^4 + 8*a^4*C + 18*a^2*b^2*C - 348*b^4*C)*Sin[c + d*x])/(1155*b^4) + (4*Sec[c + d*x]^3*(33*A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x] + 27*b^2*C*Sin[c + d*x]))/(231*b) + (8*Sec[c + d*x]^2*(132*a*A*b^2*Sin[c + d*x] - 3*a^3*C*Sin[c + d*x] + 101*a*b^2*C*Sin[c + d*x]))/(1155*b^2) + (4*Sec[c + d*x]*(33*a^2*A*b^2*Sin[c + d*x] + 275*A*b^4*Sin[c + d*x] + 8*a^4*C*Sin[c + d*x] + 19*a^2*b^2*C*Sin[c + d*x] + 225*b^4*C*Sin[c + d*x]))/(1155*b^3) + (16*a*C*Sec[c + d*x]^3*Tan[c + d*x])/33 + (4*b*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) + (8*((4*a^3*A)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (164*a*A*b)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^5*C)/(1155*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (24*a^3*C)/(385*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (464*a*b*C)/(385*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (62*a^2*A*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*A*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (10*A*b^2*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (26*a^2*C*Sqrt[Sec[c + d*x]])/(55*Sqrt[b + a*Cos[c + d*x]]) + (32*a^6*C*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]) + (64*a^4*C*Sqrt[Sec[c + d*x]])/(1155*b^2*Sqrt[b + a*Cos[c + d*x]]) + (30*b^2*C*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) - (164*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) - (464*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(385*Sqrt[b + a*Cos[c + d*x]]) + (32*a^6*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]) + (24*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(385*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(1155*b^4*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(7/2)*((4*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(1155*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (8*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/(2*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - a^2*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-16*a^4*C + 12*a^3*b*C - 6*a^2*b^2*(11*A + 8*C) + 25*b^4*(11*A + 9*C) + 3*a*b^3*(209*A + 157*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(1155*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
718,1,3537,454,24.987096,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \left(8 a^2 C+7 b^2 (9 A+7 C)\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}+\frac{2 a \left(8 a^2 C+63 A b^2+39 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^4 C+3 a^2 b^2 (21 A+11 C)+21 b^4 (9 A+7 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^3 C+6 a^2 b C+3 a b^2 (21 A+13 C)-21 b^3 (9 A+7 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}-\frac{8 a C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((4*(63*a^2*A*b^2 + 189*A*b^4 + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^3) + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b) + (8*Sec[c + d*x]*(63*a*A*b^2*Sin[c + d*x] - 2*a^3*C*Sin[c + d*x] + 44*a*b^2*C*Sin[c + d*x]))/(315*b^2) + (40*a*C*Sec[c + d*x]^2*Tan[c + d*x])/63 + (4*b*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) - (4*((-2*a^2*A)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*A*b^2)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (22*a^2*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^4*C)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b^2*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^3*A*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (2*a*A*b*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*C*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (62*a^3*C*Sqrt[Sec[c + d*x]])/(315*b*Sqrt[b + a*Cos[c + d*x]]) + (26*a*b*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (2*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) - (6*a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (22*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) - (14*a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((a + b)*((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(7/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^3*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) - (2*((a + b)*((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - a*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*(a + b)*((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a + b)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-1/2*(b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])) + (a + b)*((8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*C + 21*b^3*(9*A + 7*C) + 3*a*b^2*(21*A + 13*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))","B",0
719,1,3214,374,23.0632362,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \left(6 a^2 C-5 b^2 (7 A+5 C)\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b d}+\frac{2 (a-b) \sqrt{a+b} \left(6 a^2 C+105 a A b+57 a b C-35 A b^2-25 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}-\frac{4 a (a-b) \sqrt{a+b} \left(-3 a^2 C+70 A b^2+41 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}-\frac{4 a C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((-8*a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sin[c + d*x])/(105*b^2) + (4*Sec[c + d*x]*(35*A*b^2*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b) + (32*a*C*Sec[c + d*x]*Tan[c + d*x])/35 + (4*b*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) + (8*((-8*a*A*b)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^3*C)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (164*a*b*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*A*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^2*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (62*a^2*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*C*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (10*b^2*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (8*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (164*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-6*a^2*C + 5*b^2*(7*A + 5*C) + 3*a*b*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(7/2)*((4*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*a*(a + b)*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-6*a^2*C + 5*b^2*(7*A + 5*C) + 3*a*b*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*a*(a + b)*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-6*a^2*C + 5*b^2*(7*A + 5*C) + 3*a*b*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (8*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (a*(a + b)*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-6*a^2*C + 5*b^2*(7*A + 5*C) + 3*a*b*(35*A + 19*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (a*(a + b)*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-6*a^2*C + 5*b^2*(7*A + 5*C) + 3*a*b*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/(2*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - a^2*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-6*a^2*C + 5*b^2*(7*A + 5*C) + 3*a*b*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*(2*a*(a + b)*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(a + b)*(-6*a^2*C + 5*b^2*(7*A + 5*C) + 3*a*b*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(-70*A*b^2 + 3*a^2*C - 41*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
720,1,6117,415,26.0343342,"\int (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \left(a^2 C-2 a b (5 A+2 C)+b^2 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b d}-\frac{2 (a-b) \sqrt{a+b} \left(a^2 C+b^2 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^2 d}-\frac{2 a A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{2 a C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}",1,"Result too large to show","B",0
721,1,4010,408,25.7982183,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(6 a^2 C+a b (3 A-8 C)+2 b^2 (3 A+C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{b (3 A-2 C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{a (a-b) \sqrt{a+b} (3 A-8 C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}-\frac{3 A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((16*a*C*Sin[c + d*x])/3 + (4*b*C*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) + (2*((4*a*A*b)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a*b*C)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (2*A*b^2*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] - (2*a^2*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] - (8*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(36*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + (3*A - 8*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])))/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(7/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(36*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + (3*A - 8*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])))/(3*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(36*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + (3*A - 8*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])))/(Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((a*(a + b)*(3*A - 8*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (2*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (2*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + 2*a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 4*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + (2*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^4)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^4*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2] + a*(((3*A - 8*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 + (18*A*b*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (18*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + 36*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - a*(3*A - 8*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*A - 8*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(3*A - 8*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (18*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^4)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]))))/(3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + ((2*a*(a + b)*(3*A - 8*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + 4*(3*a^2*C + b^2*(3*A + C) + a*(-6*A*b + 4*b*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + a*(36*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2 + (3*A - 8*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
722,1,1618,414,20.6381459,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{1}{2} \left(\frac{\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(4 b C \sin (c+d x)+\frac{1}{2} a A \sin (2 (c+d x))\right)}{d (b+a \cos (c+d x))}-\frac{(a+b \sec (c+d x))^{3/2} \left(8 b^2 \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a b \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-5 A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a b \sqrt{\frac{b-a}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)-10 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 b^2 \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-8 a b \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)+5 A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+5 a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) b (5 A-8 C) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i (a-b) (b (A-4 C)+2 a (A+2 C)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{2 \sqrt{\frac{b-a}{a+b}} d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}\right)","-\frac{\sqrt{a+b} \left(4 a^2 (A+2 C)+3 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{\sqrt{a+b} (2 a A+16 a C+5 A b-8 b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} (5 A-8 C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{3 A b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}",1,"((Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(4*b*C*Sin[c + d*x] + (a*A*Sin[2*(c + d*x)])/2))/(d*(b + a*Cos[c + d*x])) - ((a + b*Sec[c + d*x])^(3/2)*(5*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 5*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 8*a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 8*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 10*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 16*a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + 5*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 5*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 8*a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + 8*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*(5*A - 8*C)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(b*(A - 4*C) + 2*a*(A + 2*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(2*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]))/2","C",0
723,1,1381,504,18.9171283,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(C \sec ^2(c+d x)+A\right) \left(\frac{1}{6} a A \sin (c+d x)+\frac{7}{12} A b \sin (2 (c+d x))+\frac{1}{6} a A \sin (3 (c+d x))\right) \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C)}+\frac{(a+b \sec (c+d x))^{3/2} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-48 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+72 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+144 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+3 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+24 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+24 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(8 (2 A+3 C) a^2+3 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a b (26 a A-7 b A+48 a C-24 b C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+72 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+144 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{12 a d (b+a \cos (c+d x))^{3/2} (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{7}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(8 a^2 (2 A+3 C)+3 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 a d}+\frac{\sqrt{a+b} \left(16 a^2 A+24 a^2 C+14 a A b+48 a b C+3 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 (2 A+3 C)+3 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a b d}+\frac{b \sqrt{a+b} \left(A b^2-12 a^2 (A+2 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}+\frac{A b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((a*A*Sin[c + d*x])/6 + (7*A*b*Sin[2*(c + d*x)])/12 + (a*A*Sin[3*(c + d*x)])/6))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] + 3*a*A*b^2*Tan[(c + d*x)/2] + 3*A*b^3*Tan[(c + d*x)/2] + 24*a^3*C*Tan[(c + d*x)/2] + 24*a^2*b*C*Tan[(c + d*x)/2] - 32*a^3*A*Tan[(c + d*x)/2]^3 - 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 48*a^3*C*Tan[(c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 + 3*a*A*b^2*Tan[(c + d*x)/2]^5 - 3*A*b^3*Tan[(c + d*x)/2]^5 + 24*a^3*C*Tan[(c + d*x)/2]^5 - 24*a^2*b*C*Tan[(c + d*x)/2]^5 + 72*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 144*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 72*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 144*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(3*A*b^2 + 8*a^2*(2*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*b*(26*a*A - 7*A*b + 48*a*C - 24*b*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(12*a*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
724,1,649,583,14.6954965,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{\left(16 a^2 A+16 a^2 C+A b^2\right) \sin (2 (c+d x))}{32 a}+\frac{1}{16} a A \sin (4 (c+d x))+\frac{3}{16} A b \sin (c+d x)+\frac{3}{16} A b \sin (3 (c+d x))\right)}{d (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 C)}-\frac{\cos ^5(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \left(-a b \left(a^2 (52 A+80 C)-3 A b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)-a b (a+b) \left(a^2 (52 A+80 C)-3 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+\left(16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)+b (a+b) \left(8 a^3 (3 A+4 C)+4 a^2 b (7 A+12 C)-6 a A b^2+3 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{32 a^3 d \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 C)}","-\frac{b \left(3 A b^2-4 a^2 (13 A+20 C)\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{64 a^2 d}+\frac{\left(4 a^2 (3 A+4 C)+A b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{32 a d}-\frac{(a-b) \sqrt{a+b} \left(3 A b^2-4 a^2 (13 A+20 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}+\frac{\sqrt{a+b} \left(8 a^3 (3 A+4 C)+a^2 (52 A b+80 b C)+2 a A b^2-3 A b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}-\frac{\sqrt{a+b} \left(16 a^4 (3 A+4 C)+24 a^2 b^2 (A+2 C)+3 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^3 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}+\frac{A b \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{8 d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*((3*A*b*Sin[c + d*x])/16 + ((16*a^2*A + A*b^2 + 16*a^2*C)*Sin[2*(c + d*x)])/(32*a) + (3*A*b*Sin[3*(c + d*x)])/16 + (a*A*Sin[4*(c + d*x)])/16))/(d*(b + a*Cos[c + d*x])*(A + 2*C + A*Cos[2*c + 2*d*x])) - (Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(-(a*b*(a + b)*(-3*A*b^2 + a^2*(52*A + 80*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(-6*a*A*b^2 + 3*A*b^3 + 8*a^3*(3*A + 4*C) + 4*a^2*b*(7*A + 12*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (3*A*b^4 + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*b*(-3*A*b^2 + a^2*(52*A + 80*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(32*a^3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2))","A",0
725,1,4418,650,26.3481948,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \left(15 a^2 C+11 b^2 (13 A+11 C)\right) \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{1287 d}+\frac{2 a \left(15 a^2 C+2717 A b^2+2209 b^2 C\right) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{9009 b d}-\frac{2 \left(90 a^4 C-15 a^2 b^2 (715 A+543 C)-539 b^4 (13 A+11 C)\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{45045 b^2 d}+\frac{2 a \left(120 a^4 C+5 a^2 b^2 (143 A+79 C)+b^4 (23309 A+18973 C)\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{45045 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(240 a^6 C+10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)-1617 b^6 (13 A+11 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{45045 b^5 d}+\frac{2 (a-b) \sqrt{a+b} \left(240 a^5 C+180 a^4 b C+10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)-6 a b^4 (2717 A+2174 C)+1617 b^5 (13 A+11 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{45045 b^4 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{13 d}+\frac{10 a C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{143 d}",1,"(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((4*(-1430*a^4*A*b^2 + 39897*a^2*A*b^4 + 21021*A*b^6 - 240*a^6*C - 760*a^4*b^2*C + 30669*a^2*b^4*C + 17787*b^6*C)*Sin[c + d*x])/(45045*b^4) + (4*Sec[c + d*x]^4*(143*A*b^2*Sin[c + d*x] + 159*a^2*C*Sin[c + d*x] + 121*b^2*C*Sin[c + d*x]))/1287 + (4*Sec[c + d*x]^3*(2717*a*A*b^2*Sin[c + d*x] + 15*a^3*C*Sin[c + d*x] + 2209*a*b^2*C*Sin[c + d*x]))/(9009*b) + (4*Sec[c + d*x]^2*(10725*a^2*A*b^2*Sin[c + d*x] + 7007*A*b^4*Sin[c + d*x] - 90*a^4*C*Sin[c + d*x] + 8145*a^2*b^2*C*Sin[c + d*x] + 5929*b^4*C*Sin[c + d*x]))/(45045*b^2) + (4*Sec[c + d*x]*(715*a^3*A*b^2*Sin[c + d*x] + 23309*a*A*b^4*Sin[c + d*x] + 120*a^5*C*Sin[c + d*x] + 395*a^3*b^2*C*Sin[c + d*x] + 18973*a*b^4*C*Sin[c + d*x]))/(45045*b^3) + (108*a*b*C*Sec[c + d*x]^4*Tan[c + d*x])/143 + (4*b^2*C*Sec[c + d*x]^5*Tan[c + d*x])/13))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + (4*((4*a^4*A)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (62*a^2*A*b)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*A*b^3)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^6*C)/(3003*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (304*a^4*C)/(9009*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (20446*a^2*b*C)/(15015*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (154*b^3*C)/(195*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (248*a^3*A*Sqrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x]]) + (4*a^5*A*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) + (76*a*A*b^2*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (27968*a^3*C*Sqrt[Sec[c + d*x]])/(45045*Sqrt[b + a*Cos[c + d*x]]) + (32*a^7*C*Sqrt[Sec[c + d*x]])/(3003*b^4*Sqrt[b + a*Cos[c + d*x]]) + (40*a^5*C*Sqrt[Sec[c + d*x]])/(1287*b^2*Sqrt[b + a*Cos[c + d*x]]) + (8696*a*b^2*C*Sqrt[Sec[c + d*x]])/(15015*Sqrt[b + a*Cos[c + d*x]]) - (62*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (4*a^5*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (14*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (20446*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15015*Sqrt[b + a*Cos[c + d*x]]) + (32*a^7*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3003*b^4*Sqrt[b + a*Cos[c + d*x]]) + (304*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(9009*b^2*Sqrt[b + a*Cos[c + d*x]]) - (154*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(195*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(45045*b^4*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(9/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(45045*b^4*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(15015*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(45045*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - a*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*(a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a + b)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*((b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])) + (a + b)*((240*a^6*C - 1617*b^6*(13*A + 11*C) + 10*a^4*b^2*(143*A + 76*C) - 3*a^2*b^4*(13299*A + 10223*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-240*a^5*C + 180*a^4*b*C + 1617*b^5*(13*A + 11*C) - 10*a^3*b^2*(143*A + 94*C) + 15*a^2*b^3*(1573*A + 1175*C) + 6*a*b^4*(2717*A + 2174*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(45045*b^4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))","B",0
726,1,3989,534,26.9970453,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \left(8 a^2 C+9 b^2 (11 A+9 C)\right) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}+\frac{2 a \left(8 a^2 C+99 A b^2+67 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{693 b^2 d}+\frac{2 \left(8 a^4 C+3 a^2 b^2 (33 A+19 C)+15 b^4 (11 A+9 C)\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{693 b^2 d}-\frac{2 a (a-b) \sqrt{a+b} \left(8 a^4 C+3 a^2 b^2 (33 A+17 C)+3 b^4 (319 A+247 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^4 C+6 a^3 b C+3 a^2 b^2 (33 A+19 C)-6 a b^3 (132 A+101 C)+15 b^4 (11 A+9 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{693 b^3 d}-\frac{8 a C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}",1,"(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((4*a*(99*a^2*A*b^2 + 957*A*b^4 + 8*a^4*C + 51*a^2*b^2*C + 741*b^4*C)*Sin[c + d*x])/(693*b^3) + (4*Sec[c + d*x]^3*(99*A*b^2*Sin[c + d*x] + 113*a^2*C*Sin[c + d*x] + 81*b^2*C*Sin[c + d*x]))/693 + (4*Sec[c + d*x]^2*(297*a*A*b^2*Sin[c + d*x] + 3*a^3*C*Sin[c + d*x] + 229*a*b^2*C*Sin[c + d*x]))/(693*b) + (4*Sec[c + d*x]*(297*a^2*A*b^2*Sin[c + d*x] + 165*A*b^4*Sin[c + d*x] - 4*a^4*C*Sin[c + d*x] + 205*a^2*b^2*C*Sin[c + d*x] + 135*b^4*C*Sin[c + d*x]))/(693*b^2) + (92*a*b*C*Sec[c + d*x]^3*Tan[c + d*x])/99 + (4*b^2*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) - (4*((-2*a^3*A)/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (58*a*A*b^2)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (34*a^3*C)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^5*C)/(693*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (494*a*b^2*C)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^4*A*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (4*a^2*A*b*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (10*A*b^3*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (16*a^6*C*Sqrt[Sec[c + d*x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (14*a^4*C*Sqrt[Sec[c + d*x]])/(99*b*Sqrt[b + a*Cos[c + d*x]]) - (52*a^2*b*C*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]) + (30*b^3*C*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) - (2*a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (58*a^2*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (16*a^6*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (34*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*b*Sqrt[b + a*Cos[c + d*x]]) - (494*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*b^3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(9/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(693*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a^2*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(693*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*(2*a*(a + b)*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^4*C - 6*a^3*b*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C) + 6*a*b^3*(132*A + 101*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(693*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
727,1,710,454,22.2022507,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 \sec (c+d x) \left(5 a^3 C \sin (c+d x)+231 a A b^2 \sin (c+d x)+163 a b^2 C \sin (c+d x)\right)}{315 b}+\frac{4}{315} \sec ^2(c+d x) \left(75 a^2 C \sin (c+d x)+63 A b^2 \sin (c+d x)+49 b^2 C \sin (c+d x)\right)+\frac{4 \left(-10 a^4 C+483 a^2 A b^2+279 a^2 b^2 C+189 A b^4+147 b^4 C\right) \sin (c+d x)}{315 b^2}+\frac{76}{63} a b C \tan (c+d x) \sec ^2(c+d x)+\frac{4}{9} b^2 C \tan (c+d x) \sec ^3(c+d x)\right)}{d (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 C)}+\frac{4 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\left(10 a^4 C-3 a^2 b^2 (161 A+93 C)-21 b^4 (9 A+7 C)\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left(\left(10 a^4 C-3 a^2 b^2 (161 A+93 C)-21 b^4 (9 A+7 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+b \left(-10 a^3 C+15 a^2 b (21 A+11 C)+6 a b^2 (28 A+19 C)+21 b^3 (9 A+7 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{315 b^2 d \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 C)}","-\frac{2 \left(10 a^2 C-7 b^2 (9 A+7 C)\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}+\frac{4 a \left(-5 a^2 C+84 A b^2+57 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}+\frac{2 (a-b) \sqrt{a+b} \left(10 a^4 C-3 a^2 b^2 (161 A+93 C)-21 b^4 (9 A+7 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(10 a^3 C+15 a^2 b (21 A+11 C)-6 a b^2 (28 A+19 C)+21 b^3 (9 A+7 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^2 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}-\frac{4 a C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d}",1,"(4*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((a + b)*((10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 21*b^3*(9*A + 7*C) + 15*a^2*b*(21*A + 11*C) + 6*a*b^2*(28*A + 19*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(9/2)) + (Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((4*(483*a^2*A*b^2 + 189*A*b^4 - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^2) + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] + 75*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/315 + (4*Sec[c + d*x]*(231*a*A*b^2*Sin[c + d*x] + 5*a^3*C*Sin[c + d*x] + 163*a*b^2*C*Sin[c + d*x]))/(315*b) + (76*a*b*C*Sec[c + d*x]^2*Tan[c + d*x])/63 + (4*b^2*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x]))","A",0
728,1,4075,481,25.6786458,"\int (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \left(3 a^2 C+b^2 (7 A+5 C)\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}-\frac{2 a (a-b) \sqrt{a+b} \left(3 a^2 C+49 A b^2+29 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b^2 d}-\frac{2 a^2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 \sqrt{a+b} \left(3 a^3 C-9 a^2 b (7 A+3 C)+a b^2 (49 A+29 C)-b^3 (7 A+5 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{21 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}+\frac{2 a C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d}",1,"(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((4*a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sin[c + d*x])/(21*b) + (4*Sec[c + d*x]*(7*A*b^2*Sin[c + d*x] + 9*a^2*C*Sin[c + d*x] + 5*b^2*C*Sin[c + d*x]))/21 + (12*a*b*C*Sec[c + d*x]*Tan[c + d*x])/7 + (4*b^2*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + (4*((2*a^3*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*a*A*b^2)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^3*C)/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (58*a*b^2*C)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^2*A*b*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^3*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (2*a^4*C*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (4*a^2*b*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (10*b^3*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (14*a^2*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (2*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (58*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(-2*a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(3*a^3*(-7*A + C) + 9*a^2*b*(7*A + 3*C) + b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 84*a^3*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*b*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(9/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(3*a^3*(-7*A + C) + 9*a^2*b*(7*A + 3*C) + b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 84*a^3*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*b*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(3*a^3*(-7*A + C) + 9*a^2*b*(7*A + 3*C) + b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 84*a^3*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(21*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*(a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - (a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(3*a^3*(-7*A + C) + 9*a^2*b*(7*A + 3*C) + b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (42*a^3*A*b*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(3*a^3*(-7*A + C) + 9*a^2*b*(7*A + 3*C) + b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (42*a^3*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + a^2*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(3*a^3*(-7*A + C) + 9*a^2*b*(7*A + 3*C) + b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (42*a^3*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(21*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(-2*a*(a + b)*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(3*a^3*(-7*A + C) + 9*a^2*b*(7*A + 3*C) + b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 84*a^3*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(21*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
729,1,6785,478,26.4248302,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{(a-b) \sqrt{a+b} \left(a^2 (15 A-46 C)-6 b^2 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}+\frac{\sqrt{a+b} \left(30 a^3 C+a^2 b (15 A-46 C)+2 a b^2 (45 A+17 C)-6 b^3 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}-\frac{b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}-\frac{a b (15 A-16 C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}-\frac{5 a A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"Result too large to show","B",0
730,1,4266,463,26.4439411,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(6 a^2 (A+12 C)+a b (27 A-56 C)+8 b^2 (3 A+C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 d}-\frac{\sqrt{a+b} \left(4 a^2 (A+2 C)+15 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}-\frac{b^2 (21 A-8 C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}+\frac{a (a-b) \sqrt{a+b} (27 A-56 C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 d}+\frac{5 A b \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{5/2}}{2 d}",1,"((Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((28*a*b*C*Sin[c + d*x])/3 + (a^2*A*Sin[2*(c + d*x)])/2 + (4*b^2*C*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2) + (((a^3*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a*A*b^2)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^3*C)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*a*b^2*C)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (11*a^2*A*b*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^3*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (4*a^2*b*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (9*a^2*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d*x]]) - (14*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]))*(a + b*Sec[c + d*x])^(5/2)*(-2*a*b*(a + b)*(27*A - 56*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 4*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 12*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*b*(27*A - 56*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(6*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*(-1/6*(Sqrt[Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]*(-2*a*b*(a + b)*(27*A - 56*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 4*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 12*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*b*(27*A - 56*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)^2) + (a*Sin[c + d*x]*(-2*a*b*(a + b)*(27*A - 56*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 4*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 12*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*b*(27*A - 56*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(12*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - (Tan[(c + d*x)/2]*(-2*a*b*(a + b)*(27*A - 56*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 4*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 12*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*b*(27*A - 56*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(12*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) + (-1/2*(a*b*(27*A - 56*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - (a*b*(a + b)*(27*A - 56*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (2*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (6*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (a*b*(a + b)*(27*A - 56*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (2*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (6*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + a^2*b*(27*A - 56*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*b*(27*A - 56*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*b*(27*A - 56*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (2*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (6*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (a*b*(a + b)*(27*A - 56*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2])/(6*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)) - ((-2*a*b*(a + b)*(27*A - 56*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 4*(4*a*b^2*(9*A - 7*C) - 4*b^3*(3*A + C) + 6*a^3*(A + 2*C) - 3*a^2*b*(A + 12*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 12*a*(15*A*b^2 + 4*a^2*(A + 2*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*b*(27*A - 56*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(12*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)))))/2","B",0
731,1,1501,507,20.0466298,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+A\right) \left(\frac{1}{6} A \sin (3 (c+d x)) a^2+\frac{13}{12} A b \sin (2 (c+d x)) a+\frac{1}{6} \left(A a^2+24 b^2 C\right) \sin (c+d x)\right) \cos ^4(c+d x)}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C)}+\frac{(a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-33 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+48 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-48 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-66 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-48 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+96 a b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+120 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+240 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+33 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+33 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+24 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-48 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-48 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+24 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(8 (2 A+3 C) a^2+3 b^2 (11 A-16 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 b \left((38 A+72 C) a^2-b (13 A+72 C) a+24 b^2 (A-C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+120 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+240 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{12 d (b+a \cos (c+d x))^{5/2} (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{9}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(16 a^2 A+24 a^2 C+26 a A b+144 a b C+33 A b^2-48 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 (2 A+3 C)+3 b^2 (11 A-16 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 b d}-\frac{5 b \sqrt{a+b} \left(4 a^2 (A+2 C)+A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}+\frac{5 A b \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d}",1,"(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(((a^2*A + 24*b^2*C)*Sin[c + d*x])/6 + (13*a*A*b*Sin[2*(c + d*x)])/12 + (a^2*A*Sin[3*(c + d*x)])/6))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] + 33*a*A*b^2*Tan[(c + d*x)/2] + 33*A*b^3*Tan[(c + d*x)/2] + 24*a^3*C*Tan[(c + d*x)/2] + 24*a^2*b*C*Tan[(c + d*x)/2] - 48*a*b^2*C*Tan[(c + d*x)/2] - 48*b^3*C*Tan[(c + d*x)/2] - 32*a^3*A*Tan[(c + d*x)/2]^3 - 66*a*A*b^2*Tan[(c + d*x)/2]^3 - 48*a^3*C*Tan[(c + d*x)/2]^3 + 96*a*b^2*C*Tan[(c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 + 33*a*A*b^2*Tan[(c + d*x)/2]^5 - 33*A*b^3*Tan[(c + d*x)/2]^5 + 24*a^3*C*Tan[(c + d*x)/2]^5 - 24*a^2*b*C*Tan[(c + d*x)/2]^5 - 48*a*b^2*C*Tan[(c + d*x)/2]^5 + 48*b^3*C*Tan[(c + d*x)/2]^5 + 120*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 240*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 240*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*b*(24*b^2*(A - C) - a*b*(13*A + 72*C) + a^2*(38*A + 72*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(12*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
732,1,693,587,20.5806371,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(\frac{1}{96} \left(48 a^2 A+48 a^2 C+59 A b^2\right) \sin (2 (c+d x))+\frac{1}{16} a^2 A \sin (4 (c+d x))+\frac{17}{48} a A b \sin (c+d x)+\frac{17}{48} a A b \sin (3 (c+d x))\right)}{d (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 C)}+\frac{\sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(a b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)+a b (a+b) \left(4 a^2 (71 A+108 C)+15 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+3 \left(-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)+b (a+b) \left(-24 a^3 (3 A+4 C)-4 a^2 b (53 A+84 C)-30 a A b^2+15 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{96 a^2 d \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 C)}","\frac{b \left(4 a^2 (71 A+108 C)+15 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\left(4 a^2 (3 A+4 C)+5 A b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{32 d}+\frac{(a-b) \sqrt{a+b} \left(4 a^2 (71 A+108 C)+15 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{\sqrt{a+b} \left(-16 a^4 (3 A+4 C)-120 a^2 b^2 (A+2 C)+5 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}+\frac{\sqrt{a+b} \left(24 a^3 (3 A+4 C)+4 a^2 b (71 A+108 C)+2 a b^2 (59 A+192 C)+15 A b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}+\frac{5 A b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{24 d}",1,"(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*((17*a*A*b*Sin[c + d*x])/48 + ((48*a^2*A + 59*A*b^2 + 48*a^2*C)*Sin[2*(c + d*x)])/96 + (17*a*A*b*Sin[3*(c + d*x)])/48 + (a^2*A*Sin[4*(c + d*x)])/16))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + A*Cos[2*c + 2*d*x])) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2)*(a*b*(a + b)*(15*A*b^2 + 4*a^2*(71*A + 108*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-30*a*A*b^2 + 15*A*b^3 - 24*a^3*(3*A + 4*C) - 4*a^2*b*(53*A + 84*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(5*A*b^4 - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + a*b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(96*a^2*d*(b + a*Cos[c + d*x])^3*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(9/2))","A",0
733,1,956,403,15.4614372,"\int (a+b \sec (c+d x))^{3/2} \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(a^2-b^2 \sec ^2(c+d x)\right) \left(-\frac{4}{5} \sec (c+d x) \tan (c+d x) b^3-\frac{8}{5} a \tan (c+d x) b^2-\frac{4}{5} \left(3 b^2-4 a^2\right) \sin (c+d x) b\right)}{d (b+a \cos (c+d x)) \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right)}-\frac{4 (a+b \sec (c+d x))^{3/2} \left(a^2-b^2 \sec ^2(c+d x)\right) \left(-3 b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a^2 b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a^3 b \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a b^3 \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 a^3 b \tan ^3\left(\frac{1}{2} (c+d x)\right)+10 a^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 b^4 \tan \left(\frac{1}{2} (c+d x)\right)+3 a b^3 \tan \left(\frac{1}{2} (c+d x)\right)-4 a^2 b^2 \tan \left(\frac{1}{2} (c+d x)\right)-4 a^3 b \tan \left(\frac{1}{2} (c+d x)\right)+b \left(-4 a^3-4 b a^2+3 b^2 a+3 b^3\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-\left(5 a^4-10 b a^3-4 b^2 a^2+4 b^3 a+3 b^4\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+10 a^4 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{5 d (b+a \cos (c+d x))^{3/2} \left(\cos (2 c+2 d x) a^2+a^2-2 b^2\right) \sec ^{\frac{7}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 a^3 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} \left(4 a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 d}+\frac{2 \sqrt{a+b} \left(10 a^3-4 a^2 b-4 a b^2+3 b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 d}-\frac{2 a b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}-\frac{2 b^2 \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"(-4*(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2)*(-4*a^3*b*Tan[(c + d*x)/2] - 4*a^2*b^2*Tan[(c + d*x)/2] + 3*a*b^3*Tan[(c + d*x)/2] + 3*b^4*Tan[(c + d*x)/2] + 8*a^3*b*Tan[(c + d*x)/2]^3 - 6*a*b^3*Tan[(c + d*x)/2]^3 - 4*a^3*b*Tan[(c + d*x)/2]^5 + 4*a^2*b^2*Tan[(c + d*x)/2]^5 + 3*a*b^3*Tan[(c + d*x)/2]^5 - 3*b^4*Tan[(c + d*x)/2]^5 + 10*a^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 10*a^4*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(-4*a^3 - 4*a^2*b + 3*a*b^2 + 3*b^3)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (5*a^4 - 10*a^3*b - 4*a^2*b^2 + 4*a*b^3 + 3*b^4)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(5*d*(b + a*Cos[c + d*x])^(3/2)*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2)*((-4*b*(-4*a^2 + 3*b^2)*Sin[c + d*x])/5 - (8*a*b^2*Tan[c + d*x])/5 - (4*b^3*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x]))","B",0
734,1,598,353,13.741833,"\int \sqrt{a+b \sec (c+d x)} \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(a^2 - b^2*Sec[c + d*x]^2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(a^2-b^2 \sec ^2(c+d x)\right) \left(-\frac{4}{3} a b \sin (c+d x)-\frac{4}{3} b^2 \tan (c+d x)\right)}{d \left(a^2 \cos (2 c+2 d x)+a^2-2 b^2\right)}-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(a^2-b^2 \sec ^2(c+d x)\right) \left(12 i a^3 \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-2 i \left(3 a^3-3 a^2 b-a b^2+b^3\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-a b \sqrt{\frac{b-a}{a+b}} \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 i a b (a-b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{3 d \sqrt{\frac{b-a}{a+b}} (a \cos (c+d x)+b) \left(a^2 \cos (2 c+2 d x)+a^2-2 b^2\right)}","\frac{2 \sqrt{a+b} \left(3 a^2+a b-b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}-\frac{2 a^2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 b^2 \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 a (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}",1,"(-4*Cos[(c + d*x)/2]^2*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(a^2 - b^2*Sec[c + d*x]^2)*((2*I)*a*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - (2*I)*(3*a^3 - 3*a^2*b - a*b^2 + b^3)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (12*I)*a^3*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - a*b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(a^2 - b^2*Sec[c + d*x]^2)*((-4*a*b*Sin[c + d*x])/3 - (4*b^2*Tan[c + d*x])/3))/(d*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x]))","C",1
735,1,3255,393,23.1296343,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{4 a (a-b) \sqrt{a+b} \left(24 a^2 C+35 A b^2+22 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^5 d}+\frac{2 \left(24 a^2 C+5 b^2 (7 A+5 C)\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b^3 d}+\frac{2 \sqrt{a+b} \left(48 a^3 C-12 a^2 b C+2 a b^2 (35 A+22 C)+5 b^3 (7 A+5 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}-\frac{12 a C \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b^2 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 b d}",1,"(Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*((-8*a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sin[c + d*x])/(105*b^4) + (4*Sec[c + d*x]*(35*A*b^2*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b^3) - (24*a*C*Sec[c + d*x]*Tan[c + d*x])/(35*b^2) + (4*C*Sec[c + d*x]^2*Tan[c + d*x])/(7*b)))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]]) + (8*((4*a*A)/(3*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^3*C)/(35*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (88*a*C)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (4*a^2*A*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) + (10*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (32*a^4*C*Sqrt[Sec[c + d*x]])/(35*b^4*Sqrt[b + a*Cos[c + d*x]]) + (64*a^2*C*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]) + (4*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) + (32*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^4*Sqrt[b + a*Cos[c + d*x]]) + (88*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) - 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((4*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) - 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) - 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (8*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + (a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(-48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) - 2*a*b^2*(35*A + 22*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/(2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]) + (a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(-48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) - 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/(2*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]) - a^2*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(-48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) - 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*(2*a*(a + b)*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-48*a^3*C - 12*a^2*b*C + 5*b^3*(7*A + 5*C) - 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
736,1,2993,320,22.3061403,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \left(8 a^2 C+3 b^2 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}-\frac{2 \sqrt{a+b} \left(8 a^2 C-2 a b C+3 b^2 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{8 a C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}",1,"(Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*((4*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*b^3) - (16*a*C*Tan[c + d*x])/(15*b^2) + (4*C*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]]) - (4*((-2*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*C)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^2*C)/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*A*Sqrt[Sec[c + d*x]])/(b*Sqrt[b + a*Cos[c + d*x]]) - (16*a^3*C*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (14*a*C*Sqrt[Sec[c + d*x]])/(15*b*Sqrt[b + a*Cos[c + d*x]]) - (2*a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*Sqrt[b + a*Cos[c + d*x]]) - (16*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (6*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(2*(a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(15*A*b^2 + (8*a^2 + 2*a*b + 9*b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (15*A*b^2 + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(15*A*b^2 + (8*a^2 + 2*a*b + 9*b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (15*A*b^2 + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(15*A*b^2 + (8*a^2 + 2*a*b + 9*b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (15*A*b^2 + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((15*A*b^2 + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(15*A*b^2 + (8*a^2 + 2*a*b + 9*b^2)*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(15*A*b^2 + (8*a^2 + 2*a*b + 9*b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (15*A*b^2 + 8*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (15*A*b^2 + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(15*A*b^2 + (8*a^2 + 2*a*b + 9*b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*(2*(a + b)*(15*A*b^2 + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(15*A*b^2 + (8*a^2 + 2*a*b + 9*b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (15*A*b^2 + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
737,1,409,253,14.6797868,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\cos (c+d x) (a \cos (c+d x)+b) \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 C \tan (c+d x)}{3 b}-\frac{8 a C \sin (c+d x)}{3 b^2}\right)}{d \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 C)}+\frac{8 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(b (C (b-2 a)+3 A b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+a C \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 a C (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 \sqrt{a+b} (C (2 a+b)+3 A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{4 a C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}",1,"(8*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(2*a*(a + b)*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(3*A*b + (-2*a + b)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + a*C*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*((-8*a*C*Sin[c + d*x])/(3*b^2) + (4*C*Tan[c + d*x])/(3*b)))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]])","A",0
738,1,914,313,17.0123526,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{4 C \cos (c+d x) (b+a \cos (c+d x)) \left(C \sec ^2(c+d x)+A\right) \sin (c+d x)}{b d (\cos (2 c+2 d x) A+A+2 C) \sqrt{a+b \sec (c+d x)}}+\frac{4 \sqrt{b+a \cos (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(a \sqrt{\frac{b-a}{a+b}} C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^3\left(\frac{1}{2} (c+d x)\right)-b \sqrt{\frac{b-a}{a+b}} C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i A b \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-a \sqrt{\frac{b-a}{a+b}} C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan \left(\frac{1}{2} (c+d x)\right)-b \sqrt{\frac{b-a}{a+b}} C \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan \left(\frac{1}{2} (c+d x)\right)-2 i A b \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i (a-b) C E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i b (A+C) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{b \sqrt{\frac{b-a}{a+b}} d (\cos (2 c+2 d x) A+A+2 C) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}-\frac{2 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}-\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(4*C*Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + C*Sec[c + d*x]^2)*Sin[c + d*x])/(b*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]]) + (4*Sqrt[b + a*Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(-(a*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]*Sqrt[1 - Tan[(c + d*x)/2]^2]) - b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]*Sqrt[1 - Tan[(c + d*x)/2]^2] + a*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3*Sqrt[1 - Tan[(c + d*x)/2]^2] - b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3*Sqrt[1 - Tan[(c + d*x)/2]^2] - (2*I)*A*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*A*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*C*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*b*(A + C)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(b*Sqrt[(-a + b)/(a + b)]*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
739,1,384,352,15.9237701,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (A \cos (c+d x)+C \sec (c+d x)) \left(A \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 A (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-4 A b \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+4 a C \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{a d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 C)}","\frac{A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{\sqrt{a+b} (2 a C+A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}+\frac{A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{a d}+\frac{A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A*Cos[c + d*x] + C*Sec[c + d*x])*(2*A*(a + b)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 4*a*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 4*A*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + A*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(a*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",1
740,1,1475,411,16.3202119,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{A (b+a \cos (c+d x)) \sec (c+d x) \sin (2 (c+d x))}{4 a d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-3 A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+3 a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-3 i A (a-b) b E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i \left(2 (A+2 C) a^2-A b a+3 A b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a^2 \sqrt{\frac{b-a}{a+b}} d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{3 A b \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}+\frac{A (2 a-3 b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{3 A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{\sqrt{a+b} \left(4 a^2 (A+2 C)+3 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"(A*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[2*(c + d*x)])/(4*a*d*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(3*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 3*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 6*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 3*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 3*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (3*I)*A*(a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(-(a*A*b) + 3*A*b^2 + 2*a^2*(A + 2*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a^2*Sqrt[(-a + b)/(a + b)]*d*Sqrt[a + b*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
741,1,1352,506,23.0859847,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(\frac{A \sin (c+d x)}{12 a}-\frac{5 A b \sin (2 (c+d x))}{24 a^2}+\frac{A \sin (3 (c+d x))}{12 a}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(15 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-24 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+24 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+30 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)+32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)+48 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-15 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-15 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)-16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)-24 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-24 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)-(a+b) \left(8 (2 A+3 C) a^2+15 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a A b (2 a+5 b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 a^3 d \sqrt{a+b \sec (c+d x)} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{5 A b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a^2 d}+\frac{b \sqrt{a+b} \left(4 a^2 (A+2 C)+5 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a^4 d}+\frac{\left(8 a^2 (2 A+3 C)+15 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 a^3 d}-\frac{\sqrt{a+b} \left(-8 a^2 (2 A+3 C)+10 a A b-15 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a^3 d}+\frac{(a-b) \sqrt{a+b} \left(8 a^2 (2 A+3 C)+15 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a^3 b d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((A*Sin[c + d*x])/(12*a) - (5*A*b*Sin[2*(c + d*x)])/(24*a^2) + (A*Sin[3*(c + d*x)])/(12*a)))/(d*Sqrt[a + b*Sec[c + d*x]]) + (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-16*a^3*A*Tan[(c + d*x)/2] - 16*a^2*A*b*Tan[(c + d*x)/2] - 15*a*A*b^2*Tan[(c + d*x)/2] - 15*A*b^3*Tan[(c + d*x)/2] - 24*a^3*C*Tan[(c + d*x)/2] - 24*a^2*b*C*Tan[(c + d*x)/2] + 32*a^3*A*Tan[(c + d*x)/2]^3 + 30*a*A*b^2*Tan[(c + d*x)/2]^3 + 48*a^3*C*Tan[(c + d*x)/2]^3 - 16*a^3*A*Tan[(c + d*x)/2]^5 + 16*a^2*A*b*Tan[(c + d*x)/2]^5 - 15*a*A*b^2*Tan[(c + d*x)/2]^5 + 15*A*b^3*Tan[(c + d*x)/2]^5 - 24*a^3*C*Tan[(c + d*x)/2]^5 + 24*a^2*b*C*Tan[(c + d*x)/2]^5 + 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (a + b)*(15*A*b^2 + 8*a^2*(2*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*A*b*(2*a + 5*b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a^3*d*Sqrt[a + b*Sec[c + d*x]]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
742,1,3853,460,29.1746389,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(6 a^2 C+5 A b^2-b^2 C\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}-\frac{2 a \left(8 a^2 C+5 A b^2-3 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^3 d \left(a^2-b^2\right)}-\frac{2 \left(16 a^4 C+2 a^2 b^2 (5 A-4 C)-b^4 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^5 d \sqrt{a+b}}-\frac{2 \left(16 a^3 C+12 a^2 b C+2 a b^2 (5 A+2 C)+b^3 (5 A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{5 b^4 d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((4*(-10*a^2*A*b^2 + 5*A*b^4 - 16*a^4*C + 8*a^2*b^2*C + 3*b^4*C)*Sin[c + d*x])/(5*b^4*(-a^2 + b^2)) + (4*(a^2*A*b^2*Sin[c + d*x] + a^4*C*Sin[c + d*x]))/(b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])) - (12*a*C*Tan[c + d*x])/(5*b^3) + (4*C*Sec[c + d*x]*Tan[c + d*x])/(5*b^2)))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) + (4*(b + a*Cos[c + d*x])*((4*a^2*A)/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*A*b)/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^4*C)/(5*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^2*C)/(5*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*b*C)/(5*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*A*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*A*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*a*C*Sqrt[Sec[c + d*x]])/(5*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (32*a^5*C*Sqrt[Sec[c + d*x]])/(5*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (24*a^3*C*Sqrt[Sec[c + d*x]])/(5*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (6*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (32*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (16*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(2*(a + b)*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*b^4*(-a^2 + b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*b^4*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(5*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(5*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(2*(a + b)*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*C + 12*a^2*b*C - 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(5*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
743,1,3312,327,25.0708356,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 a \left(a^2 C+A b^2\right) \tan (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(8 a^2 C+3 A b^2-5 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b}}+\frac{2 \left(C \left(8 a^2+6 a b+b^2\right)+3 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b}}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d}",1,"((b + a*Cos[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((4*a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)) - (4*(a*A*b^2*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (4*C*Tan[c + d*x])/(3*b^2)))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) + (4*(b + a*Cos[c + d*x])*((-2*a*A)/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (10*a*C)/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^3*C)/(3*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*A*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (16*a^4*C*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (14*a^2*C*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*b*C*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (16*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (10*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*(-2*a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*A*b^2 + (8*a^2 - 6*a*b + b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*A*b^2 + (8*a^2 - 6*a*b + b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*A*b^2 + (8*a^2 - 6*a*b + b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*(a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - (a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(3*A*b^2 + (8*a^2 - 6*a*b + b^2)*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(3*A*b^2 + (8*a^2 - 6*a*b + b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + a^2*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(3*A*b^2 + (8*a^2 - 6*a*b + b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(-2*a*(a + b)*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*A*b^2 + (8*a^2 - 6*a*b + b^2)*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
744,1,570,279,18.6043801,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a \cos (c+d x)+b)^2 \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 \left(a^2 C \sin (c+d x)+A b^2 \sin (c+d x)\right)}{b \left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{4 \left(2 a^2 C+A b^2-b^2 C\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}\right)}{d (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{4 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} (a \cos (c+d x)+b)^{3/2} \left(A+C \sec ^2(c+d x)\right) \left((a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \left(\left(2 a^2 C+A b^2-b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+b (C (b-2 a)-A b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)-\left(2 a^2 C+A b^2-b^2 C\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b\right)\right)}{b^2 d \left(b^2-a^2\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 C)}","-\frac{2 \left(a^2 C+A b^2\right) \tan (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^3 d \sqrt{a+b}}+\frac{2 (A b-C (2 a+b)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((-4*(A*b^2 + 2*a^2*C - b^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) + (4*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) + (4*(b + a*Cos[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-((A*b^2 + 2*a^2*C - b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)) + (a + b)*((A*b^2 + 2*a^2*C - b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-(A*b) + (-2*a + b)*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(b^2*(-a^2 + b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
745,1,1119,381,18.1049919,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \left(C a^2+A b^2\right) \sin (c+d x)}{a b \left(b^2-a^2\right)}+\frac{4 \left(C \sin (c+d x) a^2+A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) (b+a \cos (c+d x))}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{3/2}}-\frac{4 (b+a \cos (c+d x))^{3/2} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a^3 C \tan \left(\frac{1}{2} (c+d x)\right)+a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(C a^2+A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-a b (a+b) (A+C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a \left(b^3-a^2 b\right) d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \left(a^2 C+A b^2\right) \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2 C+A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 (A b-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((4*(A*b^2 + a^2*C)*Sin[c + d*x])/(a*b*(-a^2 + b^2)) + (4*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) - (4*(b + a*Cos[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*A*b^2*Tan[(c + d*x)/2] + A*b^3*Tan[(c + d*x)/2] + a^3*C*Tan[(c + d*x)/2] + a^2*b*C*Tan[(c + d*x)/2] - 2*a*A*b^2*Tan[(c + d*x)/2]^3 - 2*a^3*C*Tan[(c + d*x)/2]^3 + a*A*b^2*Tan[(c + d*x)/2]^5 - A*b^3*Tan[(c + d*x)/2]^5 + a^3*C*Tan[(c + d*x)/2]^5 - a^2*b*C*Tan[(c + d*x)/2]^5 + 2*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(A*b^2 + a^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - a*b*(a + b)*(A + C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*(-(a^2*b) + b^3)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
746,1,1251,431,19.9823978,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \left(C a^2+A b^2\right) \sin (c+d x)}{a^2 \left(a^2-b^2\right)}-\frac{4 \left(A \sin (c+d x) b^3+a^2 C \sin (c+d x) b\right)}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{3/2}}-\frac{2 (b+a \cos (c+d x))^{3/2} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)-2 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-2 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(a^2 (A-2 C)-3 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 a (a+b) (A b+a C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a^2 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{3 A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}-\frac{b \left(3 A b^2-a^2 (A-2 C)\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(2 a^2 C+a A b+3 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}-\frac{\left(3 A b^2-a^2 (A-2 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((4*(A*b^2 + a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) - (4*(A*b^3*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a^3*A*Tan[(c + d*x)/2] + a^2*A*b*Tan[(c + d*x)/2] - 3*a*A*b^2*Tan[(c + d*x)/2] - 3*A*b^3*Tan[(c + d*x)/2] - 2*a^3*C*Tan[(c + d*x)/2] - 2*a^2*b*C*Tan[(c + d*x)/2] - 2*a^3*A*Tan[(c + d*x)/2]^3 + 6*a*A*b^2*Tan[(c + d*x)/2]^3 + 4*a^3*C*Tan[(c + d*x)/2]^3 + a^3*A*Tan[(c + d*x)/2]^5 - a^2*A*b*Tan[(c + d*x)/2]^5 - 3*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*A*b^3*Tan[(c + d*x)/2]^5 - 2*a^3*C*Tan[(c + d*x)/2]^5 + 2*a^2*b*C*Tan[(c + d*x)/2]^5 - 6*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-3*A*b^2 + a^2*(A - 2*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*(a + b)*(A*b + a*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a^2*(a^2 - b^2)*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
747,1,2500,501,22.9032952,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{5 A b \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b} \left(4 a^2 (A+2 C)+15 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^4 d}+\frac{b^2 \left(15 A b^2-a^2 (7 A-8 C)\right) \tan (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\left(-2 a^2 (A-4 C)+5 a A b+15 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}+\frac{\left(15 A b^2-a^2 (7 A-8 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"(((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((4*b*(A*b^2 + a^2*C)*Sin[c + d*x])/(a^3*(-a^2 + b^2)) + (4*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(2*a^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 8*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + 8*a^2*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + 14*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 30*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 16*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 - 7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 15*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 8*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - 8*a^2*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - (8*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^4*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*b^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^4*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*b^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*b*(-15*A*b^2 + a^2*(7*A - 8*C))*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(10*a*A*b^2 + 15*A*b^3 + 2*a^3*(A + 2*C) + a^2*b*(A + 8*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(2*a^3*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))))/2","C",0
748,1,830,488,22.6342518,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{8 a \left(8 C a^4+A b^2 a^2-14 b^2 C a^2-3 A b^4+4 b^4 C\right) \sin (c+d x)}{3 b^4 \left(a^2-b^2\right)^2}-\frac{4 \left(C \sin (c+d x) a^3+A b^2 \sin (c+d x) a\right)}{3 b^2 \left(b^2-a^2\right) (b+a \cos (c+d x))^2}-\frac{4 \left(-7 C \sin (c+d x) a^5-A b^2 \sin (c+d x) a^3+11 b^2 C \sin (c+d x) a^3+5 A b^4 \sin (c+d x) a\right)}{3 b^3 \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}+\frac{4 C \tan (c+d x)}{3 b^3}\right) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{5/2}}+\frac{4 \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(2 a (a+b) \left(8 C a^4+b^2 (A-14 C) a^2+b^4 (4 C-3 A)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+b (a+b) \left(-16 C a^4+12 b C a^3-2 b^2 (A-8 C) a^2+3 b^3 (A-3 C) a+b^4 (3 A+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+2 a \left(8 C a^4+b^2 (A-14 C) a^2+b^4 (4 C-3 A)\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)\right) (b+a \cos (c+d x))^{5/2}}{3 b^4 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{5/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 \left(a^2 C+A b^2\right) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+A b^2-b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}+\frac{4 a \left(8 a^4 C+a^2 b^2 (A-14 C)-b^4 (3 A-4 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^5 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{4 a \left(-3 a^4 C+5 a^2 b^2 C+2 A b^4\right) \tan (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(16 a^4 C+12 a^3 b C+2 a^2 b^2 (A-8 C)+3 a b^3 (A-3 C)-b^4 (3 A+C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"(4*(b + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(2*a*(a + b)*(a^2*b^2*(A - 14*C) + 8*a^4*C + b^4*(-3*A + 4*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-2*a^2*b^2*(A - 8*C) + 3*a*b^3*(A - 3*C) - 16*a^4*C + 12*a^3*b*C + b^4*(3*A + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*(a^2*b^2*(A - 14*C) + 8*a^4*C + b^4*(-3*A + 4*C))*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(3*b^4*(a^2 - b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((-8*a*(a^2*A*b^2 - 3*A*b^4 + 8*a^4*C - 14*a^2*b^2*C + 4*b^4*C)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)^2) - (4*(a*A*b^2*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (4*(-(a^3*A*b^2*Sin[c + d*x]) + 5*a*A*b^4*Sin[c + d*x] - 7*a^5*C*Sin[c + d*x] + 11*a^3*b^2*C*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (4*C*Tan[c + d*x])/(3*b^3)))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2))","A",0
749,1,702,408,29.5613597,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec (c+d x) (a \cos (c+d x)+b)^3 \left(A+C \sec ^2(c+d x)\right) \left(\frac{4 \left(a^2 C \sin (c+d x)+A b^2 \sin (c+d x)\right)}{3 b \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{8 \left(-2 a^4 C \sin (c+d x)+a^2 A b^2 \sin (c+d x)+4 a^2 b^2 C \sin (c+d x)+A b^4 \sin (c+d x)\right)}{3 b^2 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}-\frac{4 \left(-8 a^4 C+a^2 A b^2+15 a^2 b^2 C+3 A b^4-3 b^4 C\right) \sin (c+d x)}{3 b^3 \left(b^2-a^2\right)^2}\right)}{d (a+b \sec (c+d x))^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{4 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\sec (c+d x)} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b)^2 \left(A+C \sec ^2(c+d x)\right) \left(\left(-8 a^4 C+a^2 b^2 (A+15 C)+3 b^4 (A-C)\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)-(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left(\left(8 a^4 C-a^2 b^2 (A+15 C)+3 b^4 (C-A)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+b \left(-8 a^3 C+6 a^2 b C+a b^2 (A+9 C)+3 b^3 (A-C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} (A \cos (2 c+2 d x)+A+2 C)}","\frac{2 a \left(a^2 C+A b^2\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(-5 a^4 C+a^2 b^2 (A+9 C)+3 A b^4\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-8 a^4 C+a^2 b^2 (A+15 C)+3 b^4 (A-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \left(8 a^3 C+6 a^2 b C-a b^2 (A+9 C)+3 b^3 (A-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((-4*(a^2*A*b^2 + 3*A*b^4 - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)^2) + (4*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (8*(a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] - 2*a^4*C*Sin[c + d*x] + 4*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) + (4*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*(b + a*Cos[c + d*x])^2*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2)*(-((a + b)*((8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(3*b^3*(A - C) - 8*a^3*C + 6*a^2*b*C + a*b^2*(A + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]) + (3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*(A + 2*C + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*(a + b*Sec[c + d*x])^(5/2))","A",0
750,1,759,378,26.6970661,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{4 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)^{5/2} \left(A+C \sec ^2(c+d x)\right) \left(2 a \left(a^2 (-C)+2 A b^2+3 b^2 C\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b\right)+b (a+b) \left(-2 a^2 C+3 a b (A+C)+b^2 (A+3 C)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a (a+b) \left(C \left(a^2-3 b^2\right)-2 A b^2\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 d \left(b^3-a^2 b\right)^2 \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} (a+b \sec (c+d x))^{5/2} (A \cos (2 c+2 d x)+A+2 C)}+\frac{\sec (c+d x) (a \cos (c+d x)+b)^3 \left(A+C \sec ^2(c+d x)\right) \left(-\frac{8 a \left(a^2 C-2 A b^2-3 b^2 C\right) \sin (c+d x)}{3 b^2 \left(a^2-b^2\right)^2}+\frac{4 \left(a^2 C \sin (c+d x)+A b^2 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{4 \left(a^4 C \sin (c+d x)-5 a^2 A b^2 \sin (c+d x)-5 a^2 b^2 C \sin (c+d x)+A b^4 \sin (c+d x)\right)}{3 a b \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{5/2} (A \cos (2 c+2 d x)+A+2 C)}","-\frac{4 a \left(a^2 (-C)+2 A b^2+3 b^2 C\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(a^2 C+A b^2\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+3 a b (A+C)-b^2 (A+3 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{4 a \left(a^2 (-C)+2 A b^2+3 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((-8*a*(-2*A*b^2 + a^2*C - 3*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2) + (4*(A*b^2*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(-5*a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] + a^4*C*Sin[c + d*x] - 5*a^2*b^2*C*Sin[c + d*x]))/(3*a*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) + (4*(b + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(2*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2) + 2*a*(a + b)*(-2*A*b^2 + (a^2 - 3*b^2)*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-2*a^2*C + 3*a*b*(A + C) + b^2*(A + 3*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*(-(a^2*b) + b^3)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
751,1,1715,517,23.0383109,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(b+a \cos (c+d x))^3 \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \left(-C a^4-7 A b^2 a^2-3 b^2 C a^2+3 A b^4\right) \sin (c+d x)}{3 a^2 b \left(b^2-a^2\right)^2}-\frac{4 \left(A \sin (c+d x) b^3+a^2 C \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{8 \left(C \sin (c+d x) a^4+4 A b^2 \sin (c+d x) a^2+b^2 C \sin (c+d x) a^2-2 A b^4 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{5/2}}-\frac{4 (b+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^5-2 C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^5+C \tan \left(\frac{1}{2} (c+d x)\right) a^5-b C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4+b C \tan \left(\frac{1}{2} (c+d x)\right) a^4+6 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4+6 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4+7 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3+3 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3-14 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right) a^3-6 b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^3+7 A b^2 \tan \left(\frac{1}{2} (c+d x)\right) a^3+3 b^2 C \tan \left(\frac{1}{2} (c+d x)\right) a^3-7 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2-3 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2+7 A b^3 \tan \left(\frac{1}{2} (c+d x)\right) a^2+3 b^3 C \tan \left(\frac{1}{2} (c+d x)\right) a^2-12 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2-12 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2-3 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right) a+6 A b^4 \tan ^3\left(\frac{1}{2} (c+d x)\right) a-3 A b^4 \tan \left(\frac{1}{2} (c+d x)\right) a-b (a+b) \left((3 A+C) a^2+3 b (A+C) a-2 A b^2\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a+3 A b^5 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 A b^5 \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(C a^4+b^2 (7 A+3 C) a^2-3 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{3 a^2 b \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{5/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{2 \left(a^2 C+A b^2\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(a^4 (-C)-a^2 b^2 (7 A+3 C)+3 A b^4\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(a^4 (-C)-a^2 b^2 (7 A+3 C)+3 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \left(a^3 (-C)+6 a^2 A b+3 a^2 b C-a A b^2-3 A b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((4*(-7*a^2*A*b^2 + 3*A*b^4 - a^4*C - 3*a^2*b^2*C)*Sin[c + d*x])/(3*a^2*b*(-a^2 + b^2)^2) - (4*(A*b^3*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (8*(4*a^2*A*b^2*Sin[c + d*x] - 2*A*b^4*Sin[c + d*x] + a^4*C*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (4*(b + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(7*a^3*A*b^2*Tan[(c + d*x)/2] + 7*a^2*A*b^3*Tan[(c + d*x)/2] - 3*a*A*b^4*Tan[(c + d*x)/2] - 3*A*b^5*Tan[(c + d*x)/2] + a^5*C*Tan[(c + d*x)/2] + a^4*b*C*Tan[(c + d*x)/2] + 3*a^3*b^2*C*Tan[(c + d*x)/2] + 3*a^2*b^3*C*Tan[(c + d*x)/2] - 14*a^3*A*b^2*Tan[(c + d*x)/2]^3 + 6*a*A*b^4*Tan[(c + d*x)/2]^3 - 2*a^5*C*Tan[(c + d*x)/2]^3 - 6*a^3*b^2*C*Tan[(c + d*x)/2]^3 + 7*a^3*A*b^2*Tan[(c + d*x)/2]^5 - 7*a^2*A*b^3*Tan[(c + d*x)/2]^5 - 3*a*A*b^4*Tan[(c + d*x)/2]^5 + 3*A*b^5*Tan[(c + d*x)/2]^5 + a^5*C*Tan[(c + d*x)/2]^5 - a^4*b*C*Tan[(c + d*x)/2]^5 + 3*a^3*b^2*C*Tan[(c + d*x)/2]^5 - 3*a^2*b^3*C*Tan[(c + d*x)/2]^5 + 6*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-3*A*b^4 + a^4*C + a^2*b^2*(7*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - a*b*(a + b)*(-2*A*b^2 + 3*a*b*(A + C) + a^2*(3*A + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a^2*b*(a^2 - b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
752,1,1702,559,22.8635843,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(b+a \cos (c+d x))^3 \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{8 \left(-2 C a^4-5 A b^2 a^2+3 A b^4\right) \sin (c+d x)}{3 a^3 \left(b^2-a^2\right)^2}+\frac{4 \left(A \sin (c+d x) b^4+a^2 C \sin (c+d x) b^2\right)}{3 a^3 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{4 \left(7 A \sin (c+d x) b^5-11 a^2 A \sin (c+d x) b^3+a^2 C \sin (c+d x) b^3-5 a^4 C \sin (c+d x) b\right)}{3 a^3 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right)}{d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{5/2}}-\frac{2 (b+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 A \tan ^5\left(\frac{1}{2} (c+d x)\right) a^5-8 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^5-6 A \tan ^3\left(\frac{1}{2} (c+d x)\right) a^5+16 C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^5+3 A \tan \left(\frac{1}{2} (c+d x)\right) a^5-8 C \tan \left(\frac{1}{2} (c+d x)\right) a^5-3 A b \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4+8 b C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4+3 A b \tan \left(\frac{1}{2} (c+d x)\right) a^4-8 b C \tan \left(\frac{1}{2} (c+d x)\right) a^4-30 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4-30 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4-26 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3+52 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right) a^3-26 A b^2 \tan \left(\frac{1}{2} (c+d x)\right) a^3+26 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2-26 A b^3 \tan \left(\frac{1}{2} (c+d x)\right) a^2+60 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2+60 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2+15 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right) a-30 A b^4 \tan ^3\left(\frac{1}{2} (c+d x)\right) a+15 A b^4 \tan \left(\frac{1}{2} (c+d x)\right) a+2 (a+b) \left(3 C a^3+b (6 A+C) a^2+3 A b^2 a-5 A b^3\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a-15 A b^5 \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 A b^5 \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left((3 A-8 C) a^4-26 A b^2 a^2+15 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 A b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 A b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{3 a \left(a^3-a b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C) (a+b \sec (c+d x))^{5/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{5 A b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}-\frac{b \left(5 A b^2-a^2 (3 A-2 C)\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{b \left(-\left(a^4 (3 A-8 C)\right)+26 a^2 A b^2-15 A b^4\right) \tan (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{\left(-\left(a^4 (3 A-8 C)\right)+26 a^2 A b^2-15 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{\left(6 a^4 C+a^3 b (3 A-2 C)+21 a^2 A b^2-5 a A b^3-15 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((-8*(-5*a^2*A*b^2 + 3*A*b^4 - 2*a^4*C)*Sin[c + d*x])/(3*a^3*(-a^2 + b^2)^2) + (4*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(-11*a^2*A*b^3*Sin[c + d*x] + 7*A*b^5*Sin[c + d*x] - 5*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (2*(b + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(3*a^5*A*Tan[(c + d*x)/2] + 3*a^4*A*b*Tan[(c + d*x)/2] - 26*a^3*A*b^2*Tan[(c + d*x)/2] - 26*a^2*A*b^3*Tan[(c + d*x)/2] + 15*a*A*b^4*Tan[(c + d*x)/2] + 15*A*b^5*Tan[(c + d*x)/2] - 8*a^5*C*Tan[(c + d*x)/2] - 8*a^4*b*C*Tan[(c + d*x)/2] - 6*a^5*A*Tan[(c + d*x)/2]^3 + 52*a^3*A*b^2*Tan[(c + d*x)/2]^3 - 30*a*A*b^4*Tan[(c + d*x)/2]^3 + 16*a^5*C*Tan[(c + d*x)/2]^3 + 3*a^5*A*Tan[(c + d*x)/2]^5 - 3*a^4*A*b*Tan[(c + d*x)/2]^5 - 26*a^3*A*b^2*Tan[(c + d*x)/2]^5 + 26*a^2*A*b^3*Tan[(c + d*x)/2]^5 + 15*a*A*b^4*Tan[(c + d*x)/2]^5 - 15*A*b^5*Tan[(c + d*x)/2]^5 - 8*a^5*C*Tan[(c + d*x)/2]^5 + 8*a^4*b*C*Tan[(c + d*x)/2]^5 - 30*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-26*a^2*A*b^2 + 15*A*b^4 + a^4*(3*A - 8*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*a*(a + b)*(3*a*A*b^2 - 5*A*b^3 + 3*a^3*C + a^2*b*(6*A + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a*(a^3 - a*b^2)^2*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
753,1,977,645,17.4640628,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{1}{2} \left(\frac{(b+a \cos (c+d x))^3 \left(\frac{4 b \left(-7 C a^4-13 A b^2 a^2+3 b^2 C a^2+9 A b^4\right) \sin (c+d x)}{3 a^4 \left(b^2-a^2\right)^2}-\frac{4 \left(A \sin (c+d x) b^5+a^2 C \sin (c+d x) b^3\right)}{3 a^4 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}-\frac{8 \left(5 A \sin (c+d x) b^6-7 a^2 A \sin (c+d x) b^4+2 a^2 C \sin (c+d x) b^4-4 a^4 C \sin (c+d x) b^2\right)}{3 a^4 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}+\frac{A \sin (2 (c+d x))}{2 a^3}\right) \sec ^3(c+d x)}{d (a+b \sec (c+d x))^{5/2}}+\frac{(b+a \cos (c+d x))^{5/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 (a-b)^2 \left(4 (A+2 C) a^2+35 A b^2\right) \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} (a+b)^2+a b \left((33 A-56 C) a^4+2 b^2 (12 C-85 A) a^2+105 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} (a+b)-b \left(-6 (A+12 C) a^5+b (39 A+16 C) a^4+12 b^2 (4 C-19 A) a^3+2 b^3 (29 A-12 C) a^2+210 A b^4 a-105 A b^5\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} (a+b)+a b \left((33 A-56 C) a^4+2 b^2 (12 C-85 A) a^2+105 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b\right)\right) \sec ^{\frac{5}{2}}(c+d x)}{6 a^5 \left(a^2-b^2\right)^2 d (a+b \sec (c+d x))^{5/2} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}\right)","-\frac{7 A b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}-\frac{\sqrt{a+b} \left(4 a^2 (A+2 C)+35 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^5 d}-\frac{b^2 \left(a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right) \tan (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{\left(a^4 (33 A-56 C)-2 a^2 b^2 (85 A-12 C)+105 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{b^2 \left(35 A b^2-a^2 (27 A-8 C)\right) \tan (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{\left(6 a^4 (A-8 C)-a^3 (27 A b-8 b C)-3 a^2 b^2 (45 A-8 C)+35 a A b^3+105 A b^4\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 a^4 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}",1,"(((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((4*b*(-13*a^2*A*b^2 + 9*A*b^4 - 7*a^4*C + 3*a^2*b^2*C)*Sin[c + d*x])/(3*a^4*(-a^2 + b^2)^2) - (4*(A*b^5*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (8*(-7*a^2*A*b^4*Sin[c + d*x] + 5*A*b^6*Sin[c + d*x] - 4*a^4*b^2*C*Sin[c + d*x] + 2*a^2*b^4*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(2*a^3)))/(d*(a + b*Sec[c + d*x])^(5/2)) + ((b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*b*(105*A*b^4 + a^4*(33*A - 56*C) + 2*a^2*b^2*(-85*A + 12*C))*Tan[(c + d*x)/2]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2) + a*b*(a + b)*(105*A*b^4 + a^4*(33*A - 56*C) + 2*a^2*b^2*(-85*A + 12*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(210*a*A*b^4 - 105*A*b^5 + 2*a^2*b^3*(29*A - 12*C) + 12*a^3*b^2*(-19*A + 4*C) - 6*a^5*(A + 12*C) + a^4*b*(39*A + 16*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 3*(a - b)^2*(a + b)^2*(35*A*b^2 + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(6*a^5*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))))/2","A",0
754,1,2188,626,25.2726769,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]","\text{Result too large to show}","-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{2 \left(a^2 C+A b^2\right) \tan (c+d x)}{5 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{5/2}}-\frac{2 \left(-3 a^4 C-a^2 b^2 (13 A+5 C)+5 A b^4\right) \tan (c+d x)}{15 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right) \tan (c+d x)}{15 a^3 d \left(a^2-b^2\right)^3 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^6 C-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-15 A b^6\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 b^2 d \sqrt{a+b} \left(a^2-b^2\right)^2}+\frac{2 \left(3 a^5 C-3 a^4 b (15 A+8 C)+a^3 b^2 (13 A+5 C)+36 a^2 A b^3-5 a A b^4-15 A b^5\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 a^3 b d \sqrt{a+b} \left(a^2-b^2\right)^2}",1,"((b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2)*((4*(58*a^4*A*b^2 - 41*a^2*A*b^4 + 15*A*b^6 + 3*a^6*C + 29*a^4*b^2*C)*Sin[c + d*x])/(15*a^3*b*(-a^2 + b^2)^3) + (4*(A*b^4*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(5*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^3) + (4*(-19*a^2*A*b^3*Sin[c + d*x] + 11*A*b^5*Sin[c + d*x] - 9*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(15*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (4*(74*a^4*A*b^2*Sin[c + d*x] - 65*a^2*A*b^4*Sin[c + d*x] + 23*A*b^6*Sin[c + d*x] + 9*a^6*C*Sin[c + d*x] + 25*a^4*b^2*C*Sin[c + d*x] - 2*a^2*b^4*C*Sin[c + d*x]))/(15*a^3*(a^2 - b^2)^3*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(7/2)) - (4*(b + a*Cos[c + d*x])^(7/2)*Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(58*a^5*A*b^2*Tan[(c + d*x)/2] + 58*a^4*A*b^3*Tan[(c + d*x)/2] - 41*a^3*A*b^4*Tan[(c + d*x)/2] - 41*a^2*A*b^5*Tan[(c + d*x)/2] + 15*a*A*b^6*Tan[(c + d*x)/2] + 15*A*b^7*Tan[(c + d*x)/2] + 3*a^7*C*Tan[(c + d*x)/2] + 3*a^6*b*C*Tan[(c + d*x)/2] + 29*a^5*b^2*C*Tan[(c + d*x)/2] + 29*a^4*b^3*C*Tan[(c + d*x)/2] - 116*a^5*A*b^2*Tan[(c + d*x)/2]^3 + 82*a^3*A*b^4*Tan[(c + d*x)/2]^3 - 30*a*A*b^6*Tan[(c + d*x)/2]^3 - 6*a^7*C*Tan[(c + d*x)/2]^3 - 58*a^5*b^2*C*Tan[(c + d*x)/2]^3 + 58*a^5*A*b^2*Tan[(c + d*x)/2]^5 - 58*a^4*A*b^3*Tan[(c + d*x)/2]^5 - 41*a^3*A*b^4*Tan[(c + d*x)/2]^5 + 41*a^2*A*b^5*Tan[(c + d*x)/2]^5 + 15*a*A*b^6*Tan[(c + d*x)/2]^5 - 15*A*b^7*Tan[(c + d*x)/2]^5 + 3*a^7*C*Tan[(c + d*x)/2]^5 - 3*a^6*b*C*Tan[(c + d*x)/2]^5 + 29*a^5*b^2*C*Tan[(c + d*x)/2]^5 - 29*a^4*b^3*C*Tan[(c + d*x)/2]^5 + 30*a^6*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 90*a^4*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 90*a^2*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^7*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a^6*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 90*a^4*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 90*a^2*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^7*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-41*a^2*A*b^4 + 15*A*b^6 + 3*a^6*C + 29*a^4*b^2*(2*A + C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - a*b*(a + b)*(-6*a*A*b^3 + 10*A*b^4 + 3*a^4*(5*A + C) + 6*a^3*b*(5*A + 4*C) + a^2*b^2*(-17*A + 5*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(15*a^3*b*(a^2 - b^2)^3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(7/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
755,1,3360,303,36.3133637,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{2 b \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 a \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(-4*b*Cos[c + d*x]*(b + a*Cos[c + d*x])*(a^2 - b^2*Sec[c + d*x]^2)*Sin[c + d*x])/(d*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]]) - (4*Sqrt[b + a*Cos[c + d*x]]*((2*a^2)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*b*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (2*a*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]])*(a^2 - b^2*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*((b*Tan[(c + d*x)/2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(1 + Tan[(c + d*x)/2]^2)^(3/2) + (I*(-((a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]) + (a^2 - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^4])))/(d*(a^2 - 2*b^2 + a^2*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*((-4*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((b*Tan[(c + d*x)/2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(1 + Tan[(c + d*x)/2]^2)^(3/2) + (I*(-((a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]) + (a^2 - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^4])))/Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)] - (2*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*((1 - Tan[(c + d*x)/2]^2)^(-1))^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*((b*Tan[(c + d*x)/2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(1 + Tan[(c + d*x)/2]^2)^(3/2) + (I*(-((a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]) + (a^2 - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^4])))/Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)] + (2*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*((-(a*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(1 + Tan[(c + d*x)/2]^2) - (Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(1 + Tan[(c + d*x)/2]^2)^2)*((b*Tan[(c + d*x)/2]*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(1 + Tan[(c + d*x)/2]^2)^(3/2) + (I*(-((a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]) + (a^2 - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^4])))/((a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2))^(3/2) - (4*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*((b*Tan[(c + d*x)/2]*(-(a*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(1 + Tan[(c + d*x)/2]^2)^(3/2) - (3*b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(2*(1 + Tan[(c + d*x)/2]^2)^(5/2)) + (b*Sec[(c + d*x)/2]^2*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2))/(2*(1 + Tan[(c + d*x)/2]^2)^(3/2)) + (I*(-((a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]) + (a^2 - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^3*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(-a + b)/(a + b)]*(1 - Tan[(c + d*x)/2]^4)^(3/2)) + ((I/2)*(-((a - b)*b*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]) + (a^2 - b^2)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*a^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*(-(a*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + b*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Sqrt[(-a + b)/(a + b)]*(a + b)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^4]) + (I*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]*(((I/2)*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*Sec[(c + d*x)/2]^2)/(Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (I*a^2*Sqrt[(-a + b)/(a + b)]*Sec[(c + d*x)/2]^2)/((1 - ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b))*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((I/2)*(a - b)*b*Sqrt[(-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a - b)])/Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^4])))/Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]))","C",0
756,1,143,200,6.6344879,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","-\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec (c+d x) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \left((a+b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \sqrt{a+b \sec (c+d x)}}","-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(-4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
757,1,616,338,30.4917516,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec (c+d x) (a \cos (c+d x)+b)^2 (a-b \sec (c+d x)) \left(\frac{4 b \sin (c+d x)}{b^2-a^2}-\frac{4 b^2 \sin (c+d x)}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}\right)}{d (a \cos (c+d x)-b) (a+b \sec (c+d x))^{3/2}}+\frac{4 \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b) (a-b \sec (c+d x)) \left(-i \left(a^2+2 a b-3 b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)+2 i \left(a^2-b^2\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-b \sqrt{\frac{b-a}{a+b}} \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 i b (a-b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right)}{d \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) \left(\tan ^4\left(\frac{1}{2} (c+d x)\right)-1\right) (a \cos (c+d x)-b) (a+b \sec (c+d x))^{3/2}}","\frac{4 b^2 \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{4 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}+\frac{4 \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d \sqrt{a+b}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]*(a - b*Sec[c + d*x])*((4*b*Sin[c + d*x])/(-a^2 + b^2) - (4*b^2*Sin[c + d*x])/((-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(-b + a*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)) + (4*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*(a - b*Sec[c + d*x])*((2*I)*(a - b)*b*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - I*(a^2 + 2*a*b - 3*b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + (2*I)*(a^2 - b^2)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - b*Sqrt[(-a + b)/(a + b)]*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(-b + a*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^4))","C",1
758,1,46455,445,44.4404498,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx","Integrate[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]","\text{Result too large to show}","\frac{2 b^2 \left(11 a^2-3 b^2\right) \tan (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{4 b^2 \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(9 a^2-2 a b-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a d (a-b) (a+b)^{3/2}}+\frac{2 \left(11 a^2-3 b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a d (a-b) (a+b)^{3/2}}-\frac{2 \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}",1,"Result too large to show","C",0
759,0,0,145,79.6575153,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),x]","\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","\frac{2 \left(a^2 C+A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 A b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])), x]","F",-1
760,0,0,213,21.1793083,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","-\frac{2 A b \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x]","F",-1
761,0,0,243,92.3966856,"\int (a+b \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2),x]","\int (a+b \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","A \text{Int}\left((a+b \sec (c+d x))^{2/3},x\right)+\frac{\sqrt{2} C (a+b) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}-\frac{\sqrt{2} a C \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",0,"Integrate[(a + b*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2), x]","A",-1
762,0,0,243,93.7342739,"\int \sqrt[3]{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\int \sqrt[3]{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","A \text{Int}\left(\sqrt[3]{a+b \sec (c+d x)},x\right)+\frac{\sqrt{2} C (a+b) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}-\frac{\sqrt{2} a C \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",0,"Integrate[(a + b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2), x]","A",-1
763,-1,0,240,184.0735594,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3),x]","\text{\$Aborted}","A \text{Int}\left(\frac{1}{\sqrt[3]{a+b \sec (c+d x)}},x\right)-\frac{\sqrt{2} a C \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{\sqrt{2} C \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",0,"$Aborted","F",-1
764,0,0,240,136.7066912,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3),x]","\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","A \text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{2/3}},x\right)-\frac{\sqrt{2} a C \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac{\sqrt{2} C \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",0,"Integrate[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3), x]","A",-1
765,1,106,145,1.0670624,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{45 (a C+b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 (a B+2 b C) \tan ^2(c+d x)+15 (a B+b C)+3 b C \tan ^4(c+d x)\right)+30 (a C+b B) \sec ^3(c+d x)+45 (a C+b B) \sec (c+d x)\right)}{120 d}","\frac{(5 a B+4 b C) \tan ^3(c+d x)}{15 d}+\frac{(5 a B+4 b C) \tan (c+d x)}{5 d}+\frac{3 (a C+b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(a C+b B) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 (a C+b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b C \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(45*(b*B + a*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(45*(b*B + a*C)*Sec[c + d*x] + 30*(b*B + a*C)*Sec[c + d*x]^3 + 8*(15*(a*B + b*C) + 5*(a*B + 2*b*C)*Tan[c + d*x]^2 + 3*b*C*Tan[c + d*x]^4)))/(120*d)","A",1
766,1,85,114,0.6541211,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 (4 a B+3 b C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(8 (a C+b B) (\cos (2 (c+d x))+2) \sec (c+d x)+12 a B+6 b C \sec ^2(c+d x)+9 b C\right)}{24 d}","\frac{(a C+b B) \tan ^3(c+d x)}{3 d}+\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(4 a B+3 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a B+3 b C) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(4*a*B + 3*b*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(12*a*B + 9*b*C + 8*(b*B + a*C)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*b*C*Sec[c + d*x]^2)*Tan[c + d*x])/(24*d)","A",1
767,1,67,93,0.3130213,"\int \sec (c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 (a C+b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (a C+b B) \sec (c+d x)+6 a B+2 b C \tan ^2(c+d x)+6 b C\right)}{6 d}","\frac{(3 a B+2 b C) \tan (c+d x)}{3 d}+\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a C+b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*(b*B + a*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*a*B + 6*b*C + 3*(b*B + a*C)*Sec[c + d*x] + 2*b*C*Tan[c + d*x]^2))/(6*d)","A",1
768,1,75,61,0.0285579,"\int (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tan (c+d x)}{d}+\frac{b B \tan (c+d x)}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(a C+b B) \tan (c+d x)}{d}+\frac{(2 a B+b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*B*ArcTanh[Sin[c + d*x]])/d + (b*C*ArcTanh[Sin[c + d*x]])/(2*d) + (b*B*Tan[c + d*x])/d + (a*C*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
769,1,43,35,0.0147397,"\int \cos (c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a B x+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \tan (c+d x)}{d}","\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{b C \tan (c+d x)}{d}",1,"a*B*x + (b*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (b*C*Tan[c + d*x])/d","A",1
770,1,46,35,0.0292052,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a B \sin (c) \cos (d x)}{d}+\frac{a B \cos (c) \sin (d x)}{d}+a C x+b B x+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}","x (a C+b B)+\frac{a B \sin (c+d x)}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}",1,"b*B*x + a*C*x + (b*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Cos[d*x]*Sin[c])/d + (a*B*Cos[c]*Sin[d*x])/d","A",1
771,1,51,52,0.0893301,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 (a C+b B) \sin (c+d x)+a B \sin (2 (c+d x))+2 a B c+2 a B d x+4 b C d x}{4 d}","\frac{(a C+b B) \sin (c+d x)}{d}+\frac{1}{2} x (a B+2 b C)+\frac{a B \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*a*B*c + 2*a*B*d*x + 4*b*C*d*x + 4*(b*B + a*C)*Sin[c + d*x] + a*B*Sin[2*(c + d*x)])/(4*d)","A",1
772,1,75,84,0.1716024,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 (3 a B+4 b C) \sin (c+d x)+3 (a C+b B) \sin (2 (c+d x))+a B \sin (3 (c+d x))+6 a c C+6 a C d x+6 b B c+6 b B d x}{12 d}","\frac{(2 a B+3 b C) \sin (c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a C+b B)+\frac{a B \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(6*b*B*c + 6*a*c*C + 6*b*B*d*x + 6*a*C*d*x + 3*(3*a*B + 4*b*C)*Sin[c + d*x] + 3*(b*B + a*C)*Sin[2*(c + d*x)] + a*B*Sin[3*(c + d*x)])/(12*d)","A",1
773,1,91,105,0.2389934,"\int \cos ^5(c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{-32 (a C+b B) \sin ^3(c+d x)+96 (a C+b B) \sin (c+d x)+24 (a B+b C) \sin (2 (c+d x))+3 a B \sin (4 (c+d x))+36 a B c+36 a B d x+48 b c C+48 b C d x}{96 d}","-\frac{(a C+b B) \sin ^3(c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x)}{d}+\frac{(3 a B+4 b C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (3 a B+4 b C)+\frac{a B \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(36*a*B*c + 48*b*c*C + 36*a*B*d*x + 48*b*C*d*x + 96*(b*B + a*C)*Sin[c + d*x] - 32*(b*B + a*C)*Sin[c + d*x]^3 + 24*(a*B + b*C)*Sin[2*(c + d*x)] + 3*a*B*Sin[4*(c + d*x)])/(96*d)","A",1
774,1,88,136,0.2509686,"\int \cos ^6(c+d x) (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{-160 (2 a B+b C) \sin ^3(c+d x)+480 (a B+b C) \sin (c+d x)+15 (a C+b B) (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))+96 a B \sin ^5(c+d x)}{480 d}","-\frac{(4 a B+5 b C) \sin ^3(c+d x)}{15 d}+\frac{(4 a B+5 b C) \sin (c+d x)}{5 d}+\frac{(a C+b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 (a C+b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} x (a C+b B)+\frac{a B \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(480*(a*B + b*C)*Sin[c + d*x] - 160*(2*a*B + b*C)*Sin[c + d*x]^3 + 96*a*B*Sin[c + d*x]^5 + 15*(b*B + a*C)*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/(480*d)","A",1
775,1,150,198,5.9712219,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{15 \left(4 a^2 B+6 a b C+3 b^2 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 \left(a^2 C+2 a b B+2 b^2 C\right) \tan ^2(c+d x)+15 \left(a^2 C+2 a b B+b^2 C\right)+3 b^2 C \tan ^4(c+d x)\right)+15 \left(4 a^2 B+6 a b C+3 b^2 B\right) \sec (c+d x)+30 b (2 a C+b B) \sec ^3(c+d x)\right)}{120 d}","\frac{\left(4 a^2 B+6 a b C+3 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(4 a^2 B+6 a b C+3 b^2 B\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \tan ^3(c+d x)}{15 d}+\frac{\left(5 a (a C+2 b B)+4 b^2 C\right) \tan (c+d x)}{5 d}+\frac{b (6 a C+5 b B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{b C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))}{5 d}",1,"(15*(4*a^2*B + 3*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(4*a^2*B + 3*b^2*B + 6*a*b*C)*Sec[c + d*x] + 30*b*(b*B + 2*a*C)*Sec[c + d*x]^3 + 8*(15*(2*a*b*B + a^2*C + b^2*C) + 5*(2*a*b*B + a^2*C + 2*b^2*C)*Tan[c + d*x]^2 + 3*b^2*C*Tan[c + d*x]^4)))/(120*d)","A",1
776,1,120,179,0.8033356,"\int \sec (c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 \left(4 a^2 C+8 a b B+3 b^2 C\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \left(4 a^2 C+8 a b B+3 b^2 C\right) \sec (c+d x)+24 \left(a^2 B+2 a b C+b^2 B\right)+8 b (2 a C+b B) \tan ^2(c+d x)+6 b^2 C \sec ^3(c+d x)\right)}{24 d}","\frac{\left(4 a^2 C+8 a b B+3 b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(-2 a^2 C+8 a b B+9 b^2 C\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{\left(a^3 (-C)+4 a^2 b B+8 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 b d}+\frac{(4 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d}",1,"(3*(8*a*b*B + 4*a^2*C + 3*b^2*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(a^2*B + b^2*B + 2*a*b*C) + 3*(8*a*b*B + 4*a^2*C + 3*b^2*C)*Sec[c + d*x] + 6*b^2*C*Sec[c + d*x]^3 + 8*b*(b*B + 2*a*C)*Tan[c + d*x]^2))/(24*d)","A",1
777,1,92,116,0.4782377,"\int (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 \left(2 a^2 B+2 a b C+b^2 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(2 \left(3 a^2 C+6 a b B+b^2 C \tan ^2(c+d x)+3 b^2 C\right)+3 b (2 a C+b B) \sec (c+d x)\right)}{6 d}","\frac{2 \left(a^2 C+3 a b B+b^2 C\right) \tan (c+d x)}{3 d}+\frac{\left(2 a^2 B+2 a b C+b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b (2 a C+3 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(3*(2*a^2*B + b^2*B + 2*a*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*b*(b*B + 2*a*C)*Sec[c + d*x] + 2*(6*a*b*B + 3*a^2*C + 3*b^2*C + b^2*C*Tan[c + d*x]^2)))/(6*d)","A",1
778,1,67,86,0.2722314,"\int \cos (c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\left(2 a^2 C+4 a b B+b^2 C\right) \tanh ^{-1}(\sin (c+d x))+2 a^2 B d x+b \tan (c+d x) (4 a C+2 b B+b C \sec (c+d x))}{2 d}","\frac{\left(2 a^2 C+4 a b B+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 B x+\frac{b (3 a C+2 b B) \tan (c+d x)}{2 d}+\frac{b C \tan (c+d x) (a+b \sec (c+d x))}{2 d}",1,"(2*a^2*B*d*x + (4*a*b*B + 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]] + b*(2*b*B + 4*a*C + b*C*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
779,1,109,60,0.5135985,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 B \sin (c+d x)+a (c+d x) (a C+2 b B)-b (2 a C+b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b (2 a C+b B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b^2 C \tan (c+d x)}{d}","\frac{a^2 B \sin (c+d x)}{d}+\frac{b (2 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a C+2 b B)+\frac{b^2 C \tan (c+d x)}{d}",1,"(a*(2*b*B + a*C)*(c + d*x) - b*(b*B + 2*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*(b*B + 2*a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2*B*Sin[c + d*x] + b^2*C*Tan[c + d*x])/d","A",1
780,1,120,80,0.2381314,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 (c+d x) \left(a^2 B+4 a b C+2 b^2 B\right)+a^2 B \sin (2 (c+d x))+4 a (a C+2 b B) \sin (c+d x)-4 b^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{1}{2} x \left(a^2 B+4 a b C+2 b^2 B\right)+\frac{a^2 B \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a (a C+2 b B) \sin (c+d x)}{d}+\frac{b^2 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*(a^2*B + 2*b^2*B + 4*a*b*C)*(c + d*x) - 4*b^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*b^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*a*(2*b*B + a*C)*Sin[c + d*x] + a^2*B*Sin[2*(c + d*x)])/(4*d)","A",1
781,1,90,107,0.2890167,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{6 (c+d x) \left(a^2 C+2 a b B+2 b^2 C\right)+3 \left(3 a^2 B+8 a b C+4 b^2 B\right) \sin (c+d x)+a^2 B \sin (3 (c+d x))+3 a (a C+2 b B) \sin (2 (c+d x))}{12 d}","\frac{\left(2 a^2 B+6 a b C+3 b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(a^2 C+2 a b B+2 b^2 C\right)+\frac{a^2 B \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a (a C+2 b B) \sin (c+d x) \cos (c+d x)}{2 d}",1,"(6*(2*a*b*B + a^2*C + 2*b^2*C)*(c + d*x) + 3*(3*a^2*B + 4*b^2*B + 8*a*b*C)*Sin[c + d*x] + 3*a*(2*b*B + a*C)*Sin[2*(c + d*x)] + a^2*B*Sin[3*(c + d*x)])/(12*d)","A",1
782,1,118,136,0.5569386,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{12 (c+d x) \left(3 a^2 B+8 a b C+4 b^2 B\right)+24 \left(3 a^2 C+6 a b B+4 b^2 C\right) \sin (c+d x)+24 \left(a^2 B+2 a b C+b^2 B\right) \sin (2 (c+d x))+3 a^2 B \sin (4 (c+d x))+8 a (a C+2 b B) \sin (3 (c+d x))}{96 d}","\frac{\left(a^2 C+2 a b B+b^2 C\right) \sin (c+d x)}{d}+\frac{\left(3 a^2 B+8 a b C+4 b^2 B\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2 B+8 a b C+4 b^2 B\right)+\frac{a^2 B \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{a (a C+2 b B) \sin ^3(c+d x)}{3 d}",1,"(12*(3*a^2*B + 4*b^2*B + 8*a*b*C)*(c + d*x) + 24*(6*a*b*B + 3*a^2*C + 4*b^2*C)*Sin[c + d*x] + 24*(a^2*B + b^2*B + 2*a*b*C)*Sin[2*(c + d*x)] + 8*a*(2*b*B + a*C)*Sin[3*(c + d*x)] + 3*a^2*B*Sin[4*(c + d*x)])/(96*d)","A",1
783,1,146,180,0.5977609,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{60 (c+d x) \left(3 a^2 C+6 a b B+4 b^2 C\right)+60 \left(5 a^2 B+12 a b C+6 b^2 B\right) \sin (c+d x)+120 \left(a^2 C+2 a b B+b^2 C\right) \sin (2 (c+d x))+10 \left(5 a^2 B+8 a b C+4 b^2 B\right) \sin (3 (c+d x))+6 a^2 B \sin (5 (c+d x))+15 a (a C+2 b B) \sin (4 (c+d x))}{480 d}","-\frac{\left(4 a^2 B+10 a b C+5 b^2 B\right) \sin ^3(c+d x)}{15 d}+\frac{\left(4 a^2 B+10 a b C+5 b^2 B\right) \sin (c+d x)}{5 d}+\frac{\left(3 a^2 C+6 a b B+4 b^2 C\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2 C+6 a b B+4 b^2 C\right)+\frac{a^2 B \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a (a C+2 b B) \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(60*(6*a*b*B + 3*a^2*C + 4*b^2*C)*(c + d*x) + 60*(5*a^2*B + 6*b^2*B + 12*a*b*C)*Sin[c + d*x] + 120*(2*a*b*B + a^2*C + b^2*C)*Sin[2*(c + d*x)] + 10*(5*a^2*B + 4*b^2*B + 8*a*b*C)*Sin[3*(c + d*x)] + 15*a*(2*b*B + a*C)*Sin[4*(c + d*x)] + 6*a^2*B*Sin[5*(c + d*x)])/(480*d)","A",1
784,1,214,278,5.3826388,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{15 \left(8 a^3 B+18 a^2 b C+18 a b^2 B+5 b^3 C\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(10 b \left(18 a^2 C+18 a b B+5 b^2 C\right) \sec ^3(c+d x)+80 \left(a^3 C+3 a^2 b B+6 a b^2 C+2 b^3 B\right) \tan ^2(c+d x)+15 \left(8 a^3 B+18 a^2 b C+18 a b^2 B+5 b^3 C\right) \sec (c+d x)+240 \left(a^3 C+3 a^2 b B+3 a b^2 C+b^3 B\right)+48 b^2 (3 a C+b B) \tan ^4(c+d x)+40 b^3 C \sec ^5(c+d x)\right)}{240 d}","\frac{b \left(14 a^2 C+18 a b B+5 b^2 C\right) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{\left(5 a^3 C+15 a^2 b B+12 a b^2 C+4 b^3 B\right) \tan ^3(c+d x)}{15 d}+\frac{\left(5 a^3 C+15 a^2 b B+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{5 d}+\frac{\left(8 a^3 B+18 a^2 b C+18 a b^2 B+5 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(8 a^3 B+18 a^2 b C+18 a b^2 B+5 b^3 C\right) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{b^2 (4 a C+3 b B) \tan (c+d x) \sec ^4(c+d x)}{15 d}+\frac{b C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^2}{6 d}",1,"(15*(8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(240*(3*a^2*b*B + b^3*B + a^3*C + 3*a*b^2*C) + 15*(8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*Sec[c + d*x] + 10*b*(18*a*b*B + 18*a^2*C + 5*b^2*C)*Sec[c + d*x]^3 + 40*b^3*C*Sec[c + d*x]^5 + 80*(3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*Tan[c + d*x]^2 + 48*b^2*(b*B + 3*a*C)*Tan[c + d*x]^4))/(240*d)","A",1
785,1,211,252,6.1531946,"\int \sec (c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 B \tan (c+d x)}{d}+\frac{a^2 (a C+3 b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (a C+3 b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b^2 (3 a C+b B) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 b^2 (3 a C+b B) \left(\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right)}{8 d}+\frac{a b (a C+b B) \left(\tan ^3(c+d x)+3 \tan (c+d x)\right)}{d}+\frac{b^3 C \left(3 \tan ^5(c+d x)+10 \tan ^3(c+d x)+15 \tan (c+d x)\right)}{15 d}","\frac{\left(-3 a^2 C+15 a b B+16 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}+\frac{\left(4 a^3 C+12 a^2 b B+9 a b^2 C+3 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(-6 a^3 C+30 a^2 b B+71 a b^2 C+45 b^3 B\right) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{\left(-3 a^4 C+15 a^3 b B+52 a^2 b^2 C+60 a b^3 B+16 b^4 C\right) \tan (c+d x)}{30 b d}+\frac{(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}",1,"(a^2*(3*b*B + a*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^3*B*Tan[c + d*x])/d + (a^2*(3*b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b^2*(b*B + 3*a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (3*b^2*(b*B + 3*a*C)*(ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*Tan[c + d*x]))/(8*d) + (a*b*(b*B + a*C)*(3*Tan[c + d*x] + Tan[c + d*x]^3))/d + (b^3*C*(15*Tan[c + d*x] + 10*Tan[c + d*x]^3 + 3*Tan[c + d*x]^5))/(15*d)","A",1
786,1,140,180,0.897363,"\int (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 \left(8 a^3 B+12 a^2 b C+12 a b^2 B+3 b^3 C\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(9 b \left(4 a^2 C+4 a b B+b^2 C\right) \sec (c+d x)+24 \left(a^3 C+3 a^2 b B+3 a b^2 C+b^3 B\right)+8 b^2 (3 a C+b B) \tan ^2(c+d x)+6 b^3 C \sec ^3(c+d x)\right)}{24 d}","\frac{b \left(6 a^2 C+20 a b B+9 b^2 C\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{\left(3 a^3 C+16 a^2 b B+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 d}+\frac{\left(8 a^3 B+12 a^2 b C+12 a b^2 B+3 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(3*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(3*a^2*b*B + b^3*B + a^3*C + 3*a*b^2*C) + 9*b*(4*a*b*B + 4*a^2*C + b^2*C)*Sec[c + d*x] + 6*b^3*C*Sec[c + d*x]^3 + 8*b^2*(b*B + 3*a*C)*Tan[c + d*x]^2))/(24*d)","A",1
787,1,108,137,0.7021003,"\int \cos (c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{6 a^3 B d x+3 b \tan (c+d x) \left(6 a^2 C+b (3 a C+b B) \sec (c+d x)+6 a b B+2 b^2 C\right)+3 \left(2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))+2 b^3 C \tan ^3(c+d x)}{6 d}","a^3 B x+\frac{b \left(8 a^2 C+9 a b B+2 b^2 C\right) \tan (c+d x)}{3 d}+\frac{\left(2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 (5 a C+3 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{b C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(6*a^3*B*d*x + 3*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*ArcTanh[Sin[c + d*x]] + 3*b*(6*a*b*B + 6*a^2*C + 2*b^2*C + b*(b*B + 3*a*C)*Sec[c + d*x])*Tan[c + d*x] + 2*b^3*C*Tan[c + d*x]^3)/(6*d)","A",1
788,1,277,131,4.4521075,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 a^3 B \sin (c+d x)-2 b \left(6 a^2 C+6 a b B+b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \left(6 a^2 C+6 a b B+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 a^2 (c+d x) (a C+3 b B)+\frac{4 b^2 (3 a C+b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b^2 (3 a C+b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{b^3 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^3 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 d}","-\frac{b \left(2 a^2 B-3 a b C-b^2 B\right) \tan (c+d x)}{d}+\frac{b \left(6 a^2 C+6 a b B+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 x (a C+3 b B)-\frac{b^2 (2 a B-b C) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \sin (c+d x) (a+b \sec (c+d x))^2}{d}",1,"(4*a^2*(3*b*B + a*C)*(c + d*x) - 2*b*(6*a*b*B + 6*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*b*(6*a*b*B + 6*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^3*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*b^2*(b*B + 3*a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^3*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b^2*(b*B + 3*a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*a^3*B*Sin[c + d*x])/(4*d)","B",1
789,1,217,124,1.2210124,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 B \sin (2 (c+d x))+2 a (c+d x) \left(a^2 B+6 a b C+6 b^2 B\right)+4 a^2 (a C+3 b B) \sin (c+d x)-4 b^2 (3 a C+b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2 (3 a C+b B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{4 d}","\frac{1}{2} a x \left(a^2 B+6 a b C+6 b^2 B\right)+\frac{a^2 (a C+2 b B) \sin (c+d x)}{d}-\frac{b^2 (a B-2 b C) \tan (c+d x)}{2 d}+\frac{b^2 (3 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}",1,"(2*a*(a^2*B + 6*b^2*B + 6*a*b*C)*(c + d*x) - 4*b^2*(b*B + 3*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*b^2*(b*B + 3*a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (4*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*a^2*(3*b*B + a*C)*Sin[c + d*x] + a^3*B*Sin[2*(c + d*x)])/(4*d)","A",1
790,1,159,145,0.8165935,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 B \sin (3 (c+d x))+9 a \left(a^2 B+4 a b C+4 b^2 B\right) \sin (c+d x)+3 a^2 (a C+3 b B) \sin (2 (c+d x))+6 (c+d x) \left(a^3 C+3 a^2 b B+6 a b^2 C+2 b^3 B\right)-12 b^3 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^3 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 d}","\frac{a \left(2 a^2 B+9 a b C+8 b^2 B\right) \sin (c+d x)}{3 d}+\frac{a^2 (3 a C+5 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} x \left(a^3 C+3 a^2 b B+6 a b^2 C+2 b^3 B\right)+\frac{a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^3 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(6*(3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*(c + d*x) - 12*b^3*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^3*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*a*(a^2*B + 4*b^2*B + 4*a*b*C)*Sin[c + d*x] + 3*a^2*(3*b*B + a*C)*Sin[2*(c + d*x)] + a^3*B*Sin[3*(c + d*x)])/(12*d)","A",1
791,1,140,179,0.9089791,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 a^3 B \sin (4 (c+d x))+24 a \left(a^2 B+3 a b C+3 b^2 B\right) \sin (2 (c+d x))+8 a^2 (a C+3 b B) \sin (3 (c+d x))+12 (c+d x) \left(3 a^3 B+12 a^2 b C+12 a b^2 B+8 b^3 C\right)+24 \left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right) \sin (c+d x)}{96 d}","\frac{a \left(3 a^2 B+12 a b C+10 b^2 B\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{\left(2 a^3 C+6 a^2 b B+9 a b^2 C+3 b^3 B\right) \sin (c+d x)}{3 d}+\frac{1}{8} x \left(3 a^3 B+12 a^2 b C+12 a b^2 B+8 b^3 C\right)+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"(12*(3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*(c + d*x) + 24*(9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Sin[c + d*x] + 24*a*(a^2*B + 3*b^2*B + 3*a*b*C)*Sin[2*(c + d*x)] + 8*a^2*(3*b*B + a*C)*Sin[3*(c + d*x)] + 3*a^3*B*Sin[4*(c + d*x)])/(96*d)","A",1
792,1,176,221,0.8744829,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{6 a^3 B \sin (5 (c+d x))+10 a \left(5 a^2 B+12 a b C+12 b^2 B\right) \sin (3 (c+d x))+15 a^2 (a C+3 b B) \sin (4 (c+d x))+60 (c+d x) \left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right)+60 \left(5 a^3 B+18 a^2 b C+18 a b^2 B+8 b^3 C\right) \sin (c+d x)+120 \left(a^3 C+3 a^2 b B+3 a b^2 C+b^3 B\right) \sin (2 (c+d x))}{480 d}","-\frac{a \left(4 a^2 B+15 a b C+12 b^2 B\right) \sin ^3(c+d x)}{15 d}+\frac{a^2 (5 a C+7 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{\left(4 a^3 B+15 a^2 b C+14 a b^2 B+5 b^3 C\right) \sin (c+d x)}{5 d}+\frac{\left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^3 C+9 a^2 b B+12 a b^2 C+4 b^3 B\right)+\frac{a B \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}",1,"(60*(9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*(c + d*x) + 60*(5*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 8*b^3*C)*Sin[c + d*x] + 120*(3*a^2*b*B + b^3*B + a^3*C + 3*a*b^2*C)*Sin[2*(c + d*x)] + 10*a*(5*a^2*B + 12*b^2*B + 12*a*b*C)*Sin[3*(c + d*x)] + 15*a^2*(3*b*B + a*C)*Sin[4*(c + d*x)] + 6*a^3*B*Sin[5*(c + d*x)])/(480*d)","A",1
793,1,528,187,6.648542,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{3 a^2 C \sin \left(\frac{1}{2} (c+d x)\right)-3 a b B \sin \left(\frac{1}{2} (c+d x)\right)+2 b^2 C \sin \left(\frac{1}{2} (c+d x)\right)}{3 b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{3 a^2 C \sin \left(\frac{1}{2} (c+d x)\right)-3 a b B \sin \left(\frac{1}{2} (c+d x)\right)+2 b^2 C \sin \left(\frac{1}{2} (c+d x)\right)}{3 b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 a^3 (b B-a C) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2}}+\frac{\left(2 a^3 C-2 a^2 b B+a b^2 C-b^3 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{\left(-2 a^3 C+2 a^2 b B-a b^2 C+b^3 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{-3 a C+3 b B+b C}{12 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{3 a C-3 b B-b C}{12 b^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{C \sin \left(\frac{1}{2} (c+d x)\right)}{6 b d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{C \sin \left(\frac{1}{2} (c+d x)\right)}{6 b d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}","-\frac{2 a^3 (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(2 a^2+b^2\right) (b B-a C) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{\left(-3 a^2 C+3 a b B-2 b^2 C\right) \tan (c+d x)}{3 b^3 d}+\frac{(b B-a C) \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 b d}",1,"(2*a^3*(b*B - a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*d) + ((-2*a^2*b*B - b^3*B + 2*a^3*C + a*b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*b^4*d) + ((2*a^2*b*B + b^3*B - 2*a^3*C - a*b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*b^4*d) + (3*b*B - 3*a*C + b*C)/(12*b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (C*Sin[(c + d*x)/2])/(6*b*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (C*Sin[(c + d*x)/2])/(6*b*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + (-3*b*B + 3*a*C - b*C)/(12*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (-3*a*b*B*Sin[(c + d*x)/2] + 3*a^2*C*Sin[(c + d*x)/2] + 2*b^2*C*Sin[(c + d*x)/2])/(3*b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (-3*a*b*B*Sin[(c + d*x)/2] + 3*a^2*C*Sin[(c + d*x)/2] + 2*b^2*C*Sin[(c + d*x)/2])/(3*b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))","B",1
794,1,300,143,2.2420271,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\frac{8 a^2 (a C-b B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-2 \left(2 a^2 C-2 a b B+b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(2 a^2 C-2 a b B+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 b (b B-a C) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b (b B-a C) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{b^2 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 b^3 d}","\frac{2 a^2 (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{\left(-2 a^2 C+2 a b B-b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{(b B-a C) \tan (c+d x)}{b^2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 b d}",1,"((8*a^2*(-(b*B) + a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 2*(-2*a*b*B + 2*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-2*a*b*B + 2*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*b*(b*B - a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^2*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b*(b*B - a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(4*b^3*d)","B",1
795,1,130,98,0.7817749,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 a (a C-b B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-(b B-a C) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b C \tan (c+d x)}{b^2 d}","\frac{(b B-a C) \tanh ^{-1}(\sin (c+d x))}{b^2 d}-\frac{2 a (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{C \tan (c+d x)}{b d}",1,"((-2*a*(-(b*B) + a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (b*B - a*C)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + b*C*Tan[c + d*x])/(b^2*d)","A",1
796,1,112,76,0.2239951,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 (a C-b B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+C \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{b d}","\frac{2 (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b d \sqrt{a-b} \sqrt{a+b}}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b d}",1,"((2*(-(b*B) + a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + C*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]))/(b*d)","A",1
797,1,68,67,0.1492845,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 (b B-a C) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+B (c+d x)}{a d}","\frac{B x}{a}-\frac{2 (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}",1,"(B*(c + d*x) + (2*(b*B - a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2])/(a*d)","A",1
798,1,85,90,0.2734832,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 b (b B-a C) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+(c+d x) (a C-b B)+a B \sin (c+d x)}{a^2 d}","\frac{2 b (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x (b B-a C)}{a^2}+\frac{B \sin (c+d x)}{a d}",1,"((-(b*B) + a*C)*(c + d*x) - (2*b*(b*B - a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*B*Sin[c + d*x])/(a^2*d)","A",1
799,1,121,134,0.4002685,"\int \frac{\cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{2 (c+d x) \left(a^2 B-2 a b C+2 b^2 B\right)+\frac{8 b^2 (b B-a C) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a^2 B \sin (2 (c+d x))+4 a (a C-b B) \sin (c+d x)}{4 a^3 d}","-\frac{2 b^2 (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{(b B-a C) \sin (c+d x)}{a^2 d}+\frac{x \left(a^2 B-2 a b C+2 b^2 B\right)}{2 a^3}+\frac{B \sin (c+d x) \cos (c+d x)}{2 a d}",1,"(2*(a^2*B + 2*b^2*B - 2*a*b*C)*(c + d*x) + (8*b^2*(b*B - a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 4*a*(-(b*B) + a*C)*Sin[c + d*x] + a^2*B*Sin[2*(c + d*x)])/(4*a^3*d)","A",1
800,1,174,178,8.0681571,"\int \frac{\cos ^4(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{(a C-b B) \sin (2 (c+d x))}{4 a^2 d}+\frac{\left(a^2+2 b^2\right) (c+d x) (a C-b B)}{2 a^4 d}-\frac{2 b^3 (b B-a C) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^4 d \sqrt{a^2-b^2}}+\frac{\left(3 a^2 B-4 a b C+4 b^2 B\right) \sin (c+d x)}{4 a^3 d}+\frac{B \sin (3 (c+d x))}{12 a d}","\frac{2 b^3 (b B-a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x \left(a^2+2 b^2\right) (b B-a C)}{2 a^4}+\frac{\left(2 a^2 B-3 a b C+3 b^2 B\right) \sin (c+d x)}{3 a^3 d}+\frac{B \sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"((a^2 + 2*b^2)*(-(b*B) + a*C)*(c + d*x))/(2*a^4*d) - (2*b^3*(b*B - a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]*d) + ((3*a^2*B + 4*b^2*B - 4*a*b*C)*Sin[c + d*x])/(4*a^3*d) + ((-(b*B) + a*C)*Sin[2*(c + d*x)])/(4*a^2*d) + (B*Sin[3*(c + d*x)])/(12*a*d)","A",1
801,1,438,272,7.9635987,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(-6 a^2 C+4 a b B-b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{\left(6 a^2 C-4 a b B+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{a^4 C \sin (c+d x)-a^3 b B \sin (c+d x)}{b^3 d (b-a) (a+b) (a \cos (c+d x)+b)}-\frac{2 a^2 \left(3 a^3 C-2 a^2 b B-4 a b^2 C+3 b^3 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2} \left(b^2-a^2\right)}+\frac{b B \sin \left(\frac{1}{2} (c+d x)\right)-2 a C \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{b B \sin \left(\frac{1}{2} (c+d x)\right)-2 a C \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C}{4 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{C}{4 b^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","\frac{a (b B-a C) \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{\left(-3 a^2 C+2 a b B+b^2 C\right) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left(a^2-b^2\right)}-\frac{\left(-6 a^2 C+4 a b B-b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\left(-3 a^3 C+2 a^2 b B+2 a b^2 C-b^3 B\right) \tan (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(-3 a^3 C+2 a^2 b B+4 a b^2 C-3 b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*a^2*(-2*a^2*b*B + 3*b^3*B + 3*a^3*C - 4*a*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*(-a^2 + b^2)*d) + ((4*a*b*B - 6*a^2*C - b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*b^4*d) + ((-4*a*b*B + 6*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*b^4*d) + C/(4*b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - C/(4*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (b*B*Sin[(c + d*x)/2] - 2*a*C*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (b*B*Sin[(c + d*x)/2] - 2*a*C*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (-(a^3*b*B*Sin[c + d*x]) + a^4*C*Sin[c + d*x])/(b^3*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x]))","A",0
802,1,308,164,8.2128614,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{a^2 b B \sin (c+d x)-a^3 C \sin (c+d x)}{b^2 d (b-a) (a+b) (a \cos (c+d x)+b)}+\frac{2 a \left(2 a^3 C-a^2 b B-3 a b^2 C+2 b^3 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^3 d \sqrt{a^2-b^2} \left(b^2-a^2\right)}+\frac{(2 a C-b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^3 d}+\frac{(b B-2 a C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^3 d}+\frac{C \sin \left(\frac{1}{2} (c+d x)\right)}{b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C \sin \left(\frac{1}{2} (c+d x)\right)}{b^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","-\frac{a^2 (b B-a C) \tan (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 a \left(-2 a^3 C+a^2 b B+3 a b^2 C-2 b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}",1,"(2*a*(-(a^2*b*B) + 2*b^3*B + 2*a^3*C - 3*a*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]*(-a^2 + b^2)*d) + ((-(b*B) + 2*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(b^3*d) + ((b*B - 2*a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(b^3*d) + (C*Sin[(c + d*x)/2])/(b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (C*Sin[(c + d*x)/2])/(b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (a^2*b*B*Sin[c + d*x] - a^3*C*Sin[c + d*x])/(b^2*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x]))","A",1
803,1,310,131,8.2256636,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{\cos (c+d x) (B+C \sec (c+d x)) \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)\right)}{b d (b-a) (a+b) (a \cos (c+d x)+b) (B \cos (c+d x)+C)}-\frac{2 \left(a^3 C-2 a b^2 C+b^3 B\right) \cos (c+d x) (B+C \sec (c+d x)) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^2 d \sqrt{a^2-b^2} \left(b^2-a^2\right) (B \cos (c+d x)+C)}-\frac{C \cos (c+d x) (B+C \sec (c+d x)) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^2 d (B \cos (c+d x)+C)}+\frac{C \cos (c+d x) (B+C \sec (c+d x)) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^2 d (B \cos (c+d x)+C)}","-\frac{2 \left(a^3 C-2 a b^2 C+b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (b B-a C) \tan (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"(-2*(b^3*B + a^3*C - 2*a*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Cos[c + d*x]*(B + C*Sec[c + d*x]))/(b^2*Sqrt[a^2 - b^2]*(-a^2 + b^2)*d*(C + B*Cos[c + d*x])) - (C*Cos[c + d*x]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(B + C*Sec[c + d*x]))/(b^2*d*(C + B*Cos[c + d*x])) + (C*Cos[c + d*x]*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(B + C*Sec[c + d*x]))/(b^2*d*(C + B*Cos[c + d*x])) + (Cos[c + d*x]*(B + C*Sec[c + d*x])*(-(a*b*B*Sin[c + d*x]) + a^2*C*Sin[c + d*x]))/(b*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x])*(C + B*Cos[c + d*x]))","B",1
804,1,97,100,1.7593179,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{(a C-b B) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-\frac{2 (a B-b C) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}}{d}","\frac{2 (a B-b C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{3/2} (a+b)^{3/2}}-\frac{(b B-a C) \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}",1,"((-2*(a*B - b*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + ((-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])))/d","A",1
805,1,119,124,8.2774191,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{-\frac{2 \left(a^3 C-2 a^2 b B+b^3 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a b (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}+B (c+d x)}{a^2 d}","\frac{b (b B-a C) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{B x}{a^2}-\frac{2 \left(a^3 (-C)+2 a^2 b B-b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(B*(c + d*x) - (2*(-2*a^2*b*B + b^3*B + a^3*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a*b*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])))/(a^2*d)","A",1
806,1,147,180,1.0922677,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{2 b \left(2 a^3 C-3 a^2 b B-a b^2 C+2 b^3 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a b^2 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}+(c+d x) (a C-2 b B)+a B \sin (c+d x)}{a^3 d}","-\frac{x (2 b B-a C)}{a^3}+\frac{\left(a^2 B+a b C-2 b^2 B\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (b B-a C) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{2 b \left(-2 a^3 C+3 a^2 b B+a b^2 C-2 b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((-2*b*B + a*C)*(c + d*x) + (2*b*(-3*a^2*b*B + 2*b^3*B + 2*a^3*C - a*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + a*B*Sin[c + d*x] + (a*b^2*(-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])))/(a^3*d)","A",1
807,1,184,261,3.2914628,"\int \frac{\cos ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (c+d x) \left(a^2 B-4 a b C+6 b^2 B\right)+a^2 B \sin (2 (c+d x))-\frac{8 b^2 \left(3 a^3 C-4 a^2 b B-2 a b^2 C+3 b^3 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{4 a b^3 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}+4 a (a C-2 b B) \sin (c+d x)}{4 a^4 d}","\frac{\left(a^2 B+2 a b C-3 b^2 B\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b (b B-a C) \sin (c+d x) \cos (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x \left(a^2 B-4 a b C+6 b^2 B\right)}{2 a^4}-\frac{\left(a^3 (-C)+2 a^2 b B+2 a b^2 C-3 b^3 B\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}-\frac{2 b^2 \left(-3 a^3 C+4 a^2 b B+2 a b^2 C-3 b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*(a^2*B + 6*b^2*B - 4*a*b*C)*(c + d*x) - (8*b^2*(-4*a^2*b*B + 3*b^3*B + 3*a^3*C - 2*a*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 4*a*(-2*b*B + a*C)*Sin[c + d*x] - (4*a*b^3*(-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*B*Sin[2*(c + d*x)])/(4*a^4*d)","A",1
808,1,418,289,7.941833,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{a^2 b B \sin (c+d x)-a^3 C \sin (c+d x)}{2 b^2 d (b-a) (a+b) (a \cos (c+d x)+b)^2}+\frac{4 a^5 C \sin (c+d x)-2 a^4 b B \sin (c+d x)-7 a^3 b^2 C \sin (c+d x)+5 a^2 b^3 B \sin (c+d x)}{2 b^3 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{a \left(-6 a^5 C+2 a^4 b B+15 a^3 b^2 C-5 a^2 b^3 B-12 a b^4 C+6 b^5 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2} \left(b^2-a^2\right)^2}+\frac{(3 a C-b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^4 d}+\frac{(b B-3 a C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^4 d}+\frac{C \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{C \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{a (b B-a C) \tan (c+d x) \sec ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\left(-3 a^2 C+a b B+2 b^2 C\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^3 C+a^2 b B+6 a b^2 C-4 b^3 B\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{a \left(-6 a^5 C+2 a^4 b B+15 a^3 b^2 C-5 a^2 b^3 B-12 a b^4 C+6 b^5 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(a*(2*a^4*b*B - 5*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 15*a^3*b^2*C - 12*a*b^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*(-a^2 + b^2)^2*d) + ((-(b*B) + 3*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(b^4*d) + ((b*B - 3*a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(b^4*d) + (C*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (C*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (a^2*b*B*Sin[c + d*x] - a^3*C*Sin[c + d*x])/(2*b^2*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x])^2) + (-2*a^4*b*B*Sin[c + d*x] + 5*a^2*b^3*B*Sin[c + d*x] + 4*a^5*C*Sin[c + d*x] - 7*a^3*b^2*C*Sin[c + d*x])/(2*b^3*(-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x]))","A",0
809,1,270,220,2.6756076,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\cos (c+d x) (B+C \sec (c+d x)) \left(\frac{a b \left(-2 a^3 C+5 a b^2 C-3 b^3 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{2 \left(2 a^5 C-5 a^3 b^2 C-a^2 b^3 B+6 a b^4 C-2 b^5 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^2 (a C-b B) \sin (c+d x)}{(b-a) (a+b) (a \cos (c+d x)+b)^2}-2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 b^3 d (B \cos (c+d x)+C)}","-\frac{a^2 (b B-a C) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(-3 a^3 C+a^2 b B+6 a b^2 C-4 b^3 B\right) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(-2 a^5 C+5 a^3 b^2 C+a^2 b^3 B-6 a b^4 C+2 b^5 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"(Cos[c + d*x]*(B + C*Sec[c + d*x])*((2*(-(a^2*b^3*B) - 2*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 6*a*b^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b^2*(-(b*B) + a*C)*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c + d*x])^2) + (a*b*(-3*b^3*B - 2*a^3*C + 5*a*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x]))))/(2*b^3*d*(C + B*Cos[c + d*x]))","A",1
810,1,157,180,3.0402198,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{\left(2 a^2 B-3 a b C+b^2 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}-\frac{2 \left(a^2 C-3 a b B+2 b^2 C\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{(a C-b B) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^2}}{2 d}","-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}+\frac{a (b B-a C) \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(a^3 C+a^2 b B-4 a b^2 C+2 b^3 B\right) \tan (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}",1,"((-2*(-3*a*b*B + a^2*C + 2*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + ((-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + ((2*a^2*B + b^2*B - 3*a*b*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])))/(2*d)","A",1
811,1,172,164,1.1411562,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\frac{-\frac{2 \left(2 a^2 B-3 a b C+b^2 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{\left(2 a^3 C-4 a^2 b B+a b^2 C+b^3 B\right) \sin (c+d x)}{a (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{b (b B-a C) \sin (c+d x)}{a (a-b) (a+b) (a \cos (c+d x)+b)^2}}{2 d}","\frac{\left(2 a^2 B-3 a b C+b^2 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(a^2 (-C)+3 a b B-2 b^2 C\right) \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{(b B-a C) \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}",1,"((-2*(2*a^2*B + b^2*B - 3*a*b*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + ((-4*a^2*b*B + b^3*B + 2*a^3*C + a*b^2*C)*Sin[c + d*x])/(a*(a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])))/(2*d)","A",1
812,1,203,205,1.8583378,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{a b \left(-4 a^3 C+6 a^2 b B+a b^2 C-3 b^3 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}-\frac{2 \left(2 a^5 C-6 a^4 b B+a^3 b^2 C+5 a^2 b^3 B-2 b^5 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^2 (a C-b B) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^2}+2 B (c+d x)}{2 a^3 d}","\frac{B x}{a^3}+\frac{b (b B-a C) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{b \left(-3 a^3 C+5 a^2 b B-2 b^3 B\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(-2 a^5 C+6 a^4 b B-a^3 b^2 C-5 a^2 b^3 B+2 b^5 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*B*(c + d*x) - (2*(-6*a^4*b*B + 5*a^2*b^3*B - 2*b^5*B + 2*a^5*C + a^3*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (a*b^2*(-(b*B) + a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + (a*b*(6*a^2*b*B - 3*b^3*B - 4*a^3*C + a*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])))/(2*a^3*d)","A",1
813,1,232,290,5.6698691,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{a b^2 \left(6 a^3 C-8 a^2 b B-3 a b^2 C+5 b^3 B\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}-\frac{2 b \left(-6 a^5 C+12 a^4 b B+5 a^3 b^2 C-15 a^2 b^3 B-2 a b^4 C+6 b^5 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^3 (b B-a C) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^2}+2 (c+d x) (a C-3 b B)+2 a B \sin (c+d x)}{2 a^4 d}","-\frac{x (3 b B-a C)}{a^4}+\frac{b (b B-a C) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{b \left(-4 a^3 C+6 a^2 b B+a b^2 C-3 b^3 B\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(2 a^4 B+5 a^3 b C-11 a^2 b^2 B-2 a b^3 C+6 b^4 B\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-6 a^5 C+12 a^4 b B+5 a^3 b^2 C-15 a^2 b^3 B-2 a b^4 C+6 b^5 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*(-3*b*B + a*C)*(c + d*x) - (2*b*(12*a^4*b*B - 15*a^2*b^3*B + 6*b^5*B - 6*a^5*C + 5*a^3*b^2*C - 2*a*b^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + 2*a*B*Sin[c + d*x] + (a*b^3*(b*B - a*C)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + (a*b^2*(-8*a^2*b*B + 5*b^3*B + 6*a^3*C - 3*a*b^2*C)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])))/(2*a^4*d)","A",1
814,1,746,485,20.8090138,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 \sec ^2(c+d x) \left(-6 a^2 C \sin (c+d x)+9 a b B \sin (c+d x)+49 b^2 C \sin (c+d x)\right)}{315 b^2}+\frac{2 \sec (c+d x) \left(8 a^3 C \sin (c+d x)-12 a^2 b B \sin (c+d x)+13 a b^2 C \sin (c+d x)+75 b^3 B \sin (c+d x)\right)}{315 b^3}+\frac{2 \left(-16 a^4 C+24 a^3 b B-24 a^2 b^2 C+57 a b^3 B+147 b^4 C\right) \sin (c+d x)}{315 b^4}+\frac{2 \sec ^3(c+d x) (a C \sin (c+d x)+9 b B \sin (c+d x))}{63 b}+\frac{2}{9} C \tan (c+d x) \sec ^3(c+d x)\right)}{d}+\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{a+b \sec (c+d x)} \left(b (a+b) \left(-16 a^3 C+12 a^2 b (2 B+C)-18 a b^2 (B+2 C)+3 b^3 (25 B+49 C)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+\left(16 a^4 C-24 a^3 b B+24 a^2 b^2 C-57 a b^3 B-147 b^4 C\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+b\right)+(a+b) \left(16 a^4 C-24 a^3 b B+24 a^2 b^2 C-57 a b^3 B-147 b^4 C\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{315 b^4 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+b} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \left(-6 a^2 C+9 a b B+49 b^2 C\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 \left(-8 a^3 C+12 a^2 b B-13 a b^2 C-75 b^3 B\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(-16 a^3 C+12 a^2 b (2 B-C)+18 a b^2 (B-2 C)+3 b^3 (25 B-49 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(-16 a^4 C+24 a^3 b B-24 a^2 b^2 C+57 a b^3 B+147 b^4 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^5 d}+\frac{2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}",1,"(2*Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 18*a*b^2*(B + 2*C) + 3*b^3*(25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 24*a^2*b^2*C - 147*b^4*C)*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(315*b^4*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Sec[c + d*x]]*((2*(24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^4) + (2*Sec[c + d*x]^3*(9*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(63*b) + (2*Sec[c + d*x]^2*(9*a*b*B*Sin[c + d*x] - 6*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b^2) + (2*Sec[c + d*x]*(-12*a^2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] + 8*a^3*C*Sin[c + d*x] + 13*a*b^2*C*Sin[c + d*x]))/(315*b^3) + (2*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/d","A",0
815,1,655,397,22.780144,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 \sec (c+d x) \left(-4 a^2 C \sin (c+d x)+7 a b B \sin (c+d x)+25 b^2 C \sin (c+d x)\right)}{105 b^2}+\frac{2 \left(8 a^3 C-14 a^2 b B+19 a b^2 C+63 b^3 B\right) \sin (c+d x)}{105 b^3}+\frac{2 \sec ^2(c+d x) (a C \sin (c+d x)+7 b B \sin (c+d x))}{35 b}+\frac{2}{7} C \tan (c+d x) \sec ^2(c+d x)\right)}{d}-\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{a+b \sec (c+d x)} \left(-b (a+b) \left(8 a^2 C-2 a b (7 B+3 C)+b^2 (63 B+25 C)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+\left(8 a^3 C-14 a^2 b B+19 a b^2 C+63 b^3 B\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+(a+b) \left(8 a^3 C-14 a^2 b B+19 a b^2 C+63 b^3 B\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{105 b^3 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+b} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \left(-4 a^2 C+7 a b B+25 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \left(-8 a^2 C+2 a b (7 B-3 C)+b^2 (63 B-25 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(-8 a^3 C+14 a^2 b B-19 a b^2 C-63 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}+\frac{2 (a C+7 b B) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}",1,"(-2*Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + 19*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + 19*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))))/(105*b^3*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Sec[c + d*x]]*((2*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + 19*a*b^2*C)*Sin[c + d*x])/(105*b^3) + (2*Sec[c + d*x]^2*(7*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(35*b) + (2*Sec[c + d*x]*(7*a*b*B*Sin[c + d*x] - 4*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b^2) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/d","A",0
816,1,2905,314,27.9264853,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \left(-2 a^2 C+5 a b B+9 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} (-2 a C+5 b B-9 b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}+\frac{2 (5 b B-2 a C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 b d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((2*(5*a*b*B - 2*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*b^2) + (2*Sec[c + d*x]*(5*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(15*b) + (2*C*Sec[c + d*x]*Tan[c + d*x])/5))/d + (2*(-1/3*(a*B)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*C)/(15*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*b*C)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*B*Sqrt[Sec[c + d*x]])/(3*b*Sqrt[b + a*Cos[c + d*x]]) + (b*B*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (2*a*C*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*C*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*Sqrt[b + a*Cos[c + d*x]]) - (3*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(2*(a + b)*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*b*B - 2*a*C + 9*b*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*b*B + 2*a^2*C - 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^2*d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*b*B - 2*a*C + 9*b*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*b*B + 2*a^2*C - 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*b*B - 2*a*C + 9*b*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*b*B + 2*a^2*C - 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-5*a*b*B + 2*a^2*C - 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(5*b*B - 2*a*C + 9*b*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(5*b*B - 2*a*C + 9*b*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-5*a*b*B + 2*a^2*C - 9*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-5*a*b*B + 2*a^2*C - 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(5*b*B - 2*a*C + 9*b*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-5*a*b*B + 2*a^2*C - 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*b*B - 2*a*C + 9*b*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*b*B + 2*a^2*C - 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
817,1,480,256,20.5131612,"\int \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 (a C+3 b B) \sin (c+d x)}{3 b}+\frac{2}{3} C \tan (c+d x)\right)}{d}+\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{a+b \sec (c+d x)} \left((a C+3 b B) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b\right)+b (a+b) (3 B+C) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-\left((a+b) (a C+3 b B) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{3 b d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\sec (c+d x)} \sqrt{a \cos (c+d x)+b} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 (a-b) \sqrt{a+b} (a C+3 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 (a-b) \sqrt{a+b} (3 B-C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(2*Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((3*b*B + a*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2) - (a + b)*(3*b*B + a*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(3*B + C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*b*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Sqrt[a + b*Sec[c + d*x]]*((2*(3*b*B + a*C)*Sin[c + d*x])/(3*b) + (2*C*Tan[c + d*x])/3))/d","A",0
818,1,863,320,18.0330061,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 C \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{d}+\frac{2 \sqrt{a+b \sec (c+d x)} \left(a \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-b \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a \sqrt{\frac{b-a}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 i a B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) C E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-i (a-b) (B-C) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i a B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{\sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \sqrt{a+b} (a C+b (B-C)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(2*C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d + (2*Sqrt[a + b*Sec[c + d*x]]*(a*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 2*a*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + a*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + (2*I)*a*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*a*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*C*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(B - C)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
819,1,1107,344,23.9599487,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(a \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 i a C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) B E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i (a-b) C F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 i a C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{\sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)}","\frac{\sqrt{a+b} (B+2 C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{\sqrt{a+b} (2 a C+b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}+\frac{B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(Sqrt[a + b*Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 2*a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (2*I)*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*B*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*C*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2))","C",0
820,1,1149,429,27.9960879,"\int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{B \sqrt{a+b \sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b B \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 a^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 a^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-2 b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+a b B \tan \left(\frac{1}{2} (c+d x)\right)+4 a^2 C \tan \left(\frac{1}{2} (c+d x)\right)+4 a b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (b B+4 a C) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a (2 a B-b B+4 b C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 a^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 a b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \left(4 a^2 B+4 a b C-b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{(4 a C+b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a d}+\frac{\sqrt{a+b} (2 a (B+2 C)+b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{(a-b) \sqrt{a+b} (4 a C+b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a b d}+\frac{B \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"(B*Sqrt[a + b*Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*b*B*Tan[(c + d*x)/2] + b^2*B*Tan[(c + d*x)/2] + 4*a^2*C*Tan[(c + d*x)/2] + 4*a*b*C*Tan[(c + d*x)/2] - 2*a*b*B*Tan[(c + d*x)/2]^3 - 8*a^2*C*Tan[(c + d*x)/2]^3 + a*b*B*Tan[(c + d*x)/2]^5 - b^2*B*Tan[(c + d*x)/2]^5 + 4*a^2*C*Tan[(c + d*x)/2]^5 - 4*a*b*C*Tan[(c + d*x)/2]^5 + 8*a^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(b*B + 4*a*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(2*a*B - b*B + 4*b*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
821,1,890,573,26.6054236,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left((a+b) \left(48 C a^5-88 b B a^4+108 b^2 C a^3-363 b^3 B a^2-2088 b^4 C a-1617 b^5 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+b (a+b) \left(-48 C a^4+4 b (22 B+9 C) a^3-6 b^2 (11 B+24 C) a^2+3 b^3 (143 B+471 C) a+3 b^4 (539 B+225 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(48 C a^5-88 b B a^4+108 b^2 C a^3-363 b^3 B a^2-2088 b^4 C a-1617 b^5 B\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)\right) (a+b \sec (c+d x))^{3/2}}{3465 b^4 d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}+\frac{\cos (c+d x) \left(\frac{2}{99} (11 b B \sin (c+d x)+12 a C \sin (c+d x)) \sec ^4(c+d x)+\frac{2}{11} b C \tan (c+d x) \sec ^4(c+d x)+\frac{2 \left(3 C \sin (c+d x) a^2+110 b B \sin (c+d x) a+81 b^2 C \sin (c+d x)\right) \sec ^3(c+d x)}{693 b}+\frac{2 \left(-18 C \sin (c+d x) a^3+33 b B \sin (c+d x) a^2+606 b^2 C \sin (c+d x) a+539 b^3 B \sin (c+d x)\right) \sec ^2(c+d x)}{3465 b^2}+\frac{2 \left(24 C \sin (c+d x) a^4-44 b B \sin (c+d x) a^3+57 b^2 C \sin (c+d x) a^2+968 b^3 B \sin (c+d x) a+675 b^4 C \sin (c+d x)\right) \sec (c+d x)}{3465 b^3}-\frac{2 \left(48 C a^5-88 b B a^4+108 b^2 C a^3-363 b^3 B a^2-2088 b^4 C a-1617 b^5 B\right) \sin (c+d x)}{3465 b^4}\right) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x))}","-\frac{2 \left(-24 a^2 C+44 a b B-81 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^3 d}+\frac{2 \left(-48 a^3 C+88 a^2 b B-204 a b^2 C+539 b^3 B\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{3465 b^3 d}+\frac{2 \left(-48 a^4 C+88 a^3 b B-144 a^2 b^2 C+429 a b^3 B+675 b^4 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3465 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(-48 a^4 C+4 a^3 b (22 B-9 C)+6 a^2 b^2 (11 B-24 C)+3 a b^3 (143 B-471 C)-3 b^4 (539 B-225 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(-48 a^5 C+88 a^4 b B-108 a^3 b^2 C+363 a^2 b^3 B+2088 a b^4 C+1617 b^5 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^5 d}+\frac{2 (11 b B-6 a C) \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{99 b^2 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{11 b d}",1,"(2*(a + b*Sec[c + d*x])^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C) - 6*a^2*b^2*(11*B + 24*C) + 3*b^4*(539*B + 225*C) + 3*a*b^3*(143*B + 471*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(3465*b^4*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((-2*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Sin[c + d*x])/(3465*b^4) + (2*Sec[c + d*x]^4*(11*b*B*Sin[c + d*x] + 12*a*C*Sin[c + d*x]))/99 + (2*Sec[c + d*x]^3*(110*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 81*b^2*C*Sin[c + d*x]))/(693*b) + (2*Sec[c + d*x]^2*(33*a^2*b*B*Sin[c + d*x] + 539*b^3*B*Sin[c + d*x] - 18*a^3*C*Sin[c + d*x] + 606*a*b^2*C*Sin[c + d*x]))/(3465*b^2) + (2*Sec[c + d*x]*(-44*a^3*b*B*Sin[c + d*x] + 968*a*b^3*B*Sin[c + d*x] + 24*a^4*C*Sin[c + d*x] + 57*a^2*b^2*C*Sin[c + d*x] + 675*b^4*C*Sin[c + d*x]))/(3465*b^3) + (2*b*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x]))","A",0
822,1,3766,475,38.1767463,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \left(-8 a^2 C+18 a b B-49 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(-8 a^3 C+18 a^2 b B-39 a b^2 C-75 b^3 B\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^3 C-6 a^2 b (3 B-C)-3 a b^2 (57 B-13 C)+3 b^3 (25 B-49 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(-8 a^4 C+18 a^3 b B-33 a^2 b^2 C-246 a b^3 B-147 b^4 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}+\frac{2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}",1,"(Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((2*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^3) + (2*Sec[c + d*x]^3*(9*b*B*Sin[c + d*x] + 10*a*C*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2*(72*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b) + (2*Sec[c + d*x]*(9*a^2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] - 4*a^3*C*Sin[c + d*x] + 88*a*b^2*C*Sin[c + d*x]))/(315*b^2) + (2*b*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])) - (2*((2*a^3*B)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (82*a*b*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (11*a^2*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*C)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^2*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (5*b^2*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*C*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*C*Sqrt[Sec[c + d*x]])/(315*b*Sqrt[b + a*Cos[c + d*x]]) + (13*a*b*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (82*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (11*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) - (7*a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(2*(a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*(-1/315*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(57*B + 13*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
823,1,671,387,31.5416604,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(\frac{2 \sec (c+d x) \left(3 a^2 C \sin (c+d x)+42 a b B \sin (c+d x)+25 b^2 C \sin (c+d x)\right)}{105 b}-\frac{2 \left(6 a^3 C-21 a^2 b B-82 a b^2 C-63 b^3 B\right) \sin (c+d x)}{105 b^2}+\frac{2}{35} \sec ^2(c+d x) (8 a C \sin (c+d x)+7 b B \sin (c+d x))+\frac{2}{7} b C \tan (c+d x) \sec ^2(c+d x)\right)}{d (a \cos (c+d x)+b)}+\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} (a+b \sec (c+d x))^{3/2} \left(b (a+b) \left(-6 a^2 C+3 a b (7 B+19 C)+b^2 (63 B+25 C)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+\left(6 a^3 C-21 a^2 b B-82 a b^2 C-63 b^3 B\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+(a+b) \left(6 a^3 C-21 a^2 b B-82 a b^2 C-63 b^3 B\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{105 b^2 d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \left(-6 a^2 C+21 a b B+25 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b d}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^2 C+a b (21 B-57 C)-b^2 (63 B-25 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^3 C+21 a^2 b B+82 a b^2 C+63 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 (7 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}",1,"(2*(a + b*Sec[c + d*x])^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 82*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-6*a^2*C + 3*a*b*(7*B + 19*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 82*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))))/(105*b^2*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((-2*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 82*a*b^2*C)*Sin[c + d*x])/(105*b^2) + (2*Sec[c + d*x]^2*(7*b*B*Sin[c + d*x] + 8*a*C*Sin[c + d*x]))/35 + (2*Sec[c + d*x]*(42*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b) + (2*b*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x]))","A",0
824,1,581,312,18.7511801,"\int (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(\frac{2 \left(3 a^2 C+20 a b B+9 b^2 C\right) \sin (c+d x)}{15 b}+\frac{2}{15} \sec (c+d x) (6 a C \sin (c+d x)+5 b B \sin (c+d x))+\frac{2}{5} b C \tan (c+d x) \sec (c+d x)\right)}{d (a \cos (c+d x)+b)}-\frac{2 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} (a+b \sec (c+d x))^{3/2} \left(\left(3 a^2 C+20 a b B+9 b^2 C\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)+(a+b) \left(3 a^2 C+20 a b B+9 b^2 C\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-b (a+b) (3 a (5 B+C)+b (5 B+9 C)) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{15 b d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 C+20 a b B+9 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}+\frac{2 (3 a C+5 b B) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 (a-b) \sqrt{a+b} (15 a B-3 a C-5 b B+9 b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"(-2*(a + b*Sec[c + d*x])^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(20*a*b*B + 3*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(3*a*(5*B + C) + b*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (20*a*b*B + 3*a^2*C + 9*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))))/(15*b*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((2*(20*a*b*B + 3*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*b) + (2*Sec[c + d*x]*(5*b*B*Sin[c + d*x] + 6*a*C*Sin[c + d*x]))/15 + (2*b*C*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x]))","A",0
825,1,6017,380,24.3838057,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sqrt{a+b} \left(3 a^2 C+a b (6 B-4 C)-b^2 (3 B-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{2 (a-b) \sqrt{a+b} (4 a C+3 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{2 a B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 b C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"Result too large to show","B",0
826,1,971,361,17.186955,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 b C \cos (c+d x) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x))}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 a b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a^2 B \tan \left(\frac{1}{2} (c+d x)\right)+a b B \tan \left(\frac{1}{2} (c+d x)\right)-2 b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-2 a b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (a B-2 b C) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(C a^2+2 b (B-C) a-b^2 (B+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 a^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} (a (B+4 C)+2 b (B-C)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{(a-b) \sqrt{a+b} (a B-2 b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{\sqrt{a+b} (2 a C+3 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{a B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}",1,"(2*b*C*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])) + ((a + b*Sec[c + d*x])^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a^2*B*Tan[(c + d*x)/2] + a*b*B*Tan[(c + d*x)/2] - 2*a*b*C*Tan[(c + d*x)/2] - 2*b^2*C*Tan[(c + d*x)/2] - 2*a^2*B*Tan[(c + d*x)/2]^3 + 4*a*b*C*Tan[(c + d*x)/2]^3 + a^2*B*Tan[(c + d*x)/2]^5 - a*b*B*Tan[(c + d*x)/2]^5 - 2*a*b*C*Tan[(c + d*x)/2]^5 + 2*b^2*C*Tan[(c + d*x)/2]^5 + 6*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(a*B - 2*b*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(2*a*b*(B - C) + a^2*C - b^2*(B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
827,1,1580,428,19.2881645,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a B \sqrt{a+b \sec (c+d x)} \sin (2 (c+d x))}{4 d}-\frac{\sqrt{a+b \sec (c+d x)} \left(-5 b^2 \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a^2 \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a b \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-10 a b \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 a^2 \sqrt{\frac{b-a}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 i a^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 i b^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-24 i a b C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+5 b^2 \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+5 a b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+4 a^2 \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)+4 a b \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) (5 b B+4 a C) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i (a-b) (2 a B+b (B+4 C)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i b^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-24 i a b C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 \sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \left(4 a^2 B+12 a b C+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{(4 a C+5 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{\sqrt{a+b} (2 a B+4 a C+5 b B+8 b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} (4 a C+5 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 b d}+\frac{a B \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"(a*B*Sqrt[a + b*Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) - (Sqrt[a + b*Sec[c + d*x]]*(5*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 5*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 4*a^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + 4*a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 10*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - 8*a^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + 5*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 5*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 4*a^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - 4*a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - (8*I)*a^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(5*b*B + 4*a*C)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(2*a*B + b*(B + 4*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
828,1,1516,520,19.0189631,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{1}{12} a B \sin (c+d x)+\frac{1}{24} (7 b B+6 a C) \sin (2 (c+d x))+\frac{1}{12} a B \sin (3 (c+d x))\right)}{d}+\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(16 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+30 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-32 a^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-60 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+72 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+36 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+16 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)+3 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+3 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+30 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+30 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(16 B a^2+30 b C a+3 b^2 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a \left(12 C a^2+b (26 B-6 C) a+b^2 (24 C-7 B)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+72 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+36 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 a d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(16 a^2 B+30 a b C+3 b^2 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 a d}+\frac{\sqrt{a+b} \left(16 a^2 B+12 a^2 C+14 a b B+30 a b C+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2 B+30 a b C+3 b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \left(8 a^3 C+12 a^2 b B+6 a b^2 C-b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a^2 d}+\frac{(6 a C+7 b B) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}+\frac{a B \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((a*B*Sin[c + d*x])/12 + ((7*b*B + 6*a*C)*Sin[2*(c + d*x)])/24 + (a*B*Sin[3*(c + d*x)])/12))/d + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*B*Tan[(c + d*x)/2] + 16*a^2*b*B*Tan[(c + d*x)/2] + 3*a*b^2*B*Tan[(c + d*x)/2] + 3*b^3*B*Tan[(c + d*x)/2] + 30*a^2*b*C*Tan[(c + d*x)/2] + 30*a*b^2*C*Tan[(c + d*x)/2] - 32*a^3*B*Tan[(c + d*x)/2]^3 - 6*a*b^2*B*Tan[(c + d*x)/2]^3 - 60*a^2*b*C*Tan[(c + d*x)/2]^3 + 16*a^3*B*Tan[(c + d*x)/2]^5 - 16*a^2*b*B*Tan[(c + d*x)/2]^5 + 3*a*b^2*B*Tan[(c + d*x)/2]^5 - 3*b^3*B*Tan[(c + d*x)/2]^5 + 30*a^2*b*C*Tan[(c + d*x)/2]^5 - 30*a*b^2*C*Tan[(c + d*x)/2]^5 + 72*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 36*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 72*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 36*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^2*B + 3*b^2*B + 30*a*b*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(a*b*(26*B - 6*C) + 12*a^2*C + b^2*(-7*B + 24*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
829,1,4227,565,27.025401,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \left(-8 a^2 C+22 a b B-81 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(-40 a^3 C+110 a^2 b B-335 a b^2 C-539 b^3 B\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(-40 a^4 C+110 a^3 b B-285 a^2 b^2 C-1254 a b^3 B-675 b^4 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3465 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(40 a^4 C-a^3 b (110 B-30 C)-15 a^2 b^2 (121 B-19 C)+6 a b^3 (209 B-505 C)-3 b^4 (539 B-225 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(-40 a^5 C+110 a^4 b B-255 a^3 b^2 C-3069 a^2 b^3 B-3705 a b^4 C-1617 b^5 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^4 d}+\frac{2 (11 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sin[c + d*x])/(3465*b^3) + (2*Sec[c + d*x]^4*(11*b^2*B*Sin[c + d*x] + 23*a*b*C*Sin[c + d*x]))/99 + (2*Sec[c + d*x]^3*(209*a*b*B*Sin[c + d*x] + 113*a^2*C*Sin[c + d*x] + 81*b^2*C*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^2*(825*a^2*b*B*Sin[c + d*x] + 539*b^3*B*Sin[c + d*x] + 15*a^3*C*Sin[c + d*x] + 1145*a*b^2*C*Sin[c + d*x]))/(3465*b) + (2*Sec[c + d*x]*(55*a^3*b*B*Sin[c + d*x] + 1793*a*b^3*B*Sin[c + d*x] - 20*a^4*C*Sin[c + d*x] + 1025*a^2*b^2*C*Sin[c + d*x] + 675*b^4*C*Sin[c + d*x]))/(3465*b^2) + (2*b^2*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^2) - (2*((2*a^4*B)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a^2*b*B)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^3*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (17*a^3*C)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*C)/(693*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (247*a*b^2*C)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (124*a^3*B*Sqrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*B*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) + (38*a*b^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (8*a^6*C*Sqrt[Sec[c + d*x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (7*a^4*C*Sqrt[Sec[c + d*x]])/(99*b*Sqrt[b + a*Cos[c + d*x]]) - (26*a^2*b*C*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]) + (15*b^3*C*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (7*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (8*a^6*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (17*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*b*Sqrt[b + a*Cos[c + d*x]]) - (247*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*(a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(121*B + 19*C) + 3*b^4*(539*B + 225*C) + 6*a*b^3*(209*B + 505*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*b^3*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*(-1/3465*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(121*B + 19*C) + 3*b^4*(539*B + 225*C) + 6*a*b^3*(209*B + 505*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(121*B + 19*C) + 3*b^4*(539*B + 225*C) + 6*a*b^3*(209*B + 505*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(121*B + 19*C) + 3*b^4*(539*B + 225*C) + 6*a*b^3*(209*B + 505*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(121*B + 19*C) + 3*b^4*(539*B + 225*C) + 6*a*b^3*(209*B + 505*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(121*B + 19*C) + 3*b^4*(539*B + 225*C) + 6*a*b^3*(209*B + 505*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3465*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(121*B + 19*C) + 3*b^4*(539*B + 225*C) + 6*a*b^3*(209*B + 505*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3465*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
830,1,3781,469,26.7995234,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \left(-10 a^2 C+45 a b B+49 b^2 C\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}+\frac{2 \left(-10 a^3 C+45 a^2 b B+114 a b^2 C+75 b^3 B\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}-\frac{2 (a-b) \sqrt{a+b} \left(-10 a^3 C+15 a^2 b (3 B-11 C)-6 a b^2 (60 B-19 C)+3 b^3 (25 B-49 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(-10 a^4 C+45 a^3 b B+279 a^2 b^2 C+435 a b^3 B+147 b^4 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 (9 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^2) + (2*Sec[c + d*x]^3*(9*b^2*B*Sin[c + d*x] + 19*a*b*C*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2*(135*a*b*B*Sin[c + d*x] + 75*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/315 + (2*Sec[c + d*x]*(135*a^2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] + 5*a^3*C*Sin[c + d*x] + 163*a*b^2*C*Sin[c + d*x]))/(315*b) + (2*b^2*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2) + (2*(-1/7*(a^3*B)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a*b^2*B)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^4*C)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a^2*b*C)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^3*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^4*B*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*b*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (5*b^3*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (124*a^3*C*Sqrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*C*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) + (38*a*b^2*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (29*a^2*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (7*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*(a + b)*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*C + 15*a^2*b*(3*B + 11*C) + 6*a*b^2*(60*B + 19*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*C + 15*a^2*b*(3*B + 11*C) + 6*a*b^2*(60*B + 19*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*C + 15*a^2*b*(3*B + 11*C) + 6*a*b^2*(60*B + 19*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-10*a^3*C + 15*a^2*b*(3*B + 11*C) + 6*a*b^2*(60*B + 19*C) + 3*b^3*(25*B + 49*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-10*a^3*C + 15*a^2*b*(3*B + 11*C) + 6*a*b^2*(60*B + 19*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-10*a^3*C + 15*a^2*b*(3*B + 11*C) + 6*a*b^2*(60*B + 19*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*C + 15*a^2*b*(3*B + 11*C) + 6*a*b^2*(60*B + 19*C) + 3*b^3*(25*B + 49*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 279*a^2*b^2*C - 147*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
831,1,2913,384,23.4641716,"\int (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \left(15 a^2 C+56 a b B+25 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 d}+\frac{2 (a-b) \sqrt{a+b} \left(15 a^2 (7 B-C)-8 a b (7 B-15 C)+b^2 (63 B-25 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b d}-\frac{2 (a-b) \sqrt{a+b} \left(15 a^3 C+161 a^2 b B+145 a b^2 C+63 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}+\frac{2 (5 a C+7 b B) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sin[c + d*x])/(105*b) + (2*Sec[c + d*x]^2*(7*b^2*B*Sin[c + d*x] + 15*a*b*C*Sin[c + d*x]))/35 + (2*Sec[c + d*x]*(77*a*b*B*Sin[c + d*x] + 45*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/105 + (2*b^2*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])^2) + (2*((-23*a^2*b*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*b^3*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^3*C)/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a*b^2*C)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (8*a*b^2*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (a^4*C*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*b*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (5*b^3*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (23*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (3*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (29*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(105*b*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*(-1/105*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(105*b*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[b + a*Cos[c + d*x]]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-3*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + ((Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/((b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x])))^(3/2) + (161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + ((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sec[(c + d*x)/2]^2*(-1 + Tan[(c + d*x)/2]^2))/2 - (2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*Sec[c + d*x]*(-1/2*(b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*B + C) + 8*a*b*(7*B + 15*C) + b^2*(63*B + 25*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x]*Tan[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]))/(105*b*Sqrt[Sec[(c + d*x)/2]^2])))","B",0
832,1,7094,442,25.6544018,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2 C+35 a b B+9 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}-\frac{2 a^2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 \sqrt{a+b} \left(15 a^3 C+a^2 b (45 B-23 C)-a b^2 (35 B-17 C)+b^3 (5 B-9 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}+\frac{2 b (8 a C+5 b B) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 b C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"Result too large to show","B",0
833,1,7745,433,26.1808626,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(3 a^2 (B+6 C)+2 a b (9 B-7 C)-2 b^2 (3 B-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 B-14 a b C-6 b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{b (3 a B-2 b C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{a \sqrt{a+b} (2 a C+5 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{a B \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}",1,"Result too large to show","B",0
834,1,1326,450,19.4686042,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^2(c+d x) \left(\frac{1}{4} B \sin (2 (c+d x)) a^2+2 b^2 C \sin (c+d x)\right) (a+b \sec (c+d x))^{5/2}}{d (b+a \cos (c+d x))^2}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-9 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+9 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-18 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+16 a b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+8 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+30 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+40 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+9 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+9 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+4 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-8 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-8 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+4 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(4 C a^2+9 b B a-8 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(2 B a^3-b (B-12 C) a^2+12 b^2 (B-C) a-4 b^3 (B+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+40 a^2 b C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{5/2}}{4 d (b+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \left(2 a^2 (B+2 C)+3 a b (3 B+8 C)+8 b^2 (B-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(4 a^2 C+9 a b B-8 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 b d}-\frac{\sqrt{a+b} \left(4 a^2 B+20 a b C+15 b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{a (4 a C+7 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{a B \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(2*b^2*C*Sin[c + d*x] + (a^2*B*Sin[2*(c + d*x)])/4))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x])^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(9*a^2*b*B*Tan[(c + d*x)/2] + 9*a*b^2*B*Tan[(c + d*x)/2] + 4*a^3*C*Tan[(c + d*x)/2] + 4*a^2*b*C*Tan[(c + d*x)/2] - 8*a*b^2*C*Tan[(c + d*x)/2] - 8*b^3*C*Tan[(c + d*x)/2] - 18*a^2*b*B*Tan[(c + d*x)/2]^3 - 8*a^3*C*Tan[(c + d*x)/2]^3 + 16*a*b^2*C*Tan[(c + d*x)/2]^3 + 9*a^2*b*B*Tan[(c + d*x)/2]^5 - 9*a*b^2*B*Tan[(c + d*x)/2]^5 + 4*a^3*C*Tan[(c + d*x)/2]^5 - 4*a^2*b*C*Tan[(c + d*x)/2]^5 - 8*a*b^2*C*Tan[(c + d*x)/2]^5 + 8*b^3*C*Tan[(c + d*x)/2]^5 + 8*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 40*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 40*a^2*b*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(9*a*b*B + 4*a^2*C - 8*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(2*a^3*B - a^2*b*(B - 12*C) + 12*a*b^2*(B - C) - 4*b^3*(B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
835,1,1530,518,19.5510739,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{1}{12} B \sin (c+d x) a^2+\frac{1}{12} B \sin (3 (c+d x)) a^2+\frac{1}{24} (13 b B+6 a C) \sin (2 (c+d x)) a\right)}{d}+\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(16 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-33 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-54 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+54 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-32 a^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-66 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-108 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+120 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+180 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+16 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)+33 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+33 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+54 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+54 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(16 B a^2+54 b C a+33 b^2 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(12 C a^3+b (38 B-6 C) a^2+b^2 (72 C-13 B) a+24 b^3 (B-C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+120 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+180 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(16 a^2 B+54 a b C+33 b^2 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{\sqrt{a+b} \left(4 a^2 (4 B+3 C)+a b (26 B+54 C)+3 b^2 (11 B+16 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2 B+54 a b C+33 b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 b d}-\frac{\sqrt{a+b} \left(8 a^3 C+20 a^2 b B+30 a b^2 C+5 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a d}+\frac{a (2 a C+3 b B) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{a B \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((a^2*B*Sin[c + d*x])/12 + (a*(13*b*B + 6*a*C)*Sin[2*(c + d*x)])/24 + (a^2*B*Sin[3*(c + d*x)])/12))/d + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*B*Tan[(c + d*x)/2] + 16*a^2*b*B*Tan[(c + d*x)/2] + 33*a*b^2*B*Tan[(c + d*x)/2] + 33*b^3*B*Tan[(c + d*x)/2] + 54*a^2*b*C*Tan[(c + d*x)/2] + 54*a*b^2*C*Tan[(c + d*x)/2] - 32*a^3*B*Tan[(c + d*x)/2]^3 - 66*a*b^2*B*Tan[(c + d*x)/2]^3 - 108*a^2*b*C*Tan[(c + d*x)/2]^3 + 16*a^3*B*Tan[(c + d*x)/2]^5 - 16*a^2*b*B*Tan[(c + d*x)/2]^5 + 33*a*b^2*B*Tan[(c + d*x)/2]^5 - 33*b^3*B*Tan[(c + d*x)/2]^5 + 54*a^2*b*C*Tan[(c + d*x)/2]^5 - 54*a*b^2*C*Tan[(c + d*x)/2]^5 + 120*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 180*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 180*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^2*B + 33*b^2*B + 54*a*b*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(a^2*b*(38*B - 6*C) + 24*b^3*(B - C) + 12*a^3*C + a*b^2*(-13*B + 72*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
836,1,5172,617,24.2929682,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\left(36 a^2 B+104 a b C+59 b^2 B\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\left(128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\sqrt{a+b} \left(8 a^3 (9 B+16 C)+4 a^2 b (71 B+52 C)+2 a b^2 (59 B+132 C)+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{(a-b) \sqrt{a+b} \left(128 a^3 C+284 a^2 b B+264 a b^2 C+15 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \left(48 a^4 B+160 a^3 b C+120 a^2 b^2 B+40 a b^3 C-5 b^4 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}+\frac{a (8 a C+11 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}",1,"Result too large to show","B",0
837,1,3426,411,24.6989628,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{2 \left(-24 a^2 C+28 a b B-25 b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{105 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(-48 a^3 C+56 a^2 b B-44 a b^2 C+63 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^5 d}-\frac{2 \sqrt{a+b} \left(-48 a^3 C+4 a^2 b (14 B+3 C)-2 a b^2 (7 B+22 C)+b^3 (63 B-25 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}+\frac{2 (7 b B-6 a C) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b^2 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 b d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((2*(56*a^2*b*B + 63*b^3*B - 48*a^3*C - 44*a*b^2*C)*Sin[c + d*x])/(105*b^4) + (2*Sec[c + d*x]^2*(7*b*B*Sin[c + d*x] - 6*a*C*Sin[c + d*x]))/(35*b^2) + (2*Sec[c + d*x]*(-28*a*b*B*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b^3) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(7*b)))/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*((-3*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^2*B)/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^3*C)/(35*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (44*a*C)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*B*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (7*a*B*Sqrt[Sec[c + d*x]])/(15*b*Sqrt[b + a*Cos[c + d*x]]) + (5*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (16*a^4*C*Sqrt[Sec[c + d*x]])/(35*b^4*Sqrt[b + a*Cos[c + d*x]]) + (32*a^2*C*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (3*a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (16*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^4*Sqrt[b + a*Cos[c + d*x]]) + (44*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(2*a*b^2*(7*B - 22*C) + 4*a^2*b*(14*B - 3*C) - 48*a^3*C + b^3*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(2*a*b^2*(7*B - 22*C) + 4*a^2*b*(14*B - 3*C) - 48*a^3*C + b^3*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(2*a*b^2*(7*B - 22*C) + 4*a^2*b*(14*B - 3*C) - 48*a^3*C + b^3*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(2*a*b^2*(7*B - 22*C) + 4*a^2*b*(14*B - 3*C) - 48*a^3*C + b^3*(63*B + 25*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(2*a*b^2*(7*B - 22*C) + 4*a^2*b*(14*B - 3*C) - 48*a^3*C + b^3*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(2*a*b^2*(7*B - 22*C) + 4*a^2*b*(14*B - 3*C) - 48*a^3*C + b^3*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(2*a*b^2*(7*B - 22*C) + 4*a^2*b*(14*B - 3*C) - 48*a^3*C + b^3*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 44*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
838,1,3000,329,21.96774,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{2 (a-b) \sqrt{a+b} \left(-8 a^2 C+10 a b B-9 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}+\frac{2 \sqrt{a+b} \left(-8 a^2 C+2 a b (5 B+C)+b^2 (5 B-9 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}+\frac{2 (5 b B-4 a C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((2*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*b^3) + (2*Sec[c + d*x]*(5*b*B*Sin[c + d*x] - 4*a*C*Sin[c + d*x]))/(15*b^2) + (2*C*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/(d*Sqrt[a + b*Sec[c + d*x]]) - (2*((2*a*B)/(3*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*C)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^2*C)/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (B*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*B*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^3*C*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (7*a*C*Sqrt[Sec[c + d*x]])/(15*b*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (3*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]*(-1/15*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-10*a*b*B + 8*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
839,1,372,261,16.1839426,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(\frac{2 (3 b B-2 a C) \sin (c+d x)}{3 b^2}+\frac{2 C \tan (c+d x)}{3 b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left((3 b B-2 a C) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)+2 b (b (3 B+C)-2 a C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 (a+b) (2 a C-3 b B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} (3 b B-2 a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d}-\frac{2 \sqrt{a+b} (-2 a C+3 b B-b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}",1,"(2*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(-2*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (3*b*B - 2*a*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^2*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*((2*(3*b*B - 2*a*C)*Sin[c + d*x])/(3*b^2) + (2*C*Tan[c + d*x])/(3*b)))/(d*Sqrt[a + b*Sec[c + d*x]])","A",0
840,1,312,210,14.3689791,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 C \tan (c+d x) (a \cos (c+d x)+b)}{b d \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\sec (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-2 b (B+C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+C \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 C (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{b d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{a+b} (B-C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}",1,"(-2*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*C*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(B + C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + C*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])*Tan[c + d*x])/(b*d*Sqrt[a + b*Sec[c + d*x]])","A",0
841,1,145,208,2.3330465,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec (c+d x) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \left((C-B) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((-B + C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
842,1,1027,348,16.6432731,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(-a \sqrt{\frac{b-a}{a+b}} B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^3\left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{b-a}{a+b}} B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 i b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 i a C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a \sqrt{\frac{b-a}{a+b}} B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan \left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{b-a}{a+b}} B \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan \left(\frac{1}{2} (c+d x)\right)+2 i b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 i a C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-i (a-b) B E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i (b B-a C) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a \sqrt{\frac{b-a}{a+b}} d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} (b B-2 a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{a d}+\frac{B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}+\frac{B (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}",1,"(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]*Sqrt[1 - Tan[(c + d*x)/2]^2] + b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]*Sqrt[1 - Tan[(c + d*x)/2]^2] - a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3*Sqrt[1 - Tan[(c + d*x)/2]^2] + b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3*Sqrt[1 - Tan[(c + d*x)/2]^2] + (2*I)*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*B*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(b*B - a*C)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*d*Sqrt[a + b*Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
843,1,3953,471,26.0299943,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 a (b B-a C) \tan (c+d x) \sec ^2(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-6 a^2 C+5 a b B+b^2 C\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \left(-24 a^3 C+20 a^2 b B+9 a b^2 C-5 b^3 B\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \left(-48 a^3 C+4 a^2 b (10 B-9 C)+6 a b^2 (5 B-2 C)+b^3 (5 B-9 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d \sqrt{a+b}}+\frac{2 \left(-48 a^4 C+40 a^3 b B+24 a^2 b^2 C-25 a b^3 B+9 b^4 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^5 d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(40*a^3*b*B - 25*a*b^3*B - 48*a^4*C + 24*a^2*b^2*C + 9*b^4*C)*Sin[c + d*x])/(15*b^4*(-a^2 + b^2)) + (2*Sec[c + d*x]*(5*b*B*Sin[c + d*x] - 9*a*C*Sin[c + d*x]))/(15*b^3) - (2*(a^3*b*B*Sin[c + d*x] - a^4*C*Sin[c + d*x]))/(b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*C*Sec[c + d*x]*Tan[c + d*x])/(5*b^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) + (2*(b + a*Cos[c + d*x])*((5*a*B)/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*B)/(3*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^4*C)/(5*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^2*C)/(5*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*b*C)/(5*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*B*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (7*a^2*B*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (b*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (4*a*C*Sqrt[Sec[c + d*x]])/(5*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (16*a^5*C*Sqrt[Sec[c + d*x]])/(5*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (12*a^3*C*Sqrt[Sec[c + d*x]])/(5*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (5*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (3*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (16*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*B + 2*C) + b^3*(5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^4*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*B + 2*C) + b^3*(5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^4*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*B + 2*C) + b^3*(5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*B + 2*C) + b^3*(5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*B + 2*C) + b^3*(5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*B + 2*C) + b^3*(5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*B + 2*C) + b^3*(5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-40*a^3*b*B + 25*a*b^3*B + 48*a^4*C - 24*a^2*b^2*C - 9*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b^4*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
844,1,3460,329,24.5981962,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 a^2 (b B-a C) \tan (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-8 a^3 C+6 a^2 b B+5 a b^2 C-3 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b}}-\frac{2 (2 a+b) (-4 a C+3 b B-b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b}}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)) + (2*(a^2*b*B*Sin[c + d*x] - a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*C*Tan[c + d*x])/(3*b^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])*((2*a^2*B)/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b*B)/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a*C)/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*C)/(3*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*B*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*C*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (7*a^2*C*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (b*C*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (5*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(-4*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2)*(-1/3*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(-4*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(-4*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(-2*a^2 - a*b + b^2)*(-4*a*C + b*(3*B + C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(-2*a^2 - a*b + b^2)*(-4*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(-2*a^2 - a*b + b^2)*(-4*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(-4*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
845,1,466,275,17.5667387,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}-\frac{2 \left(a b B \sin (c+d x)-a^2 C \sin (c+d x)\right)}{b \left(b^2-a^2\right) (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b) \left(\left(2 a^2 C-a b B-b^2 C\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 (a+b) \left(2 a^2 C-a b B-b^2 C\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 b (a+b) (b (B+C)-2 a C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{b^2 d \left(b^2-a^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{3/2}}","\frac{2 a (b B-a C) \tan (c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^3 d \sqrt{a+b}}+\frac{2 (b (B-C)-2 a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(a*b*B - 2*a^2*C + b^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) - (2*(a*b*B*Sin[c + d*x] - a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) + (2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-(a*b*B) + 2*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-2*a*C + b*(B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-(a*b*B) + 2*a^2*C - b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^2*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2))","A",0
846,1,426,254,14.8711511,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(\frac{2 (b B \sin (c+d x)-a C \sin (c+d x))}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{2 (b B-a C) \sin (c+d x)}{b \left(b^2-a^2\right)}\right)}{d (a+b \sec (c+d x))^{3/2}}-\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b) \left(-\left((b B-a C) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)+2 b (a+b) (B-C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 (a+b) (a C-b B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \left(b^3-a^2 b\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{3/2}}","-\frac{2 (b B-a C) \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}+\frac{2 (B+C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((-2*(b*B - a*C)*Sin[c + d*x])/(b*(-a^2 + b^2)) + (2*(b*B*Sin[c + d*x] - a*C*Sin[c + d*x]))/((-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-(b*B) + a*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(B - C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (b*B - a*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/((-(a^2*b) + b^3)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2))","A",0
847,1,1445,376,14.4370932,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(\frac{2 (a C-b B) \sin (c+d x)}{a \left(a^2-b^2\right)}-\frac{2 \left(a b C \sin (c+d x)-b^2 B \sin (c+d x)\right)}{a \left(a^2-b^2\right) (b+a \cos (c+d x))}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-b^2 \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 a^2 \sqrt{\frac{b-a}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i a^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+2 i b^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-a^2 \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)+i (a-b) (a C-b B) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i (a-b) (2 b B+a (B-C)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i a^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i b^2 B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{2 b (b B-a C) \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}+\frac{2 (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((2*(-(b*B) + a*C)*Sin[c + d*x])/(a*(a^2 - b^2)) - (2*(-(b^2*B*Sin[c + d*x]) + a*b*C*Sin[c + d*x]))/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - a^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 2*a^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - a^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - (2*I)*a^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*b^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(-(b*B) + a*C)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(2*b*B + a*(B - C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",1
848,1,1597,427,19.3013461,"\int \frac{\cos ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \left(\frac{2 \left(a b^2 C \sin (c+d x)-b^3 B \sin (c+d x)\right)}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))}-\frac{2 b (b B-a C) \sin (c+d x)}{a^2 \left(b^2-a^2\right)}\right)}{d (a+b \sec (c+d x))^{3/2}}-\frac{(b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-4 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a^3 B \tan \left(\frac{1}{2} (c+d x)\right)-3 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)-3 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+2 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+2 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(B a^2+2 b C a-3 b^2 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a (a+b) (a C-b B) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 b^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 a b^2 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a^2 \left(a^2-b^2\right) d (a+b \sec (c+d x))^{3/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\sqrt{a+b} (3 b B-2 a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{b \left(a^2 B+2 a b C-3 b^2 B\right) \tan (c+d x)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(a^2 B+2 a b C-3 b^2 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{(a (B-2 C)+3 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d \sqrt{a+b}}+\frac{B \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((-2*b*(b*B - a*C)*Sin[c + d*x])/(a^2*(-a^2 + b^2)) + (2*(-(b^3*B*Sin[c + d*x]) + a*b^2*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) - ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a^3*B*Tan[(c + d*x)/2] + a^2*b*B*Tan[(c + d*x)/2] - 3*a*b^2*B*Tan[(c + d*x)/2] - 3*b^3*B*Tan[(c + d*x)/2] + 2*a^2*b*C*Tan[(c + d*x)/2] + 2*a*b^2*C*Tan[(c + d*x)/2] - 2*a^3*B*Tan[(c + d*x)/2]^3 + 6*a*b^2*B*Tan[(c + d*x)/2]^3 - 4*a^2*b*C*Tan[(c + d*x)/2]^3 + a^3*B*Tan[(c + d*x)/2]^5 - a^2*b*B*Tan[(c + d*x)/2]^5 - 3*a*b^2*B*Tan[(c + d*x)/2]^5 + 3*b^3*B*Tan[(c + d*x)/2]^5 + 2*a^2*b*C*Tan[(c + d*x)/2]^5 - 2*a*b^2*C*Tan[(c + d*x)/2]^5 - 6*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*b^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*b^2*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(a^2*B - 3*b^2*B + 2*a*b*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(a + b)*(-(b*B) + a*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
849,1,4342,509,26.9026362,"\int \frac{\sec ^3(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 a (b B-a C) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(-2 a^2 C+a b B+b^2 C\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 a^2 \left(-6 a^3 C+3 a^2 b B+10 a b^2 C-7 b^3 B\right) \tan (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-16 a^4 C+a^3 b (8 B-12 C)+2 a^2 b^2 (3 B+8 C)-9 a b^3 (B-C)-b^4 (3 B-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \left(-16 a^5 C+8 a^4 b B+28 a^3 b^2 C-15 a^2 b^3 B-8 a b^4 C+3 b^5 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^5 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 16*a^5*C + 28*a^3*b^2*C - 8*a*b^4*C)*Sin[c + d*x])/(3*b^4*(-a^2 + b^2)^2) + (2*(a^2*b*B*Sin[c + d*x] - a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (2*(-4*a^4*b*B*Sin[c + d*x] + 8*a^2*b^3*B*Sin[c + d*x] + 7*a^5*C*Sin[c + d*x] - 11*a^3*b^2*C*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (2*C*Tan[c + d*x])/(3*b^3)))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*((5*a^2*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*B)/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^2*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^5*C)/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (28*a^3*C)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*a*b*C)/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*B*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (17*a^3*B*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (3*a*b*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a^2*C*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (16*a^6*C*Sqrt[Sec[c + d*x]])/(3*b^4*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (32*a^4*C*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (b^2*C*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (8*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (16*a^6*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^4*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (28*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*C - 9*a*b^3*(B + C) + b^4*(3*B + C) + 4*a^3*b*(2*B + 3*C) + 2*a^2*b^2*(-3*B + 8*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*C - 9*a*b^3*(B + C) + b^4*(3*B + C) + 4*a^3*b*(2*B + 3*C) + 2*a^2*b^2*(-3*B + 8*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*C - 9*a*b^3*(B + C) + b^4*(3*B + C) + 4*a^3*b*(2*B + 3*C) + 2*a^2*b^2*(-3*B + 8*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-16*a^4*C - 9*a*b^3*(B + C) + b^4*(3*B + C) + 4*a^3*b*(2*B + 3*C) + 2*a^2*b^2*(-3*B + 8*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-16*a^4*C - 9*a*b^3*(B + C) + b^4*(3*B + C) + 4*a^3*b*(2*B + 3*C) + 2*a^2*b^2*(-3*B + 8*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-16*a^4*C - 9*a*b^3*(B + C) + b^4*(3*B + C) + 4*a^3*b*(2*B + 3*C) + 2*a^2*b^2*(-3*B + 8*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*C - 9*a*b^3*(B + C) + b^4*(3*B + C) + 4*a^3*b*(2*B + 3*C) + 2*a^2*b^2*(-3*B + 8*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
850,1,3920,417,26.3673506,"\int \frac{\sec ^2(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 a^2 (b B-a C) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(-5 a^3 C+2 a^2 b B+9 a b^2 C-6 b^3 B\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-8 a^3 C+2 a^2 b (B-3 C)+3 a b^2 (B+3 C)-3 b^3 (B-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-8 a^4 C+2 a^3 b B+15 a^2 b^2 C-6 a b^3 B-3 b^4 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)^2) - (2*(a*b*B*Sin[c + d*x] - a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (2*(-(a^3*b*B*Sin[c + d*x]) + 5*a*b^3*B*Sin[c + d*x] + 4*a^4*C*Sin[c + d*x] - 8*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) - (2*(b + a*Cos[c + d*x])^2*((2*a^3*B)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*b*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a^2*C)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*C)/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^2*C)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (5*a^2*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (b^2*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*C*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (17*a^3*C*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (3*a*b*C*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(B - 3*C) + 8*a^3*C + 3*b^3*(B + C) - 2*a^2*b*(B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*(-1/3*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(B - 3*C) + 8*a^3*C + 3*b^3*(B + C) - 2*a^2*b*(B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(B - 3*C) + 8*a^3*C + 3*b^3*(B + C) - 2*a^2*b*(B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(3*a*b^2*(B - 3*C) + 8*a^3*C + 3*b^3*(B + C) - 2*a^2*b*(B + 3*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(3*a*b^2*(B - 3*C) + 8*a^3*C + 3*b^3*(B + C) - 2*a^2*b*(B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(3*a*b^2*(B - 3*C) + 8*a^3*C + 3*b^3*(B + C) - 2*a^2*b*(B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(B - 3*C) + 8*a^3*C + 3*b^3*(B + C) - 2*a^2*b*(B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C - 15*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
851,1,3514,387,23.9042597,"\int \frac{\sec (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 a (b B-a C) \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(2 a^2 C+a b (B+3 C)-3 b^2 (B+C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(2 a^3 C+a^2 b B-6 a b^2 C+3 b^3 B\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(2 a^3 C+a^2 b B-6 a b^2 C+3 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sin[c + d*x])/(3*b^2*(-a^2 + b^2)^2) + (2*(b*B*Sin[c + d*x] - a*C*Sin[c + d*x]))/(3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (2*(2*a^2*b*B*Sin[c + d*x] + 2*b^3*B*Sin[c + d*x] + a^3*C*Sin[c + d*x] - 5*a*b^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*((a^2*B)/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b^2*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^3*C)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*b*C)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^3*B*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*b*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (5*a^2*C*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*C*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (b^2*C*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(B - 3*C) + 3*b^2*(B - C) + 2*a^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(B - 3*C) + 3*b^2*(B - C) + 2*a^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(B - 3*C) + 3*b^2*(B - C) + 2*a^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(a*b*(B - 3*C) + 3*b^2*(B - C) + 2*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(a*b*(B - 3*C) + 3*b^2*(B - C) + 2*a^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(a*b*(B - 3*C) + 3*b^2*(B - C) + 2*a^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(B - 3*C) + 3*b^2*(B - C) + 2*a^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
852,1,559,353,18.251341,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b)^2 \left(\left(a^2 C-4 a b B+3 b^2 C\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 (a+b) \left(a^2 C-4 a b B+3 b^2 C\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 b (a+b) (3 a B-a C+b B-3 b C) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{5/2}}+\frac{\sec ^3(c+d x) (a \cos (c+d x)+b)^3 \left(-\frac{2 \left(a^2 C-4 a b B+3 b^2 C\right) \sin (c+d x)}{3 b \left(b^2-a^2\right)^2}+\frac{2 \left(b^2 B \sin (c+d x)-a b C \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{2 \left(2 a^3 C \sin (c+d x)-5 a^2 b B \sin (c+d x)+2 a b^2 C \sin (c+d x)+b^3 B \sin (c+d x)\right)}{3 a \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{5/2}}","-\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 (b B-a C) \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(a^2 (-C)+4 a b B-3 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a B+a C-b B-3 b C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-2*(-4*a*b*B + a^2*C + 3*b^2*C)*Sin[c + d*x])/(3*b*(-a^2 + b^2)^2) + (2*(b^2*B*Sin[c + d*x] - a*b*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (2*(-5*a^2*b*B*Sin[c + d*x] + b^3*B*Sin[c + d*x] + 2*a^3*C*Sin[c + d*x] + 2*a*b^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-4*a*b*B + a^2*C + 3*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*a*B + b*B - a*C - 3*b*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-4*a*b*B + a^2*C + 3*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2))","A",0
853,1,2039,495,16.6335922,"\int \frac{\cos (c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 B \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{2 b (b B-a C) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(-4 a^3 C+7 a^2 b B-3 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b d (a-b) (a+b)^{3/2}}+\frac{2 b \left(-4 a^3 C+7 a^2 b B-3 b^3 B\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^3 C+6 a^2 b B+a^2 b C-a b^2 B-3 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(-7*a^2*b*B + 3*b^3*B + 4*a^3*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2) - (2*(b^3*B*Sin[c + d*x] - a*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-8*a^2*b^2*B*Sin[c + d*x] + 4*b^4*B*Sin[c + d*x] + 5*a^3*b*C*Sin[c + d*x] - a*b^3*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(7*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 3*a*b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 3*b^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 4*a^4*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 4*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 14*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 6*a*b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 8*a^4*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + 7*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 7*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 3*a*b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 3*b^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*a^4*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + 4*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - (6*I)*a^4*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (12*I)*a^2*b^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^4*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*a^4*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (12*I)*a^2*b^2*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*b^4*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(-7*a^2*b*B + 3*b^3*B + 4*a^3*C)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(-4*a*b^2*B - 6*b^3*B + 3*a^3*(B - C) + a^2*b*(9*B + C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a^2*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
854,1,3729,446,24.4711708,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]","\text{Result too large to show}","-\frac{2 \left(-3 a^2 C+8 a b B-5 b^2 C\right) \tan (c+d x)}{15 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}-\frac{2 (b B-a C) \tan (c+d x)}{5 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{5/2}}+\frac{2 \left(3 a^2 (5 B+C)-8 a b (B+3 C)+b^2 (9 B+5 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d \sqrt{a+b} \left(a^2-b^2\right)^2}-\frac{2 \left(-3 a^3 C+23 a^2 b B-29 a b^2 C+9 b^3 B\right) \tan (c+d x)}{15 d \left(a^2-b^2\right)^3 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^3 C+23 a^2 b B-29 a b^2 C+9 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d (a-b)^2 (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])^4*Sec[c + d*x]^4*((-2*(23*a^2*b*B + 9*b^3*B - 3*a^3*C - 29*a*b^2*C)*Sin[c + d*x])/(15*b*(-a^2 + b^2)^3) - (2*(b^3*B*Sin[c + d*x] - a*b^2*C*Sin[c + d*x]))/(5*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^3) - (2*(-14*a^2*b^2*B*Sin[c + d*x] + 6*b^4*B*Sin[c + d*x] + 9*a^3*b*C*Sin[c + d*x] - a*b^3*C*Sin[c + d*x]))/(15*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (2*(-34*a^4*b*B*Sin[c + d*x] + 5*a^2*b^3*B*Sin[c + d*x] - 3*b^5*B*Sin[c + d*x] + 9*a^5*C*Sin[c + d*x] + 25*a^3*b^2*C*Sin[c + d*x] - 2*a*b^4*C*Sin[c + d*x]))/(15*a^2*(a^2 - b^2)^3*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(7/2)) - (2*(b + a*Cos[c + d*x])^3*((23*a^2*b*B)/(15*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*b^3*B)/(5*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^3*C)/(5*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a*b^2*C)/(15*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*a^3*B*Sqrt[Sec[c + d*x]])/(15*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) - (8*a*b^2*B*Sqrt[Sec[c + d*x]])/(15*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) - (a^4*C*Sqrt[Sec[c + d*x]])/(5*b*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*b*C*Sqrt[Sec[c + d*x]])/(15*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) + (b^3*C*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) + (23*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) + (3*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) - (a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]) - (29*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(7/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(b^2*(9*B - 5*C) + 8*a*b*(B - 3*C) + 3*a^2*(5*B - C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b*(-a^2 + b^2)^3*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(7/2)*(-1/15*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(b^2*(9*B - 5*C) + 8*a*b*(B - 3*C) + 3*a^2*(5*B - C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b*(-a^2 + b^2)^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(b^2*(9*B - 5*C) + 8*a*b*(B - 3*C) + 3*a^2*(5*B - C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(b^2*(9*B - 5*C) + 8*a*b*(B - 3*C) + 3*a^2*(5*B - C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(b^2*(9*B - 5*C) + 8*a*b*(B - 3*C) + 3*a^2*(5*B - C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(b^2*(9*B - 5*C) + 8*a*b*(B - 3*C) + 3*a^2*(5*B - C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(b^2*(9*B - 5*C) + 8*a*b*(B - 3*C) + 3*a^2*(5*B - C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-23*a^2*b*B - 9*b^3*B + 3*a^3*C + 29*a*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b*(-a^2 + b^2)^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
855,1,76,101,0.6919575,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left((b B-a C) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a C F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a b d}","\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}",1,"(2*Cot[c + d*x]*(a*C*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (b*B - a*C)*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(a*b*d)","A",1
856,1,91,138,0.2740026,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(B F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+C \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 B \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(B*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + C*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
857,1,21744,229,26.9799809,"\int (a+b \sec (c+d x))^{2/3} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(2/3)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{2} (b B-a C) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{\sqrt{2} C (a+b) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",1,"Result too large to show","B",0
858,1,21684,229,26.6371344,"\int \sqrt[3]{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(1/3)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{2} (b B-a C) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}+\frac{\sqrt{2} C (a+b) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",1,"Result too large to show","B",0
859,1,12792,226,27.0111975,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3),x]","\text{Result too large to show}","\frac{\sqrt{2} (b B-a C) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{\sqrt{2} C \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",1,"Result too large to show","B",0
860,1,12774,226,26.950222,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","\frac{\sqrt{2} (b B-a C) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac{\sqrt{2} C \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",1,"Result too large to show","B",0
861,1,124,165,1.1902078,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{15 (4 a A+3 a C+3 b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 \tan ^2(c+d x) (a B+A b+2 b C)+15 (a B+A b+b C)+3 b C \tan ^4(c+d x)\right)+15 \sec (c+d x) (4 a A+3 a C+3 b B)+30 (a C+b B) \sec ^3(c+d x)\right)}{120 d}","\frac{\tan ^3(c+d x) (5 a B+5 A b+4 b C)}{15 d}+\frac{\tan (c+d x) (5 a B+5 A b+4 b C)}{5 d}+\frac{(4 a A+3 a C+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (4 a A+3 a C+3 b B)}{8 d}+\frac{(a C+b B) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{b C \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(15*(4*a*A + 3*b*B + 3*a*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(4*a*A + 3*b*B + 3*a*C)*Sec[c + d*x] + 30*(b*B + a*C)*Sec[c + d*x]^3 + 8*(15*(A*b + a*B + b*C) + 5*(A*b + a*B + 2*b*C)*Tan[c + d*x]^2 + 3*b*C*Tan[c + d*x]^4)))/(120*d)","A",1
862,1,100,137,0.6797146,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 (4 a B+4 A b+3 b C) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(3 a (A+C)+(a C+b B) \tan ^2(c+d x)+3 b B\right)+3 \sec (c+d x) (4 a B+4 A b+3 b C)+6 b C \sec ^3(c+d x)\right)}{24 d}","\frac{\tan (c+d x) (3 a A+2 a C+2 b B)}{3 d}+\frac{(4 a B+4 A b+3 b C) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) (4 a B+4 A b+3 b C)}{8 d}+\frac{(a C+b B) \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{b C \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(4*A*b + 4*a*B + 3*b*C)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*(4*A*b + 4*a*B + 3*b*C)*Sec[c + d*x] + 6*b*C*Sec[c + d*x]^3 + 8*(3*b*B + 3*a*(A + C) + (b*B + a*C)*Tan[c + d*x]^2)))/(24*d)","A",1
863,1,75,101,0.3864858,"\int \sec (c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 (a (2 A+C)+b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (a C+b B) \sec (c+d x)+6 a B+6 A b+2 b C \tan ^2(c+d x)+6 b C\right)}{6 d}","\frac{\tan (c+d x) (3 a B+3 A b+2 b C)}{3 d}+\frac{(a (2 A+C)+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a C+b B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b C \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*(b*B + a*(2*A + C))*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*A*b + 6*a*B + 6*b*C + 3*(b*B + a*C)*Sec[c + d*x] + 2*b*C*Tan[c + d*x]^2))/(6*d)","A",1
864,1,92,69,0.0215038,"\int (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a C \tan (c+d x)}{d}+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \tan (c+d x)}{d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(2 a B+2 A b+b C) \tanh ^{-1}(\sin (c+d x))}{2 d}+a A x+\frac{(a C+b B) \tan (c+d x)}{d}+\frac{b C \tan (c+d x) \sec (c+d x)}{2 d}",1,"a*A*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d + (b*C*ArcTanh[Sin[c + d*x]])/(2*d) + (b*B*Tan[c + d*x])/d + (a*C*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
865,1,71,52,0.0225531,"\int \cos (c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a B x+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}+A b x+\frac{b B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \tan (c+d x)}{d}","x (a B+A b)+\frac{a A \sin (c+d x)}{d}+\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b C \tan (c+d x)}{d}",1,"A*b*x + a*B*x + (b*B*ArcTanh[Sin[c + d*x]])/d + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Cos[d*x]*Sin[c])/d + (a*A*Cos[c]*Sin[d*x])/d + (b*C*Tan[c + d*x])/d","A",1
866,1,68,69,0.1303646,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 (a B+A b) \sin (c+d x)+a A \sin (2 (c+d x))+2 a A c+2 a A d x+4 a C d x+4 b B d x+4 b C \tanh ^{-1}(\sin (c+d x))}{4 d}","\frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (a (A+2 C)+2 b B)+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*a*A*c + 2*a*A*d*x + 4*b*B*d*x + 4*a*C*d*x + 4*b*C*ArcTanh[Sin[c + d*x]] + 4*(A*b + a*B)*Sin[c + d*x] + a*A*Sin[2*(c + d*x)])/(4*d)","A",1
867,1,85,92,0.1875758,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 \sin (c+d x) (3 a A+4 a C+4 b B)+3 (a B+A b) \sin (2 (c+d x))+a A \sin (3 (c+d x))+6 a B c+6 a B d x+6 A b c+6 A b d x+12 b C d x}{12 d}","\frac{\sin (c+d x) (2 a A+3 a C+3 b B)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b+2 b C)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(6*A*b*c + 6*a*B*c + 6*A*b*d*x + 6*a*B*d*x + 12*b*C*d*x + 3*(3*a*A + 4*b*B + 4*a*C)*Sin[c + d*x] + 3*(A*b + a*B)*Sin[2*(c + d*x)] + a*A*Sin[3*(c + d*x)])/(12*d)","A",1
868,1,117,116,0.3318439,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{24 \sin (c+d x) (3 a B+3 A b+4 b C)+24 \sin (2 (c+d x)) (a (A+C)+b B)+3 a A \sin (4 (c+d x))+36 a A c+36 a A d x+8 a B \sin (3 (c+d x))+48 a c C+48 a C d x+8 A b \sin (3 (c+d x))+48 b B c+48 b B d x}{96 d}","\frac{\sin (c+d x) (a B+A b+b C)}{d}+\frac{\sin (c+d x) \cos (c+d x) (3 a A+4 a C+4 b B)}{8 d}-\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{1}{8} x (3 a A+4 a C+4 b B)+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(36*a*A*c + 48*b*B*c + 48*a*c*C + 36*a*A*d*x + 48*b*B*d*x + 48*a*C*d*x + 24*(3*A*b + 3*a*B + 4*b*C)*Sin[c + d*x] + 24*(b*B + a*(A + C))*Sin[2*(c + d*x)] + 8*A*b*Sin[3*(c + d*x)] + 8*a*B*Sin[3*(c + d*x)] + 3*a*A*Sin[4*(c + d*x)])/(96*d)","A",1
869,1,117,156,0.4711678,"\int \cos ^5(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{-160 \sin ^3(c+d x) (a (2 A+C)+b B)+480 \sin (c+d x) (a (A+C)+b B)+15 (4 (c+d x) (3 a B+3 A b+4 b C)+8 \sin (2 (c+d x)) (a B+A b+b C)+(a B+A b) \sin (4 (c+d x)))+96 a A \sin ^5(c+d x)}{480 d}","-\frac{\sin ^3(c+d x) (4 a A+5 a C+5 b B)}{15 d}+\frac{\sin (c+d x) (4 a A+5 a C+5 b B)}{5 d}+\frac{\sin (c+d x) \cos (c+d x) (3 a B+3 A b+4 b C)}{8 d}+\frac{(a B+A b) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{1}{8} x (3 a B+3 A b+4 b C)+\frac{a A \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(480*(b*B + a*(A + C))*Sin[c + d*x] - 160*(b*B + a*(2*A + C))*Sin[c + d*x]^3 + 96*a*A*Sin[c + d*x]^5 + 15*(4*(3*A*b + 3*a*B + 4*b*C)*(c + d*x) + 8*(A*b + a*B + b*C)*Sin[2*(c + d*x)] + (A*b + a*B)*Sin[4*(c + d*x)]))/(480*d)","A",1
870,1,371,233,2.1481756,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^5(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(2 \sin (c+d x) \left(15 \cos (c+d x) \left(12 a^2 B+24 a A b+34 a b C+17 b^2 B\right)+48 \cos (2 (c+d x)) \left(5 a^2 (A+C)+10 a b B+b^2 (5 A+4 C)\right)+60 a^2 A \cos (4 (c+d x))+180 a^2 A+60 a^2 B \cos (3 (c+d x))+40 a^2 C \cos (4 (c+d x))+200 a^2 C+120 a A b \cos (3 (c+d x))+80 a b B \cos (4 (c+d x))+400 a b B+90 a b C \cos (3 (c+d x))+40 A b^2 \cos (4 (c+d x))+200 A b^2+45 b^2 B \cos (3 (c+d x))+32 b^2 C \cos (4 (c+d x))+256 b^2 C\right)-120 \cos ^5(c+d x) \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{480 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\tan (c+d x) \left(5 a^2 (3 A+2 C)+20 a b B+2 b^2 (5 A+4 C)\right)}{15 d}+\frac{\left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(2 a^2 C+10 a b B+5 A b^2+4 b^2 C\right)}{15 d}+\frac{\tan (c+d x) \sec (c+d x) \left(4 a^2 B+8 a A b+6 a b C+3 b^2 B\right)}{8 d}+\frac{b (2 a C+5 b B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{C \tan (c+d x) \sec ^2(c+d x) (a+b \sec (c+d x))^2}{5 d}",1,"((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(-120*(8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(180*a^2*A + 200*A*b^2 + 400*a*b*B + 200*a^2*C + 256*b^2*C + 15*(24*a*A*b + 12*a^2*B + 17*b^2*B + 34*a*b*C)*Cos[c + d*x] + 48*(10*a*b*B + 5*a^2*(A + C) + b^2*(5*A + 4*C))*Cos[2*(c + d*x)] + 120*a*A*b*Cos[3*(c + d*x)] + 60*a^2*B*Cos[3*(c + d*x)] + 45*b^2*B*Cos[3*(c + d*x)] + 90*a*b*C*Cos[3*(c + d*x)] + 60*a^2*A*Cos[4*(c + d*x)] + 40*A*b^2*Cos[4*(c + d*x)] + 80*a*b*B*Cos[4*(c + d*x)] + 40*a^2*C*Cos[4*(c + d*x)] + 32*b^2*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(480*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
871,1,300,200,1.5833666,"\int \sec (c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{\sec ^4(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(12 \cos ^4(c+d x) \left(4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2 \sin (c+d x) \left(4 \cos (c+d x) \left(9 a^2 B+2 a b (9 A+10 C)+10 b^2 B\right)+3 \cos (2 (c+d x)) \left(4 a^2 C+8 a b B+4 A b^2+3 b^2 C\right)+12 a^2 B \cos (3 (c+d x))+12 a^2 C+24 a A b \cos (3 (c+d x))+24 a b B+16 a b C \cos (3 (c+d x))+12 A b^2+8 b^2 B \cos (3 (c+d x))+21 b^2 C\right)\right)}{48 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\left(4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{\tan (c+d x) \left(a^3 (-C)+4 a^2 b B+4 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 b d}+\frac{(4 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d}",1,"-1/48*((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^4*(12*(8*a*b*B + 4*a^2*(2*A + C) + b^2*(4*A + 3*C))*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*(12*A*b^2 + 24*a*b*B + 12*a^2*C + 21*b^2*C + 4*(9*a^2*B + 10*b^2*B + 2*a*b*(9*A + 10*C))*Cos[c + d*x] + 3*(4*A*b^2 + 8*a*b*B + 4*a^2*C + 3*b^2*C)*Cos[2*(c + d*x)] + 24*a*A*b*Cos[3*(c + d*x)] + 12*a^2*B*Cos[3*(c + d*x)] + 8*b^2*B*Cos[3*(c + d*x)] + 16*a*b*C*Cos[3*(c + d*x)])*Sin[c + d*x]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
872,1,322,134,1.6958678,"\int (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^3(c+d x) \left(4 \sin (c+d x) \left(\cos (2 (c+d x)) \left(3 a^2 C+6 a b B+3 A b^2+2 b^2 C\right)+3 a^2 C+3 b (2 a C+b B) \cos (c+d x)+6 a b B+3 A b^2+4 b^2 C\right)+9 \cos (c+d x) \left(-\left(2 a^2 B+2 a b (2 A+C)+b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2 B+2 a b (2 A+C)+b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2 A (c+d x)\right)+3 \cos (3 (c+d x)) \left(-\left(2 a^2 B+2 a b (2 A+C)+b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2 B+2 a b (2 A+C)+b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2 A (c+d x)\right)\right)}{24 d}","\frac{\tan (c+d x) \left(2 a^2 C+6 a b B+3 A b^2+2 b^2 C\right)}{3 d}+\frac{\left(2 a^2 B+2 a b (2 A+C)+b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac{b (2 a C+3 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(Sec[c + d*x]^3*(9*Cos[c + d*x]*(2*a^2*A*(c + d*x) - (2*a^2*B + b^2*B + 2*a*b*(2*A + C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (2*a^2*B + b^2*B + 2*a*b*(2*A + C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 3*Cos[3*(c + d*x)]*(2*a^2*A*(c + d*x) - (2*a^2*B + b^2*B + 2*a*b*(2*A + C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (2*a^2*B + b^2*B + 2*a*b*(2*A + C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 4*(3*A*b^2 + 6*a*b*B + 3*a^2*C + 4*b^2*C + 3*b*(b*B + 2*a*C)*Cos[c + d*x] + (3*A*b^2 + 6*a*b*B + 3*a^2*C + 2*b^2*C)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(24*d)","B",1
873,1,453,126,1.2709701,"\int \cos (c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^2(c+d x) \left(\cos (2 (c+d x)) \left(-\left(2 a^2 C+4 a b B+2 A b^2+b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+\left(2 a^2 C+4 a b B+2 A b^2+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 a (c+d x) (a B+2 A b)\right)+\left(a^2 A+2 b^2 C\right) \sin (c+d x)+a^2 A \sin (3 (c+d x))+2 a^2 B c+2 a^2 B d x-2 a^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 a A b c+4 a A b d x-4 a b B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 a b B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 a b C \sin (2 (c+d x))-2 A b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 A b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 b^2 B \sin (2 (c+d x))-b^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d}","\frac{\left(2 a^2 C+4 a b B+2 A b^2+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b \tan (c+d x) (2 a A-2 a C-b B)}{d}+a x (a B+2 A b)+\frac{A \sin (c+d x) (a+b \sec (c+d x))^2}{d}-\frac{b^2 (2 A-C) \tan (c+d x) \sec (c+d x)}{2 d}",1,"(Sec[c + d*x]^2*(4*a*A*b*c + 2*a^2*B*c + 4*a*A*b*d*x + 2*a^2*B*d*x - 2*A*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4*a*b*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 2*a^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - b^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*A*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*a*b*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 2*a^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Cos[2*(c + d*x)]*(2*a*(2*A*b + a*B)*(c + d*x) - (2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (a^2*A + 2*b^2*C)*Sin[c + d*x] + 2*b^2*B*Sin[2*(c + d*x)] + 4*a*b*C*Sin[2*(c + d*x)] + a^2*A*Sin[3*(c + d*x)]))/(4*d)","B",1
874,1,153,118,1.1302547,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 (c+d x) \left(a^2 (A+2 C)+4 a b B+2 A b^2\right)+\tan (c+d x) \left(a^2 A \cos (2 (c+d x))+a^2 A+4 a (a B+2 A b) \cos (c+d x)+4 b^2 C\right)-4 b (2 a C+b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b (2 a C+b B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{1}{2} x \left(a^2 (A+2 C)+4 a b B+2 A b^2\right)+\frac{a (a B+A b) \sin (c+d x)}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{b (2 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{b^2 (A-2 C) \tan (c+d x)}{2 d}",1,"(2*(2*A*b^2 + 4*a*b*B + a^2*(A + 2*C))*(c + d*x) - 4*b*(b*B + 2*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*b*(b*B + 2*a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*A + 4*b^2*C + 4*a*(2*A*b + a*B)*Cos[c + d*x] + a^2*A*Cos[2*(c + d*x)])*Tan[c + d*x])/(4*d)","A",1
875,1,157,141,0.5178625,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{6 (c+d x) \left(a^2 B+2 a b (A+2 C)+2 b^2 B\right)+3 \sin (c+d x) \left(a^2 (3 A+4 C)+8 a b B+4 A b^2\right)+a^2 A \sin (3 (c+d x))+3 a (a B+2 A b) \sin (2 (c+d x))-12 b^2 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^2 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 d}","\frac{\sin (c+d x) \left(a^2 (2 A+3 C)+6 a b B+2 A b^2\right)}{3 d}+\frac{1}{2} x \left(a^2 B+2 a b (A+2 C)+2 b^2 B\right)+\frac{a (3 a B+2 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^2 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(6*(a^2*B + 2*b^2*B + 2*a*b*(A + 2*C))*(c + d*x) - 12*b^2*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^2*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 3*(4*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Sin[c + d*x] + 3*a*(2*A*b + a*B)*Sin[2*(c + d*x)] + a^2*A*Sin[3*(c + d*x)])/(12*d)","A",1
876,1,134,175,0.7140517,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{12 (c+d x) \left(a^2 (3 A+4 C)+8 a b B+4 b^2 (A+2 C)\right)+24 \sin (c+d x) \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)+24 \sin (2 (c+d x)) \left(a^2 (A+C)+2 a b B+A b^2\right)+3 a^2 A \sin (4 (c+d x))+8 a (a B+2 A b) \sin (3 (c+d x))}{96 d}","\frac{\sin (c+d x) \left(2 a^2 B+4 a A b+6 a b C+3 b^2 B\right)}{3 d}+\frac{\sin (c+d x) \cos (c+d x) \left(a^2 (3 A+4 C)+8 a b B+2 A b^2\right)}{8 d}+\frac{1}{8} x \left(a^2 (3 A+4 C)+8 a b B+4 b^2 (A+2 C)\right)+\frac{a (2 a B+A b) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"(12*(8*a*b*B + 4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*(c + d*x) + 24*(6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*Sin[c + d*x] + 24*(A*b^2 + 2*a*b*B + a^2*(A + C))*Sin[2*(c + d*x)] + 8*a*(2*A*b + a*B)*Sin[3*(c + d*x)] + 3*a^2*A*Sin[4*(c + d*x)])/(96*d)","A",1
877,1,169,215,0.7660523,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{60 (c+d x) \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)+60 \sin (c+d x) \left(a^2 (5 A+6 C)+12 a b B+2 b^2 (3 A+4 C)\right)+120 \sin (2 (c+d x)) \left(a^2 B+2 a b (A+C)+b^2 B\right)+10 \sin (3 (c+d x)) \left(a^2 (5 A+4 C)+8 a b B+4 A b^2\right)+6 a^2 A \sin (5 (c+d x))+15 a (a B+2 A b) \sin (4 (c+d x))}{480 d}","-\frac{\sin ^3(c+d x) \left(a^2 (4 A+5 C)+10 a b B+2 A b^2\right)}{15 d}+\frac{\sin (c+d x) \left(a^2 (4 A+5 C)+10 a b B+b^2 (4 A+5 C)\right)}{5 d}+\frac{\sin (c+d x) \cos (c+d x) \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)}{8 d}+\frac{1}{8} x \left(3 a^2 B+6 a A b+8 a b C+4 b^2 B\right)+\frac{a (5 a B+2 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}",1,"(60*(6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*(c + d*x) + 60*(12*a*b*B + 2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Sin[c + d*x] + 120*(a^2*B + b^2*B + 2*a*b*(A + C))*Sin[2*(c + d*x)] + 10*(4*A*b^2 + 8*a*b*B + a^2*(5*A + 4*C))*Sin[3*(c + d*x)] + 15*a*(2*A*b + a*B)*Sin[4*(c + d*x)] + 6*a^2*A*Sin[5*(c + d*x)])/(480*d)","A",1
878,1,384,381,3.0328998,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{\sec ^5(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(-b \left(5 \sin (2 (c+d x)) \left(18 a^2 C+18 a b B+6 A b^2+5 b^2 C\right)+48 b (3 a C+b B) \sin (c+d x)+40 b^2 C \tan (c+d x)\right)-16 \sin (c+d x) \cos ^4(c+d x) \left(5 a^3 (3 A+2 C)+30 a^2 b B+6 a b^2 (5 A+4 C)+8 b^3 B\right)-15 \sin (c+d x) \cos ^3(c+d x) \left(8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right)-16 \sin (c+d x) \cos ^2(c+d x) \left(5 a^3 C+15 a^2 b B+3 a b^2 (5 A+4 C)+4 b^3 B\right)+15 \cos ^5(c+d x) \left(8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{120 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\tan (c+d x) \left(2 a^2 C-6 a b B+30 A b^2+25 b^2 C\right) (a+b \sec (c+d x))^3}{120 b^2 d}+\frac{\left(8 a^3 B+6 a^2 b (4 A+3 C)+18 a b^2 B+b^3 (6 A+5 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\tan (c+d x) \left(-2 a^3 C+6 a^2 b B-3 a b^2 (10 A+7 C)-32 b^3 B\right) (a+b \sec (c+d x))^2}{120 b^2 d}-\frac{\tan (c+d x) \sec (c+d x) \left(-4 a^4 C+12 a^3 b B-12 a^2 b^2 (5 A+3 C)-142 a b^3 B-15 b^4 (6 A+5 C)\right)}{240 b d}-\frac{\tan (c+d x) \left(-2 a^5 C+6 a^4 b B-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B-24 a b^4 (5 A+4 C)-32 b^5 B\right)}{60 b^2 d}+\frac{(3 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{15 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^4}{6 b d}",1,"-1/120*((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(15*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 16*(15*a^2*b*B + 4*b^3*B + 5*a^3*C + 3*a*b^2*(5*A + 4*C))*Cos[c + d*x]^2*Sin[c + d*x] - 15*(8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*Cos[c + d*x]^3*Sin[c + d*x] - 16*(30*a^2*b*B + 8*b^3*B + 5*a^3*(3*A + 2*C) + 6*a*b^2*(5*A + 4*C))*Cos[c + d*x]^4*Sin[c + d*x] - b*(48*b*(b*B + 3*a*C)*Sin[c + d*x] + 5*(6*A*b^2 + 18*a*b*B + 18*a^2*C + 5*b^2*C)*Sin[2*(c + d*x)] + 40*b^2*C*Tan[c + d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
879,1,451,286,2.8771608,"\int \sec (c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{\sec ^5(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(120 \cos ^5(c+d x) \left(4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-2 \sin (c+d x) \left(60 a^3 B \cos (4 (c+d x))+180 a^3 B+60 a^3 C \cos (3 (c+d x))+180 a^2 A b \cos (4 (c+d x))+540 a^2 A b+180 a^2 b B \cos (3 (c+d x))+120 a^2 b C \cos (4 (c+d x))+600 a^2 b C+15 \cos (c+d x) \left(12 a^3 C+36 a^2 b B+3 a b^2 (12 A+17 C)+17 b^3 B\right)+48 \cos (2 (c+d x)) \left(5 a^3 B+15 a^2 b (A+C)+15 a b^2 B+b^3 (5 A+4 C)\right)+180 a A b^2 \cos (3 (c+d x))+120 a b^2 B \cos (4 (c+d x))+600 a b^2 B+135 a b^2 C \cos (3 (c+d x))+40 A b^3 \cos (4 (c+d x))+200 A b^3+45 b^3 B \cos (3 (c+d x))+32 b^3 C \cos (4 (c+d x))+256 b^3 C\right)\right)}{480 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\left(4 a^3 (2 A+C)+12 a^2 b B+3 a b^2 (4 A+3 C)+3 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\tan (c+d x) \sec (c+d x) \left(-6 a^3 C+30 a^2 b B+a b^2 (100 A+71 C)+45 b^3 B\right)}{120 d}+\frac{\tan (c+d x) \left(-3 a^4 C+15 a^3 b B+4 a^2 b^2 (20 A+13 C)+60 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 b d}+\frac{\tan (c+d x) \left(3 a (5 b B-a C)+4 b^2 (5 A+4 C)\right) (a+b \sec (c+d x))^2}{60 b d}+\frac{(5 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}",1,"-1/480*((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(120*(12*a^2*b*B + 3*b^3*B + 4*a^3*(2*A + C) + 3*a*b^2*(4*A + 3*C))*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 2*(540*a^2*A*b + 200*A*b^3 + 180*a^3*B + 600*a*b^2*B + 600*a^2*b*C + 256*b^3*C + 15*(36*a^2*b*B + 17*b^3*B + 12*a^3*C + 3*a*b^2*(12*A + 17*C))*Cos[c + d*x] + 48*(5*a^3*B + 15*a*b^2*B + 15*a^2*b*(A + C) + b^3*(5*A + 4*C))*Cos[2*(c + d*x)] + 180*a*A*b^2*Cos[3*(c + d*x)] + 180*a^2*b*B*Cos[3*(c + d*x)] + 45*b^3*B*Cos[3*(c + d*x)] + 60*a^3*C*Cos[3*(c + d*x)] + 135*a*b^2*C*Cos[3*(c + d*x)] + 180*a^2*A*b*Cos[4*(c + d*x)] + 40*A*b^3*Cos[4*(c + d*x)] + 60*a^3*B*Cos[4*(c + d*x)] + 120*a*b^2*B*Cos[4*(c + d*x)] + 120*a^2*b*C*Cos[4*(c + d*x)] + 32*b^3*C*Cos[4*(c + d*x)])*Sin[c + d*x]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
880,1,525,207,5.3084651,"\int (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos (c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(36 a^3 A (c+d x)+48 a^3 A (c+d x) \cos (2 (c+d x))+12 a^3 A (c+d x) \cos (4 (c+d x))+24 a^3 C \sin (2 (c+d x))+12 a^3 C \sin (4 (c+d x))+72 a^2 b B \sin (2 (c+d x))+36 a^2 b B \sin (4 (c+d x))+36 a^2 b C \sin (c+d x)+36 a^2 b C \sin (3 (c+d x))-12 \cos ^4(c+d x) \left(8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 \cos ^4(c+d x) \left(8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+72 a A b^2 \sin (2 (c+d x))+36 a A b^2 \sin (4 (c+d x))+36 a b^2 B \sin (c+d x)+36 a b^2 B \sin (3 (c+d x))+96 a b^2 C \sin (2 (c+d x))+24 a b^2 C \sin (4 (c+d x))+12 A b^3 \sin (c+d x)+12 A b^3 \sin (3 (c+d x))+32 b^3 B \sin (2 (c+d x))+8 b^3 B \sin (4 (c+d x))+33 b^3 C \sin (c+d x)+9 b^3 C \sin (3 (c+d x))\right)}{48 d (a \cos (c+d x)+b)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","a^3 A x+\frac{b \tan (c+d x) \sec (c+d x) \left(6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right)}{24 d}+\frac{\tan (c+d x) \left(3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{\left(8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(36*a^3*A*(c + d*x) + 48*a^3*A*(c + d*x)*Cos[2*(c + d*x)] + 12*a^3*A*(c + d*x)*Cos[4*(c + d*x)] - 12*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*Cos[c + d*x]^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 12*A*b^3*Sin[c + d*x] + 36*a*b^2*B*Sin[c + d*x] + 36*a^2*b*C*Sin[c + d*x] + 33*b^3*C*Sin[c + d*x] + 72*a*A*b^2*Sin[2*(c + d*x)] + 72*a^2*b*B*Sin[2*(c + d*x)] + 32*b^3*B*Sin[2*(c + d*x)] + 24*a^3*C*Sin[2*(c + d*x)] + 96*a*b^2*C*Sin[2*(c + d*x)] + 12*A*b^3*Sin[3*(c + d*x)] + 36*a*b^2*B*Sin[3*(c + d*x)] + 36*a^2*b*C*Sin[3*(c + d*x)] + 9*b^3*C*Sin[3*(c + d*x)] + 36*a*A*b^2*Sin[4*(c + d*x)] + 36*a^2*b*B*Sin[4*(c + d*x)] + 8*b^3*B*Sin[4*(c + d*x)] + 12*a^3*C*Sin[4*(c + d*x)] + 24*a*b^2*C*Sin[4*(c + d*x)]))/(48*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
881,1,509,192,6.3695144,"\int \cos (c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^5(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(12 a^3 A \sin (c+d x)+\frac{4 b \sin \left(\frac{1}{2} (c+d x)\right) \left(9 a^2 C+9 a b B+3 A b^2+2 b^2 C\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b \sin \left(\frac{1}{2} (c+d x)\right) \left(9 a^2 C+9 a b B+3 A b^2+2 b^2 C\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+12 a^2 (c+d x) (a B+3 A b)-6 \left(2 a^3 C+6 a^2 b B+3 a b^2 (2 A+C)+b^3 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \left(2 a^3 C+6 a^2 b B+3 a b^2 (2 A+C)+b^3 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b^2 (9 a C+b (3 B+C))}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 (9 a C+b (3 B+C))}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)}{6 d (a \cos (c+d x)+b)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{b \tan (c+d x) \left(-\left(a^2 (6 A-8 C)\right)+9 a b B+b^2 (3 A+2 C)\right)}{3 d}+a^2 x (a B+3 A b)+\frac{\left(2 a^3 C+6 a^2 b B+3 a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \tan (c+d x) \sec (c+d x) (6 a A-5 a C-3 b B)}{6 d}-\frac{b (3 A-C) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^3}{d}",1,"(Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(12*a^2*(3*A*b + a*B)*(c + d*x) - 6*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*(2*A + C))*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*(2*A + C))*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*(9*a*C + b*(3*B + C)))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*b*(3*A*b^2 + 9*a*b*B + 9*a^2*C + 2*b^2*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (b^2*(9*a*C + b*(3*B + C)))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b*(3*A*b^2 + 9*a*b*B + 9*a^2*C + 2*b^2*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 12*a^3*A*Sin[c + d*x]))/(6*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
882,1,320,204,2.8464835,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 A \sin (2 (c+d x))+2 a (c+d x) \left(a^2 (A+2 C)+6 a b B+6 A b^2\right)-2 b \left(6 a^2 C+6 a b B+2 A b^2+b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b \left(6 a^2 C+6 a b B+2 A b^2+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+4 a^2 (a B+3 A b) \sin (c+d x)+\frac{4 b^2 (3 a C+b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b^2 (3 a C+b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{b^3 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^3 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 d}","-\frac{b \tan (c+d x) \left(4 a^2 B+9 a A b-6 a b C-2 b^2 B\right)}{2 d}+\frac{b \left(6 a^2 C+6 a b B+2 A b^2+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{1}{2} a x \left(a^2 (A+2 C)+6 a b B+6 A b^2\right)-\frac{b^2 \tan (c+d x) \sec (c+d x) (2 a B+4 A b-b C)}{2 d}+\frac{(2 a B+3 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^3}{2 d}",1,"(2*a*(6*A*b^2 + 6*a*b*B + a^2*(A + 2*C))*(c + d*x) - 2*b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^3*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*b^2*(b*B + 3*a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^3*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b^2*(b*B + 3*a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*a^2*(3*A*b + a*B)*Sin[c + d*x] + a^3*A*Sin[2*(c + d*x)])/(4*d)","A",1
883,1,263,196,1.3225136,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^3 A \sin (3 (c+d x))+3 a \sin (c+d x) \left(a^2 (3 A+4 C)+12 a b B+12 A b^2\right)+3 a^2 (a B+3 A b) \sin (2 (c+d x))+6 (c+d x) \left(a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right)-12 b^2 (3 a C+b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^2 (3 a C+b B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{12 b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 b^3 C \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{12 d}","\frac{a \sin (c+d x) \left(a^2 (2 A+3 C)+6 a b B+3 A b^2\right)}{3 d}+\frac{1}{2} x \left(a^3 B+3 a^2 b (A+2 C)+6 a b^2 B+2 A b^3\right)-\frac{b^2 \tan (c+d x) (3 a B+5 A b-6 b C)}{6 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac{b^2 (3 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}",1,"(6*(2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*(c + d*x) - 12*b^2*(b*B + 3*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^2*(b*B + 3*a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (12*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (12*b^3*C*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*a*(12*A*b^2 + 12*a*b*B + a^2*(3*A + 4*C))*Sin[c + d*x] + 3*a^2*(3*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*d)","A",1
884,1,215,223,0.9395082,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{3 a^3 A \sin (4 (c+d x))+24 a \sin (2 (c+d x)) \left(a^2 (A+C)+3 a b B+3 A b^2\right)+8 a^2 (a B+3 A b) \sin (3 (c+d x))+12 (c+d x) \left(a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right)+24 \sin (c+d x) \left(3 a^3 B+3 a^2 b (3 A+4 C)+12 a b^2 B+4 A b^3\right)-96 b^3 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 b^3 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{96 d}","\frac{a \sin (c+d x) \cos (c+d x) \left(3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right)}{24 d}+\frac{\sin (c+d x) \left(4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right)}{6 d}+\frac{1}{8} x \left(a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right)+\frac{(4 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{b^3 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(12*(12*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*(c + d*x) - 96*b^3*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*b^3*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(4*A*b^3 + 3*a^3*B + 12*a*b^2*B + 3*a^2*b*(3*A + 4*C))*Sin[c + d*x] + 24*a*(3*A*b^2 + 3*a*b*B + a^2*(A + C))*Sin[2*(c + d*x)] + 8*a^2*(3*A*b + a*B)*Sin[3*(c + d*x)] + 3*a^3*A*Sin[4*(c + d*x)])/(96*d)","A",1
885,1,288,269,1.0729925,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{50 a^3 A \sin (3 (c+d x))+6 a^3 A \sin (5 (c+d x))+15 a^3 B \sin (4 (c+d x))+180 a^3 B c+180 a^3 B d x+40 a^3 C \sin (3 (c+d x))+45 a^2 A b \sin (4 (c+d x))+540 a^2 A b c+540 a^2 A b d x+120 a^2 b B \sin (3 (c+d x))+720 a^2 b c C+720 a^2 b C d x+60 \sin (c+d x) \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right)+120 \sin (2 (c+d x)) \left(a^3 B+3 a^2 b (A+C)+3 a b^2 B+A b^3\right)+120 a A b^2 \sin (3 (c+d x))+720 a b^2 B c+720 a b^2 B d x+240 A b^3 c+240 A b^3 d x+480 b^3 c C+480 b^3 C d x}{480 d}","\frac{a \sin (c+d x) \cos ^2(c+d x) \left(2 a^2 (4 A+5 C)+15 a b B+3 A b^2\right)}{30 d}+\frac{\sin (c+d x) \left(2 a^3 (4 A+5 C)+30 a^2 b B+15 a b^2 (2 A+3 C)+15 b^3 B\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(15 a^3 B+15 a^2 b (3 A+4 C)+50 a b^2 B+6 A b^3\right)}{40 d}+\frac{1}{8} x \left(3 a^3 B+3 a^2 b (3 A+4 C)+12 a b^2 B+4 b^3 (A+2 C)\right)+\frac{(5 a B+3 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}",1,"(540*a^2*A*b*c + 240*A*b^3*c + 180*a^3*B*c + 720*a*b^2*B*c + 720*a^2*b*c*C + 480*b^3*c*C + 540*a^2*A*b*d*x + 240*A*b^3*d*x + 180*a^3*B*d*x + 720*a*b^2*B*d*x + 720*a^2*b*C*d*x + 480*b^3*C*d*x + 60*(18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*Sin[c + d*x] + 120*(A*b^3 + a^3*B + 3*a*b^2*B + 3*a^2*b*(A + C))*Sin[2*(c + d*x)] + 50*a^3*A*Sin[3*(c + d*x)] + 120*a*A*b^2*Sin[3*(c + d*x)] + 120*a^2*b*B*Sin[3*(c + d*x)] + 40*a^3*C*Sin[3*(c + d*x)] + 45*a^2*A*b*Sin[4*(c + d*x)] + 15*a^3*B*Sin[4*(c + d*x)] + 6*a^3*A*Sin[5*(c + d*x)])/(480*d)","A",1
886,1,369,320,1.1814033,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{45 a^3 A \sin (4 (c+d x))+5 a^3 A \sin (6 (c+d x))+300 a^3 A c+300 a^3 A d x+100 a^3 B \sin (3 (c+d x))+12 a^3 B \sin (5 (c+d x))+30 a^3 C \sin (4 (c+d x))+360 a^3 c C+360 a^3 C d x+300 a^2 A b \sin (3 (c+d x))+36 a^2 A b \sin (5 (c+d x))+90 a^2 b B \sin (4 (c+d x))+1080 a^2 b B c+1080 a^2 b B d x+240 a^2 b C \sin (3 (c+d x))+120 \sin (c+d x) \left(5 a^3 B+3 a^2 b (5 A+6 C)+18 a b^2 B+2 b^3 (3 A+4 C)\right)+15 \sin (2 (c+d x)) \left(a^3 (15 A+16 C)+48 a^2 b B+48 a b^2 (A+C)+16 b^3 B\right)+90 a A b^2 \sin (4 (c+d x))+1080 a A b^2 c+1080 a A b^2 d x+240 a b^2 B \sin (3 (c+d x))+1440 a b^2 c C+1440 a b^2 C d x+80 A b^3 \sin (3 (c+d x))+480 b^3 B c+480 b^3 B d x}{960 d}","\frac{a \sin (c+d x) \cos ^3(c+d x) \left(5 a^2 (5 A+6 C)+42 a b B+6 A b^2\right)}{120 d}-\frac{\sin ^3(c+d x) \left(4 a^3 B+3 a^2 b (4 A+5 C)+12 a b^2 B+A b^3\right)}{15 d}+\frac{\sin (c+d x) \left(12 a^3 B+9 a^2 b (4 A+5 C)+42 a b^2 B+b^3 (11 A+15 C)\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right)}{16 d}+\frac{1}{16} x \left(a^3 (5 A+6 C)+18 a^2 b B+6 a b^2 (3 A+4 C)+8 b^3 B\right)+\frac{(2 a B+A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}",1,"(300*a^3*A*c + 1080*a*A*b^2*c + 1080*a^2*b*B*c + 480*b^3*B*c + 360*a^3*c*C + 1440*a*b^2*c*C + 300*a^3*A*d*x + 1080*a*A*b^2*d*x + 1080*a^2*b*B*d*x + 480*b^3*B*d*x + 360*a^3*C*d*x + 1440*a*b^2*C*d*x + 120*(5*a^3*B + 18*a*b^2*B + 2*b^3*(3*A + 4*C) + 3*a^2*b*(5*A + 6*C))*Sin[c + d*x] + 15*(48*a^2*b*B + 16*b^3*B + 48*a*b^2*(A + C) + a^3*(15*A + 16*C))*Sin[2*(c + d*x)] + 300*a^2*A*b*Sin[3*(c + d*x)] + 80*A*b^3*Sin[3*(c + d*x)] + 100*a^3*B*Sin[3*(c + d*x)] + 240*a*b^2*B*Sin[3*(c + d*x)] + 240*a^2*b*C*Sin[3*(c + d*x)] + 45*a^3*A*Sin[4*(c + d*x)] + 90*a*A*b^2*Sin[4*(c + d*x)] + 90*a^2*b*B*Sin[4*(c + d*x)] + 30*a^3*C*Sin[4*(c + d*x)] + 36*a^2*A*b*Sin[5*(c + d*x)] + 12*a^3*B*Sin[5*(c + d*x)] + 5*a^3*A*Sin[6*(c + d*x)])/(960*d)","A",1
887,1,486,491,4.4810532,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{\sec ^6(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(-8 b^2 \left(3 \sin (2 (c+d x)) \left(42 a^2 C+28 a b B+7 A b^2+6 b^2 C\right)+35 b (4 a C+b B) \sin (c+d x)+30 b^2 C \tan (c+d x)\right)-70 b \sin (c+d x) \cos ^2(c+d x) \left(24 a^3 C+36 a^2 b B+4 a b^2 (6 A+5 C)+5 b^3 B\right)-16 \sin (c+d x) \cos ^5(c+d x) \left(35 a^4 (3 A+2 C)+280 a^3 b B+84 a^2 b^2 (5 A+4 C)+224 a b^3 B+8 b^4 (7 A+6 C)\right)-105 \sin (c+d x) \cos ^4(c+d x) \left(8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right)-16 \sin (c+d x) \cos ^3(c+d x) \left(35 a^4 C+140 a^3 b B+42 a^2 b^2 (5 A+4 C)+112 a b^3 B+4 b^4 (7 A+6 C)\right)+105 \cos ^6(c+d x) \left(8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{840 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\tan (c+d x) \left(2 a^2 C-7 a b B+42 A b^2+36 b^2 C\right) (a+b \sec (c+d x))^4}{210 b^2 d}-\frac{\tan (c+d x) \left(-8 a^3 C+28 a^2 b B-4 a b^2 (42 A+31 C)-175 b^3 B\right) (a+b \sec (c+d x))^3}{840 b^2 d}+\frac{\left(8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\frac{\tan (c+d x) \left(-8 a^4 C+28 a^3 b B-12 a^2 b^2 (14 A+9 C)-371 a b^3 B-32 b^4 (7 A+6 C)\right) (a+b \sec (c+d x))^2}{840 b^2 d}-\frac{\tan (c+d x) \sec (c+d x) \left(-16 a^5 C+56 a^4 b B-48 a^3 b^2 (7 A+4 C)-1246 a^2 b^3 B-4 a b^4 (406 A+333 C)-525 b^5 B\right)}{1680 b d}-\frac{\tan (c+d x) \left(-8 a^6 C+28 a^5 b B-4 a^4 b^2 (42 A+23 C)-847 a^3 b^3 B-32 a^2 b^4 (49 A+39 C)-896 a b^5 B-32 b^6 (7 A+6 C)\right)}{420 b^2 d}+\frac{(7 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^5}{42 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d}",1,"-1/840*((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^6*(105*(8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 70*b*(36*a^2*b*B + 5*b^3*B + 24*a^3*C + 4*a*b^2*(6*A + 5*C))*Cos[c + d*x]^2*Sin[c + d*x] - 16*(140*a^3*b*B + 112*a*b^3*B + 35*a^4*C + 42*a^2*b^2*(5*A + 4*C) + 4*b^4*(7*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x] - 105*(8*a^4*B + 36*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]^4*Sin[c + d*x] - 16*(280*a^3*b*B + 224*a*b^3*B + 35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C) + 8*b^4*(7*A + 6*C))*Cos[c + d*x]^5*Sin[c + d*x] - 8*b^2*(35*b*(b*B + 4*a*C)*Sin[c + d*x] + 3*(7*A*b^2 + 28*a*b*B + 42*a^2*C + 6*b^2*C)*Sin[2*(c + d*x)] + 30*b^2*C*Tan[c + d*x])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
888,1,424,384,3.5368631,"\int \sec (c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{\sec ^6(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(-10 b^2 \sin (c+d x) \cos ^2(c+d x) \left(36 a^2 C+24 a b B+6 A b^2+5 b^2 C\right)-32 b \sin (c+d x) \cos ^3(c+d x) \left(10 a^3 C+15 a^2 b B+2 a b^2 (5 A+4 C)+2 b^3 B\right)-16 \sin (c+d x) \cos ^5(c+d x) \left(15 a^4 B+20 a^3 b (3 A+2 C)+60 a^2 b^2 B+8 a b^3 (5 A+4 C)+8 b^4 B\right)-15 \sin (c+d x) \cos ^4(c+d x) \left(8 a^4 C+32 a^3 b B+12 a^2 b^2 (4 A+3 C)+24 a b^3 B+b^4 (6 A+5 C)\right)+15 \cos ^6(c+d x) \left(8 a^4 (2 A+C)+32 a^3 b B+12 a^2 b^2 (4 A+3 C)+24 a b^3 B+b^4 (6 A+5 C)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-8 b^3 \sin (c+d x) (6 (4 a C+b B) \cos (c+d x)+5 b C)\right)}{120 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\tan (c+d x) \left(-4 a^3 C+24 a^2 b B+a b^2 (70 A+53 C)+32 b^3 B\right) (a+b \sec (c+d x))^2}{120 b d}+\frac{\left(8 a^4 (2 A+C)+32 a^3 b B+12 a^2 b^2 (4 A+3 C)+24 a b^3 B+b^4 (6 A+5 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\tan (c+d x) \sec (c+d x) \left(-8 a^4 C+48 a^3 b B+2 a^2 b^2 (130 A+89 C)+232 a b^3 B+15 b^4 (6 A+5 C)\right)}{240 d}+\frac{\tan (c+d x) \left(-4 a^5 C+24 a^4 b B+a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+32 a b^4 (5 A+4 C)+32 b^5 B\right)}{60 b d}+\frac{\tan (c+d x) \left(4 a (6 b B-a C)+5 b^2 (6 A+5 C)\right) (a+b \sec (c+d x))^3}{120 b d}+\frac{(6 b B-a C) \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}",1,"-1/120*((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^6*(15*(32*a^3*b*B + 24*a*b^3*B + 8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - 10*b^2*(6*A*b^2 + 24*a*b*B + 36*a^2*C + 5*b^2*C)*Cos[c + d*x]^2*Sin[c + d*x] - 32*b*(15*a^2*b*B + 2*b^3*B + 10*a^3*C + 2*a*b^2*(5*A + 4*C))*Cos[c + d*x]^3*Sin[c + d*x] - 15*(32*a^3*b*B + 24*a*b^3*B + 8*a^4*C + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*Cos[c + d*x]^4*Sin[c + d*x] - 16*(15*a^4*B + 60*a^2*b^2*B + 8*b^4*B + 20*a^3*b*(3*A + 2*C) + 8*a*b^3*(5*A + 4*C))*Cos[c + d*x]^5*Sin[c + d*x] - 8*b^3*(5*b*C + 6*(b*B + 4*a*C)*Cos[c + d*x])*Sin[c + d*x]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","A",1
889,1,690,290,3.9504902,"\int (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^5(c+d x) \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(600 a^4 A (c+d x) \cos (c+d x)+300 a^4 A (c+d x) \cos (3 (c+d x))+60 a^4 A c \cos (5 (c+d x))+60 a^4 A d x \cos (5 (c+d x))+120 a^4 C \sin (c+d x)+180 a^4 C \sin (3 (c+d x))+60 a^4 C \sin (5 (c+d x))+480 a^3 b B \sin (c+d x)+720 a^3 b B \sin (3 (c+d x))+240 a^3 b B \sin (5 (c+d x))+480 a^3 b C \sin (2 (c+d x))+240 a^3 b C \sin (4 (c+d x))+720 a^2 A b^2 \sin (c+d x)+1080 a^2 A b^2 \sin (3 (c+d x))+360 a^2 A b^2 \sin (5 (c+d x))+720 a^2 b^2 B \sin (2 (c+d x))+360 a^2 b^2 B \sin (4 (c+d x))+960 a^2 b^2 C \sin (c+d x)+1200 a^2 b^2 C \sin (3 (c+d x))+240 a^2 b^2 C \sin (5 (c+d x))-120 \cos ^5(c+d x) \left(8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+480 a A b^3 \sin (2 (c+d x))+240 a A b^3 \sin (4 (c+d x))+640 a b^3 B \sin (c+d x)+800 a b^3 B \sin (3 (c+d x))+160 a b^3 B \sin (5 (c+d x))+840 a b^3 C \sin (2 (c+d x))+180 a b^3 C \sin (4 (c+d x))+160 A b^4 \sin (c+d x)+200 A b^4 \sin (3 (c+d x))+40 A b^4 \sin (5 (c+d x))+210 b^4 B \sin (2 (c+d x))+45 b^4 B \sin (4 (c+d x))+320 b^4 C \sin (c+d x)+160 b^4 C \sin (3 (c+d x))+32 b^4 C \sin (5 (c+d x))\right)}{480 d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","a^4 A x+\frac{\tan (c+d x) \left(12 a^2 C+35 a b B+20 A b^2+16 b^2 C\right) (a+b \sec (c+d x))^2}{60 d}+\frac{b \tan (c+d x) \sec (c+d x) \left(24 a^3 C+130 a^2 b B+4 a b^2 (40 A+29 C)+45 b^3 B\right)}{120 d}+\frac{\tan (c+d x) \left(12 a^4 C+95 a^3 b B+2 a^2 b^2 (85 A+56 C)+80 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 d}+\frac{\left(8 a^4 B+16 a^3 b (2 A+C)+24 a^2 b^2 B+4 a b^3 (4 A+3 C)+3 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a C+5 b B) \tan (c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac{C \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}",1,"((C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*Sec[c + d*x]^5*(600*a^4*A*(c + d*x)*Cos[c + d*x] + 300*a^4*A*(c + d*x)*Cos[3*(c + d*x)] + 60*a^4*A*c*Cos[5*(c + d*x)] + 60*a^4*A*d*x*Cos[5*(c + d*x)] - 120*(8*a^4*B + 24*a^2*b^2*B + 3*b^4*B + 16*a^3*b*(2*A + C) + 4*a*b^3*(4*A + 3*C))*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 720*a^2*A*b^2*Sin[c + d*x] + 160*A*b^4*Sin[c + d*x] + 480*a^3*b*B*Sin[c + d*x] + 640*a*b^3*B*Sin[c + d*x] + 120*a^4*C*Sin[c + d*x] + 960*a^2*b^2*C*Sin[c + d*x] + 320*b^4*C*Sin[c + d*x] + 480*a*A*b^3*Sin[2*(c + d*x)] + 720*a^2*b^2*B*Sin[2*(c + d*x)] + 210*b^4*B*Sin[2*(c + d*x)] + 480*a^3*b*C*Sin[2*(c + d*x)] + 840*a*b^3*C*Sin[2*(c + d*x)] + 1080*a^2*A*b^2*Sin[3*(c + d*x)] + 200*A*b^4*Sin[3*(c + d*x)] + 720*a^3*b*B*Sin[3*(c + d*x)] + 800*a*b^3*B*Sin[3*(c + d*x)] + 180*a^4*C*Sin[3*(c + d*x)] + 1200*a^2*b^2*C*Sin[3*(c + d*x)] + 160*b^4*C*Sin[3*(c + d*x)] + 240*a*A*b^3*Sin[4*(c + d*x)] + 360*a^2*b^2*B*Sin[4*(c + d*x)] + 45*b^4*B*Sin[4*(c + d*x)] + 240*a^3*b*C*Sin[4*(c + d*x)] + 180*a*b^3*C*Sin[4*(c + d*x)] + 360*a^2*A*b^2*Sin[5*(c + d*x)] + 40*A*b^4*Sin[5*(c + d*x)] + 240*a^3*b*B*Sin[5*(c + d*x)] + 160*a*b^3*B*Sin[5*(c + d*x)] + 60*a^4*C*Sin[5*(c + d*x)] + 240*a^2*b^2*C*Sin[5*(c + d*x)] + 32*b^4*C*Sin[5*(c + d*x)]))/(480*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","B",1
890,1,813,273,6.8639293,"\int \cos (c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\left(-8 C a^4-32 b B a^3-48 A b^2 a^2-24 b^2 C a^2-16 b^3 B a-4 A b^4-3 b^4 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6(c+d x)}{4 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{\left(8 C a^4+32 b B a^3+48 A b^2 a^2+24 b^2 C a^2+16 b^3 B a+4 A b^4+3 b^4 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^6(c+d x)}{4 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x))^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(36 B (c+d x) a^4+48 B (c+d x) \cos (2 (c+d x)) a^4+12 B (c+d x) \cos (4 (c+d x)) a^4+12 A \sin (c+d x) a^4+18 A \sin (3 (c+d x)) a^4+6 A \sin (5 (c+d x)) a^4+144 A b (c+d x) a^3+192 A b (c+d x) \cos (2 (c+d x)) a^3+48 A b (c+d x) \cos (4 (c+d x)) a^3+96 b C \sin (2 (c+d x)) a^3+48 b C \sin (4 (c+d x)) a^3+72 b^2 C \sin (c+d x) a^2+144 b^2 B \sin (2 (c+d x)) a^2+72 b^2 C \sin (3 (c+d x)) a^2+72 b^2 B \sin (4 (c+d x)) a^2+48 b^3 B \sin (c+d x) a+96 A b^3 \sin (2 (c+d x)) a+128 b^3 C \sin (2 (c+d x)) a+48 b^3 B \sin (3 (c+d x)) a+48 A b^3 \sin (4 (c+d x)) a+32 b^3 C \sin (4 (c+d x)) a+12 A b^4 \sin (c+d x)+33 b^4 C \sin (c+d x)+32 b^4 B \sin (2 (c+d x))+12 A b^4 \sin (3 (c+d x))+9 b^4 C \sin (3 (c+d x))+8 b^4 B \sin (4 (c+d x))\right) \cos ^2(c+d x)}{48 d (b+a \cos (c+d x))^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}","a^3 x (a B+4 A b)+\frac{b^2 \tan (c+d x) \sec (c+d x) \left(-\left(a^2 (24 A-26 C)\right)+32 a b B+3 b^2 (4 A+3 C)\right)}{24 d}+\frac{b \tan (c+d x) \left(-\left(a^3 (12 A-19 C)\right)+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{\left(8 a^4 C+32 a^3 b B+24 a^2 b^2 (2 A+C)+16 a b^3 B+b^4 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}-\frac{b \tan (c+d x) (12 a A-7 a C-4 b B) (a+b \sec (c+d x))^2}{12 d}-\frac{b (4 A-C) \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^4}{d}",1,"((-48*a^2*A*b^2 - 4*A*b^4 - 32*a^3*b*B - 16*a*b^3*B - 8*a^4*C - 24*a^2*b^2*C - 3*b^4*C)*Cos[c + d*x]^6*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((48*a^2*A*b^2 + 4*A*b^4 + 32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*C + 3*b^4*C)*Cos[c + d*x]^6*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(4*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(144*a^3*A*b*(c + d*x) + 36*a^4*B*(c + d*x) + 192*a^3*A*b*(c + d*x)*Cos[2*(c + d*x)] + 48*a^4*B*(c + d*x)*Cos[2*(c + d*x)] + 48*a^3*A*b*(c + d*x)*Cos[4*(c + d*x)] + 12*a^4*B*(c + d*x)*Cos[4*(c + d*x)] + 12*a^4*A*Sin[c + d*x] + 12*A*b^4*Sin[c + d*x] + 48*a*b^3*B*Sin[c + d*x] + 72*a^2*b^2*C*Sin[c + d*x] + 33*b^4*C*Sin[c + d*x] + 96*a*A*b^3*Sin[2*(c + d*x)] + 144*a^2*b^2*B*Sin[2*(c + d*x)] + 32*b^4*B*Sin[2*(c + d*x)] + 96*a^3*b*C*Sin[2*(c + d*x)] + 128*a*b^3*C*Sin[2*(c + d*x)] + 18*a^4*A*Sin[3*(c + d*x)] + 12*A*b^4*Sin[3*(c + d*x)] + 48*a*b^3*B*Sin[3*(c + d*x)] + 72*a^2*b^2*C*Sin[3*(c + d*x)] + 9*b^4*C*Sin[3*(c + d*x)] + 48*a*A*b^3*Sin[4*(c + d*x)] + 72*a^2*b^2*B*Sin[4*(c + d*x)] + 8*b^4*B*Sin[4*(c + d*x)] + 48*a^3*b*C*Sin[4*(c + d*x)] + 32*a*b^3*C*Sin[4*(c + d*x)] + 6*a^4*A*Sin[5*(c + d*x)]))/(48*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","B",1
891,1,348,274,2.3925919,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^3(c+d x) \left(36 a^2 (c+d x) \cos (c+d x) \left(a^2 (A+2 C)+8 a b B+12 A b^2\right)+12 a^2 (c+d x) \cos (3 (c+d x)) \left(a^2 (A+2 C)+8 a b B+12 A b^2\right)-48 b \cos ^3(c+d x) \left(8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 \sin (c+d x) \left(3 a^4 A \cos (4 (c+d x))+9 a^4 A+12 a^4 B \cos (3 (c+d x))+48 a^3 A b \cos (3 (c+d x))+144 a^2 b^2 C+12 \cos (c+d x) \left(3 a^4 B+12 a^3 A b+8 a b^3 C+2 b^4 B\right)+4 \cos (2 (c+d x)) \left(3 a^4 A+36 a^2 b^2 C+24 a b^3 B+6 A b^4+4 b^4 C\right)+96 a b^3 B+24 A b^4+32 b^4 C\right)\right)}{96 d}","-\frac{b^2 \tan (c+d x) \sec (c+d x) \left(6 a^2 B+2 a b (9 A-4 C)-3 b^2 B\right)}{6 d}+\frac{1}{2} a^2 x \left(a^2 (A+2 C)+8 a b B+12 A b^2\right)-\frac{b \tan (c+d x) \left(12 a^3 B+a^2 b (39 A-34 C)-24 a b^2 B-2 b^3 (3 A+2 C)\right)}{6 d}+\frac{b \left(8 a^3 C+12 a^2 b B+4 a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b \tan (c+d x) (6 a B+15 A b-2 b C) (a+b \sec (c+d x))^2}{6 d}+\frac{(a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^4}{2 d}",1,"(Sec[c + d*x]^3*(36*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[c + d*x] + 12*a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[3*(c + d*x)] - 48*b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*Cos[c + d*x]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + 2*(9*a^4*A + 24*A*b^4 + 96*a*b^3*B + 144*a^2*b^2*C + 32*b^4*C + 12*(12*a^3*A*b + 3*a^4*B + 2*b^4*B + 8*a*b^3*C)*Cos[c + d*x] + 4*(3*a^4*A + 6*A*b^4 + 24*a*b^3*B + 36*a^2*b^2*C + 4*b^4*C)*Cos[2*(c + d*x)] + 48*a^3*A*b*Cos[3*(c + d*x)] + 12*a^4*B*Cos[3*(c + d*x)] + 3*a^4*A*Cos[4*(c + d*x)])*Sin[c + d*x]))/(96*d)","A",1
892,1,370,303,4.778938,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^4 A \sin (3 (c+d x))+3 a^3 (a B+4 A b) \sin (2 (c+d x))+3 a^2 \sin (c+d x) \left(a^2 (3 A+4 C)+16 a b B+24 A b^2\right)-6 b^2 \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 b^2 \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 a (c+d x) \left(a^3 B+4 a^2 b (A+2 C)+12 a b^2 B+8 A b^3\right)+\frac{12 b^3 (4 a C+b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 b^3 (4 a C+b B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{3 b^4 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{3 b^4 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{12 d}","\frac{b^2 \left(12 a^2 C+8 a b B+2 A b^2+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{\sin (c+d x) \left(a^2 (4 A+6 C)+15 a b B+12 A b^2\right) (a+b \sec (c+d x))^2}{6 d}-\frac{b^2 \tan (c+d x) \sec (c+d x) \left(a^2 (4 A+6 C)+18 a b B+3 b^2 (6 A-C)\right)}{6 d}-\frac{b \tan (c+d x) \left(4 a^3 (2 A+3 C)+39 a^2 b B+4 a b^2 (11 A-6 C)-6 b^3 B\right)}{6 d}+\frac{1}{2} a x \left(a^3 B+4 a^2 b (A+2 C)+12 a b^2 B+8 A b^3\right)+\frac{(3 a B+4 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^3}{6 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^4}{3 d}",1,"(6*a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*(c + d*x) - 6*b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*b^4*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*b^3*(b*B + 4*a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (3*b^4*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (12*b^3*(b*B + 4*a*C)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*a^2*(24*A*b^2 + 16*a*b*B + a^2*(3*A + 4*C))*Sin[c + d*x] + 3*a^3*(4*A*b + a*B)*Sin[2*(c + d*x)] + a^4*A*Sin[3*(c + d*x)])/(12*d)","A",1
893,1,382,293,3.8187654,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec (c+d x) \left(3 \sin (3 (c+d x)) \left(a^2 (9 A+8 C)+32 a b B+48 A b^2\right)+a (8 (a B+4 A b) \sin (4 (c+d x))+3 a A \sin (5 (c+d x)))\right)+32 a \sin (c+d x) \left(5 a^3 B+4 a^2 b (5 A+6 C)+36 a b^2 B+24 A b^3\right)+24 \left(3 a^4 A c+3 a^4 A d x+4 a^4 c C+4 a^4 C d x+16 a^3 b B c+16 a^3 b B d x+24 a^2 A b^2 c+24 a^2 A b^2 d x+48 a^2 b^2 c C+48 a^2 b^2 C d x+\tan (c+d x) \left(a^4 (A+C)+4 a^3 b B+6 a^2 A b^2+8 b^4 C\right)-8 b^3 (4 a C+b B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+32 a b^3 B c+32 a b^3 B d x+32 a b^3 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 A b^4 c+8 A b^4 d x+8 b^4 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 d}","-\frac{b^2 \tan (c+d x) \left(3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right)}{24 d}+\frac{\sin (c+d x) \cos (c+d x) \left(a^2 (3 A+4 C)+8 a b B+4 A b^2\right) (a+b \sec (c+d x))^2}{8 d}+\frac{a \sin (c+d x) \left(8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right)}{12 d}+\frac{1}{8} x \left(a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right)+\frac{(a B+A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^4}{4 d}+\frac{b^3 (4 a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}",1,"(32*a*(24*A*b^3 + 5*a^3*B + 36*a*b^2*B + 4*a^2*b*(5*A + 6*C))*Sin[c + d*x] + a^2*Sec[c + d*x]*(3*(48*A*b^2 + 32*a*b*B + a^2*(9*A + 8*C))*Sin[3*(c + d*x)] + a*(8*(4*A*b + a*B)*Sin[4*(c + d*x)] + 3*a*A*Sin[5*(c + d*x)])) + 24*(3*a^4*A*c + 24*a^2*A*b^2*c + 8*A*b^4*c + 16*a^3*b*B*c + 32*a*b^3*B*c + 4*a^4*c*C + 48*a^2*b^2*c*C + 3*a^4*A*d*x + 24*a^2*A*b^2*d*x + 8*A*b^4*d*x + 16*a^3*b*B*d*x + 32*a*b^3*B*d*x + 4*a^4*C*d*x + 48*a^2*b^2*C*d*x - 8*b^3*(b*B + 4*a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*b^4*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 32*a*b^3*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (6*a^2*A*b^2 + 4*a^3*b*B + 8*b^4*C + a^4*(A + C))*Tan[c + d*x]))/(192*d)","A",1
894,1,382,314,1.2701857,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{50 a^4 A \sin (3 (c+d x))+6 a^4 A \sin (5 (c+d x))+15 a^4 B \sin (4 (c+d x))+180 a^4 B c+180 a^4 B d x+40 a^4 C \sin (3 (c+d x))+60 a^3 A b \sin (4 (c+d x))+720 a^3 A b c+720 a^3 A b d x+160 a^3 b B \sin (3 (c+d x))+960 a^3 b c C+960 a^3 b C d x+240 a^2 A b^2 \sin (3 (c+d x))+1440 a^2 b^2 B c+1440 a^2 b^2 B d x+120 a \sin (2 (c+d x)) \left(a^3 B+4 a^2 b (A+C)+6 a b^2 B+4 A b^3\right)+60 \sin (c+d x) \left(a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 A b^4\right)+960 a A b^3 c+960 a A b^3 d x+1920 a b^3 c C+1920 a b^3 C d x+480 b^4 B c+480 b^4 B d x-480 b^4 C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+480 b^4 C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{480 d}","\frac{\sin (c+d x) \cos ^2(c+d x) \left(4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right) (a+b \sec (c+d x))^2}{60 d}+\frac{a \sin (c+d x) \cos (c+d x) \left(45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right)}{120 d}+\frac{\sin (c+d x) \left(4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right)}{30 d}+\frac{1}{8} x \left(3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right)+\frac{(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac{b^4 C \tanh ^{-1}(\sin (c+d x))}{d}",1,"(720*a^3*A*b*c + 960*a*A*b^3*c + 180*a^4*B*c + 1440*a^2*b^2*B*c + 480*b^4*B*c + 960*a^3*b*c*C + 1920*a*b^3*c*C + 720*a^3*A*b*d*x + 960*a*A*b^3*d*x + 180*a^4*B*d*x + 1440*a^2*b^2*B*d*x + 480*b^4*B*d*x + 960*a^3*b*C*d*x + 1920*a*b^3*C*d*x - 480*b^4*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 480*b^4*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 60*(8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*Sin[c + d*x] + 120*a*(4*A*b^3 + a^3*B + 6*a*b^2*B + 4*a^2*b*(A + C))*Sin[2*(c + d*x)] + 50*a^4*A*Sin[3*(c + d*x)] + 240*a^2*A*b^2*Sin[3*(c + d*x)] + 160*a^3*b*B*Sin[3*(c + d*x)] + 40*a^4*C*Sin[3*(c + d*x)] + 60*a^3*A*b*Sin[4*(c + d*x)] + 15*a^4*B*Sin[4*(c + d*x)] + 6*a^4*A*Sin[5*(c + d*x)])/(480*d)","A",1
895,1,432,372,1.5700269,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{45 a^4 A \sin (4 (c+d x))+5 a^4 A \sin (6 (c+d x))+300 a^4 A c+300 a^4 A d x+100 a^4 B \sin (3 (c+d x))+12 a^4 B \sin (5 (c+d x))+30 a^4 C \sin (4 (c+d x))+360 a^4 c C+360 a^4 C d x+400 a^3 A b \sin (3 (c+d x))+48 a^3 A b \sin (5 (c+d x))+120 a^3 b B \sin (4 (c+d x))+1440 a^3 b B c+1440 a^3 b B d x+320 a^3 b C \sin (3 (c+d x))+180 a^2 A b^2 \sin (4 (c+d x))+2160 a^2 A b^2 c+2160 a^2 A b^2 d x+480 a^2 b^2 B \sin (3 (c+d x))+2880 a^2 b^2 c C+2880 a^2 b^2 C d x+120 \sin (c+d x) \left(5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)+15 \sin (2 (c+d x)) \left(a^4 (15 A+16 C)+64 a^3 b B+96 a^2 b^2 (A+C)+64 a b^3 B+16 A b^4\right)+320 a A b^3 \sin (3 (c+d x))+1920 a b^3 B c+1920 a b^3 B d x+480 A b^4 c+480 A b^4 d x+960 b^4 c C+960 b^4 C d x}{960 d}","\frac{\sin (c+d x) \cos ^3(c+d x) \left(5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right) (a+b \sec (c+d x))^2}{120 d}+\frac{a \sin (c+d x) \cos ^2(c+d x) \left(16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right)}{60 d}+\frac{\sin (c+d x) \left(8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right)}{15 d}+\frac{\sin (c+d x) \cos (c+d x) \left(15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right)}{240 d}+\frac{1}{16} x \left(a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right)+\frac{(3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{15 d}+\frac{A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}",1,"(300*a^4*A*c + 2160*a^2*A*b^2*c + 480*A*b^4*c + 1440*a^3*b*B*c + 1920*a*b^3*B*c + 360*a^4*c*C + 2880*a^2*b^2*c*C + 960*b^4*c*C + 300*a^4*A*d*x + 2160*a^2*A*b^2*d*x + 480*A*b^4*d*x + 1440*a^3*b*B*d*x + 1920*a*b^3*B*d*x + 360*a^4*C*d*x + 2880*a^2*b^2*C*d*x + 960*b^4*C*d*x + 120*(5*a^4*B + 36*a^2*b^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Sin[c + d*x] + 15*(16*A*b^4 + 64*a^3*b*B + 64*a*b^3*B + 96*a^2*b^2*(A + C) + a^4*(15*A + 16*C))*Sin[2*(c + d*x)] + 400*a^3*A*b*Sin[3*(c + d*x)] + 320*a*A*b^3*Sin[3*(c + d*x)] + 100*a^4*B*Sin[3*(c + d*x)] + 480*a^2*b^2*B*Sin[3*(c + d*x)] + 320*a^3*b*C*Sin[3*(c + d*x)] + 45*a^4*A*Sin[4*(c + d*x)] + 180*a^2*A*b^2*Sin[4*(c + d*x)] + 120*a^3*b*B*Sin[4*(c + d*x)] + 30*a^4*C*Sin[4*(c + d*x)] + 48*a^3*A*b*Sin[5*(c + d*x)] + 12*a^4*B*Sin[5*(c + d*x)] + 5*a^4*A*Sin[6*(c + d*x)])/(960*d)","A",1
896,1,528,438,1.4617704,"\int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^7*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{735 a^4 A \sin (3 (c+d x))+147 a^4 A \sin (5 (c+d x))+15 a^4 A \sin (7 (c+d x))+315 a^4 B \sin (4 (c+d x))+35 a^4 B \sin (6 (c+d x))+2100 a^4 B c+2100 a^4 B d x+700 a^4 C \sin (3 (c+d x))+84 a^4 C \sin (5 (c+d x))+1260 a^3 A b \sin (4 (c+d x))+140 a^3 A b \sin (6 (c+d x))+8400 a^3 A b c+8400 a^3 A b d x+2800 a^3 b B \sin (3 (c+d x))+336 a^3 b B \sin (5 (c+d x))+840 a^3 b C \sin (4 (c+d x))+10080 a^3 b c C+10080 a^3 b C d x+4200 a^2 A b^2 \sin (3 (c+d x))+504 a^2 A b^2 \sin (5 (c+d x))+1260 a^2 b^2 B \sin (4 (c+d x))+15120 a^2 b^2 B c+15120 a^2 b^2 B d x+3360 a^2 b^2 C \sin (3 (c+d x))+105 \sin (c+d x) \left(5 a^4 (7 A+8 C)+160 a^3 b B+48 a^2 b^2 (5 A+6 C)+192 a b^3 B+16 b^4 (3 A+4 C)\right)+105 \sin (2 (c+d x)) \left(15 a^4 B+a^3 (60 A b+64 b C)+96 a^2 b^2 B+64 a b^3 (A+C)+16 b^4 B\right)+840 a A b^3 \sin (4 (c+d x))+10080 a A b^3 c+10080 a A b^3 d x+2240 a b^3 B \sin (3 (c+d x))+13440 a b^3 c C+13440 a b^3 C d x+560 A b^4 \sin (3 (c+d x))+3360 b^4 B c+3360 b^4 B d x}{6720 d}","\frac{\sin (c+d x) \cos ^4(c+d x) \left(2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right) (a+b \sec (c+d x))^2}{70 d}+\frac{a \sin (c+d x) \cos ^3(c+d x) \left(175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right)}{840 d}-\frac{\sin ^3(c+d x) \left(4 a^4 (6 A+7 C)+112 a^3 b B+3 a^2 b^2 (50 A+63 C)+91 a b^3 B+4 A b^4\right)}{105 d}+\frac{\sin (c+d x) \left(12 a^4 (6 A+7 C)+336 a^3 b B+3 a^2 b^2 (162 A+203 C)+371 a b^3 B+b^4 (74 A+105 C)\right)}{105 d}+\frac{\sin (c+d x) \cos (c+d x) \left(5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)}{16 d}+\frac{1}{16} x \left(5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)+\frac{(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{42 d}+\frac{A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}",1,"(8400*a^3*A*b*c + 10080*a*A*b^3*c + 2100*a^4*B*c + 15120*a^2*b^2*B*c + 3360*b^4*B*c + 10080*a^3*b*c*C + 13440*a*b^3*c*C + 8400*a^3*A*b*d*x + 10080*a*A*b^3*d*x + 2100*a^4*B*d*x + 15120*a^2*b^2*B*d*x + 3360*b^4*B*d*x + 10080*a^3*b*C*d*x + 13440*a*b^3*C*d*x + 105*(160*a^3*b*B + 192*a*b^3*B + 16*b^4*(3*A + 4*C) + 48*a^2*b^2*(5*A + 6*C) + 5*a^4*(7*A + 8*C))*Sin[c + d*x] + 105*(15*a^4*B + 96*a^2*b^2*B + 16*b^4*B + 64*a*b^3*(A + C) + a^3*(60*A*b + 64*b*C))*Sin[2*(c + d*x)] + 735*a^4*A*Sin[3*(c + d*x)] + 4200*a^2*A*b^2*Sin[3*(c + d*x)] + 560*A*b^4*Sin[3*(c + d*x)] + 2800*a^3*b*B*Sin[3*(c + d*x)] + 2240*a*b^3*B*Sin[3*(c + d*x)] + 700*a^4*C*Sin[3*(c + d*x)] + 3360*a^2*b^2*C*Sin[3*(c + d*x)] + 1260*a^3*A*b*Sin[4*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 315*a^4*B*Sin[4*(c + d*x)] + 1260*a^2*b^2*B*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] + 147*a^4*A*Sin[5*(c + d*x)] + 504*a^2*A*b^2*Sin[5*(c + d*x)] + 336*a^3*b*B*Sin[5*(c + d*x)] + 84*a^4*C*Sin[5*(c + d*x)] + 140*a^3*A*b*Sin[6*(c + d*x)] + 35*a^4*B*Sin[6*(c + d*x)] + 15*a^4*A*Sin[7*(c + d*x)])/(6720*d)","A",1
897,1,170,214,1.3038186,"\int (a+b \sec (c+d x))^3 \left(a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^3*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2),x]","\frac{24 a^4 d x (b B-a C)+3 b^2 \tan (c+d x) \left(b \left(8 a^2 C+16 a b B+3 b^2 C\right) \sec (c+d x)+8 \left(-2 a^3 C+6 a^2 b B+3 a b^2 C+b^3 B\right)+2 b^3 C \sec ^3(c+d x)\right)+3 b \left(-24 a^4 C+32 a^3 b B+8 a^2 b^2 C+16 a b^3 B+3 b^4 C\right) \tanh ^{-1}(\sin (c+d x))+8 b^4 (3 a C+b B) \tan ^3(c+d x)}{24 d}","a^4 x (b B-a C)+\frac{b^3 \left(-6 a^2 C+32 a b B+9 b^2 C\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{b^2 \left(-15 a^3 C+34 a^2 b B+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 d}+\frac{b \left(-24 a^4 C+32 a^3 b B+8 a^2 b^2 C+16 a b^3 B+3 b^4 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 (3 a C+4 b B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{b^2 C \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(24*a^4*(b*B - a*C)*d*x + 3*b*(32*a^3*b*B + 16*a*b^3*B - 24*a^4*C + 8*a^2*b^2*C + 3*b^4*C)*ArcTanh[Sin[c + d*x]] + 3*b^2*(8*(6*a^2*b*B + b^3*B - 2*a^3*C + 3*a*b^2*C) + b*(16*a*b*B + 8*a^2*C + 3*b^2*C)*Sec[c + d*x] + 2*b^3*C*Sec[c + d*x]^3)*Tan[c + d*x] + 8*b^4*(b*B + 3*a*C)*Tan[c + d*x]^3)/(24*d)","A",1
898,1,114,149,0.9273987,"\int (a+b \sec (c+d x))^2 \left(a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^2*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2),x]","\frac{6 a^3 d x (b B-a C)+3 b \left(-4 a^3 C+6 a^2 b B+2 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))+3 b^3 \tan (c+d x) \sec (c+d x) (2 (3 a B+b C) \cos (c+d x)+2 a C+b B)+2 b^4 C \tan ^3(c+d x)}{6 d}","a^3 x (b B-a C)+\frac{b^2 \left(a^2 (-C)+9 a b B+2 b^2 C\right) \tan (c+d x)}{3 d}+\frac{b \left(-4 a^3 C+6 a^2 b B+2 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^3 (2 a C+3 b B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{b^2 C \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(6*a^3*(b*B - a*C)*d*x + 3*b*(6*a^2*b*B + b^3*B - 4*a^3*C + 2*a*b^2*C)*ArcTanh[Sin[c + d*x]] + 3*b^3*(b*B + 2*a*C + 2*(3*a*B + b*C)*Cos[c + d*x])*Sec[c + d*x]*Tan[c + d*x] + 2*b^4*C*Tan[c + d*x]^3)/(6*d)","A",1
899,1,77,97,0.5804425,"\int (a+b \sec (c+d x)) \left(a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2),x]","\frac{b \left(-2 a^2 C+4 a b B+b^2 C\right) \tanh ^{-1}(\sin (c+d x))+2 a^2 d x (b B-a C)+b^2 \tan (c+d x) (2 a C+2 b B+b C \sec (c+d x))}{2 d}","\frac{b \left(-2 a^2 C+4 a b B+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 x (b B-a C)+\frac{b^2 (a C+2 b B) \tan (c+d x)}{2 d}+\frac{b^2 C \tan (c+d x) (a+b \sec (c+d x))}{2 d}",1,"(2*a^2*(b*B - a*C)*d*x + b*(4*a*b*B - 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]] + b^2*(2*b*B + 2*a*C + b*C*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
900,1,512,215,3.6626793,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(b \sec (c) \left(-6 \sin (2 c+d x) \left(a^2 C-a b B+A b^2\right)+12 \sin (d x) \left(a^2 C-a b B+A b^2+b^2 C\right)+6 a^2 C \sin (2 c+3 d x)-6 a b B \sin (2 c+3 d x)-3 a b C \sin (c+2 d x)-3 a b C \sin (3 c+2 d x)+6 A b^2 \sin (2 c+3 d x)+3 b^2 B \sin (c+2 d x)+3 b^2 B \sin (3 c+2 d x)+4 b^2 C \sin (2 c+3 d x)\right)-\frac{48 i a^2 (\cos (c)-i \sin (c)) \cos ^3(c+d x) \left(a (a C-b B)+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}+12 \cos ^3(c+d x) \left(2 a^3 C-2 a^2 b B+a b^2 (2 A+C)-b^3 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-12 \cos ^3(c+d x) \left(2 a^3 C-2 a^2 b B+a b^2 (2 A+C)-b^3 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{12 b^4 d (a+b \sec (c+d x)) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 a^2 \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{\tan (c+d x) \left(3 a^2 C-3 a b B+3 A b^2+2 b^2 C\right)}{3 b^3 d}+\frac{\left(-2 a^3 C+2 a^2 b B-a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{(b B-a C) \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{C \tan (c+d x) \sec ^2(c+d x)}{3 b d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(12*(-2*a^2*b*B - b^3*B + 2*a^3*C + a*b^2*(2*A + C))*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 12*(-2*a^2*b*B - b^3*B + 2*a^3*C + a*b^2*(2*A + C))*Cos[c + d*x]^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((48*I)*a^2*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*Cos[c + d*x]^3*(Cos[c] - I*Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) + b*Sec[c]*(12*(A*b^2 - a*b*B + a^2*C + b^2*C)*Sin[d*x] - 6*(A*b^2 - a*b*B + a^2*C)*Sin[2*c + d*x] + 3*b^2*B*Sin[c + 2*d*x] - 3*a*b*C*Sin[c + 2*d*x] + 3*b^2*B*Sin[3*c + 2*d*x] - 3*a*b*C*Sin[3*c + 2*d*x] + 6*A*b^2*Sin[2*c + 3*d*x] - 6*a*b*B*Sin[2*c + 3*d*x] + 6*a^2*C*Sin[2*c + 3*d*x] + 4*b^2*C*Sin[2*c + 3*d*x])))/(12*b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x]))","C",1
901,1,472,153,2.6050364,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\cos (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-2 \left(2 a^2 C-2 a b B+2 A b^2+b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(2 a^2 C-2 a b B+2 A b^2+b^2 C\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 a (\sin (c)+i \cos (c)) \left(a (a C-b B)+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{4 b \sin \left(\frac{d x}{2}\right) (b B-a C)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 b \sin \left(\frac{d x}{2}\right) (b B-a C)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{b^2 C}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 C}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{2 b^3 d (a+b \sec (c+d x)) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\left(b^2 (2 A+C)-2 a (b B-a C)\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}-\frac{2 a \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}+\frac{(b B-a C) \tan (c+d x)}{b^2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 b d}",1,"(Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-2*(2*A*b^2 - 2*a*b*B + 2*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 - 2*a*b*B + 2*a^2*C + b^2*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (8*a*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(I*Cos[c] + Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b^2*C)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*b*(b*B - a*C)*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - (b^2*C)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b*(b*B - a*C)*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(2*b^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x]))","C",1
902,1,365,106,2.6372925,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{2 \cos (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{2 i (\cos (c)-i \sin (c)) \left(a (a C-b B)+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}-\left((b B-a C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+(b B-a C) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b C \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{b C \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{b^2 d (a+b \sec (c+d x)) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d \sqrt{a-b} \sqrt{a+b}}+\frac{(b B-a C) \tanh ^{-1}(\sin (c+d x))}{b^2 d}+\frac{C \tan (c+d x)}{b d}",1,"(2*Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-((b*B - a*C)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + (b*B - a*C)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - ((2*I)*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(Cos[c] - I*Sin[c]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b*C*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (b*C*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(b^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x]))","C",1
903,1,261,94,0.5106633,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","\frac{2 \left(A \cos ^2(c+d x)+B \cos (c+d x)+C\right) \left(2 (\sin (c)+i \cos (c)) \left(a (a C-b B)+A b^2\right) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)+\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2} \left(-a C \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+a C \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+A b d x\right)\right)}{a b d \sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{2 \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a b d \sqrt{a-b} \sqrt{a+b}}+\frac{A x}{a}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b d}",1,"(2*(C + B*Cos[c + d*x] + A*Cos[c + d*x]^2)*(Sqrt[a^2 - b^2]*(A*b*d*x - a*C*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + a*C*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sqrt[(Cos[c] - I*Sin[c])^2] + 2*(A*b^2 + a*(-(b*B) + a*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(I*Cos[c] + Sin[c])))/(a*b*Sqrt[a^2 - b^2]*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sqrt[(Cos[c] - I*Sin[c])^2])","C",1
904,1,92,98,0.2393469,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+(c+d x) (a B-A b)+a A \sin (c+d x)}{a^2 d}","\frac{2 \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x (A b-a B)}{a^2}+\frac{A \sin (c+d x)}{a d}",1,"((-(A*b) + a*B)*(c + d*x) - (2*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*A*Sin[c + d*x])/(a^2*d)","A",1
905,1,131,145,0.4403223,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{2 (c+d x) \left(a^2 (A+2 C)-2 a b B+2 A b^2\right)+\frac{8 b \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a^2 A \sin (2 (c+d x))+4 a (a B-A b) \sin (c+d x)}{4 a^3 d}","-\frac{2 b \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \sin (c+d x)}{a^2 d}+\frac{x \left(a^2 (A+2 C)-2 a b B+2 A b^2\right)}{2 a^3}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d}",1,"(2*(2*A*b^2 - 2*a*b*B + a^2*(A + 2*C))*(c + d*x) + (8*b*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 4*a*(-(A*b) + a*B)*Sin[c + d*x] + a^2*A*Sin[2*(c + d*x)])/(4*a^3*d)","A",1
906,1,178,205,0.6248832,"\int \frac{\cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{a^3 A \sin (3 (c+d x))+3 a \sin (c+d x) \left(a^2 (3 A+4 C)-4 a b B+4 A b^2\right)-\frac{24 b^2 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+3 a^2 (a B-A b) \sin (2 (c+d x))+6 (c+d x) \left(a^3 B-a^2 b (A+2 C)+2 a b^2 B-2 A b^3\right)}{12 a^4 d}","\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{\sin (c+d x) \left(a^2 (2 A+3 C)-3 a b B+3 A b^2\right)}{3 a^3 d}-\frac{x \left(a^3 (-B)+a^2 b (A+2 C)-2 a b^2 B+2 A b^3\right)}{2 a^4}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"(6*(-2*A*b^3 + a^3*B + 2*a*b^2*B - a^2*b*(A + 2*C))*(c + d*x) - (24*b^2*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 3*a*(4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*Sin[c + d*x] + 3*a^2*(-(A*b) + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^4*d)","A",1
907,1,235,276,0.857783,"\int \frac{\cos ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{3 a^4 A \sin (4 (c+d x))+8 a^3 (a B-A b) \sin (3 (c+d x))+24 a^2 \sin (2 (c+d x)) \left(a^2 (A+C)-a b B+A b^2\right)+\frac{192 b^3 \left(a (a C-b B)+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+24 a \sin (c+d x) \left(3 a^3 B-a^2 b (3 A+4 C)+4 a b^2 B-4 A b^3\right)+12 (c+d x) \left(a^4 (3 A+4 C)-4 a^3 b B+4 a^2 b^2 (A+2 C)-8 a b^3 B+8 A b^4\right)}{96 a^5 d}","-\frac{2 b^3 \left(A b^2-a (b B-a C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{\sin (c+d x) \cos (c+d x) \left(a^2 (3 A+4 C)-4 a b B+4 A b^2\right)}{8 a^3 d}-\frac{\sin (c+d x) \left(-2 a^3 B+a^2 b (2 A+3 C)-3 a b^2 B+3 A b^3\right)}{3 a^4 d}+\frac{x \left(a^4 (3 A+4 C)-4 a^3 b B+4 a^2 b^2 (A+2 C)-8 a b^3 B+8 A b^4\right)}{8 a^5}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"(12*(8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*(c + d*x) + (192*b^3*(A*b^2 + a*(-(b*B) + a*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 24*a*(-4*A*b^3 + 3*a^3*B + 4*a*b^2*B - a^2*b*(3*A + 4*C))*Sin[c + d*x] + 24*a^2*(A*b^2 - a*b*B + a^2*(A + C))*Sin[2*(c + d*x)] + 8*a^3*(-(A*b) + a*B)*Sin[3*(c + d*x)] + 3*a^4*A*Sin[4*(c + d*x)])/(96*a^5*d)","A",1
908,1,605,407,4.0328757,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{(a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(6 \left(8 a^3 C-6 a^2 b B+2 a b^2 (2 A+C)-b^3 B\right) (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 \left(-8 a^3 C+6 a^2 b B-2 a b^2 (2 A+C)+b^3 B\right) (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\frac{24 a^2 \left(4 a^4 C-3 a^3 b B+a^2 b^2 (2 A-5 C)+4 a b^3 B-3 A b^4\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b \tan (c+d x) \sec ^2(c+d x) \left(-12 a^5 C \cos (3 (c+d x))+9 a^4 b B \cos (3 (c+d x))-12 a^4 b C-6 a^3 A b^2 \cos (3 (c+d x))+9 a^3 b^2 B+7 a^3 b^2 C \cos (3 (c+d x))-6 a^2 A b^3+b \left(b^2-a^2\right) \cos (2 (c+d x)) \left(12 a^2 C-9 a b B+6 A b^2+4 b^2 C\right)-6 a^2 b^3 B \cos (3 (c+d x))+4 a^2 b^3 C+\cos (c+d x) \left(-36 a^5 C+27 a^4 b B+a^3 b^2 (29 C-18 A)-24 a^2 b^3 B+a b^4 (9 A-2 C)+6 b^5 B\right)+3 a A b^4 \cos (3 (c+d x))-9 a b^4 B+2 a b^4 C \cos (3 (c+d x))+6 A b^5+8 b^5 C\right)}{b^2-a^2}\right)}{6 b^5 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{\tan (c+d x) \sec ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(4 a^2 C-3 a b B+3 A b^2-b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{\tan (c+d x) \sec (c+d x) \left(-4 a^3 C+3 a^2 b B-2 a b^2 (A-C)-b^3 B\right)}{2 b^3 d \left(a^2-b^2\right)}+\frac{\left(-8 a^3 C+6 a^2 b B-2 a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}-\frac{\tan (c+d x) \left(-12 a^4 C+9 a^3 b B-a^2 b^2 (6 A-7 C)-6 a b^3 B+b^4 (3 A+2 C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(4 a^4 C-3 a^3 b B+2 a^2 A b^2-5 a^2 b^2 C+4 a b^3 B-3 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-24*a^2*(-3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + a^2*b^2*(2*A - 5*C) + 4*a^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) + 6*(-6*a^2*b*B - b^3*B + 8*a^3*C + 2*a*b^2*(2*A + C))*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b*(-6*a^2*A*b^3 + 6*A*b^5 + 9*a^3*b^2*B - 9*a*b^4*B - 12*a^4*b*C + 4*a^2*b^3*C + 8*b^5*C + (27*a^4*b*B - 24*a^2*b^3*B + 6*b^5*B + a*b^4*(9*A - 2*C) - 36*a^5*C + a^3*b^2*(-18*A + 29*C))*Cos[c + d*x] + b*(-a^2 + b^2)*(6*A*b^2 - 9*a*b*B + 12*a^2*C + 4*b^2*C)*Cos[2*(c + d*x)] - 6*a^3*A*b^2*Cos[3*(c + d*x)] + 3*a*A*b^4*Cos[3*(c + d*x)] + 9*a^4*b*B*Cos[3*(c + d*x)] - 6*a^2*b^3*B*Cos[3*(c + d*x)] - 12*a^5*C*Cos[3*(c + d*x)] + 7*a^3*b^2*C*Cos[3*(c + d*x)] + 2*a*b^4*C*Cos[3*(c + d*x)])*Sec[c + d*x]^2*Tan[c + d*x])/(-a^2 + b^2)))/(6*b^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","A",0
909,1,519,312,3.0212916,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{(a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 a^2 b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(b-a) (a+b)}-2 \left(6 a^2 C-4 a b B+2 A b^2+b^2 C\right) (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(6 a^2 C-4 a b B+2 A b^2+b^2 C\right) (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{8 a \left(3 a^4 C-2 a^3 b B+a^2 b^2 (A-4 C)+3 a b^3 B-2 A b^4\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{b^2 C (a \cos (c+d x)+b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 C (a \cos (c+d x)+b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{4 b (b B-2 a C) \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b (b B-2 a C) \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{2 b^4 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{\tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\tan (c+d x) \sec (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right)}{2 b^2 d \left(a^2-b^2\right)}+\frac{\left(6 a^2 C-4 a b B+2 A b^2+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\tan (c+d x) \left(-3 a^3 C+2 a^2 b B-a b^2 (A-2 C)-b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{2 a \left(3 a^4 C-2 a^3 b B+a^2 A b^2-4 a^2 b^2 C+3 a b^3 B-2 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*a*(-2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + a^2*b^2*(A - 4*C) + 3*a^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) - 2*(2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*C*(b + a*Cos[c + d*x]))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*b*(b*B - 2*a*C)*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^2*C*(b + a*Cos[c + d*x]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b*(b*B - 2*a*C)*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (4*a^2*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((-a + b)*(a + b))))/(2*b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","A",0
910,1,382,177,3.0029852,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(a \left(-2 a^3 C+a^2 b B+3 a b^2 C-2 b^3 B\right)+A b^4\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b)}-(b B-2 a C) (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+(b B-2 a C) (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{b C \sin \left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}\right)}{b^3 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 \left(-2 a^4 C+a^3 b B+3 a^2 b^2 C-2 a b^3 B+A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}",1,"(2*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(A*b^4 + a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) - (b*B - 2*a*C)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + (b*B - 2*a*C)*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (b*C*(b + a*Cos[c + d*x])*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + (a*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b))))/(b^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","B",0
911,1,356,148,3.1386806,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 (\sin (c)+i \cos (c)) \left(a^3 C-a b^2 (A+2 C)+b^3 B\right) (a \cos (c+d x)+b) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{3/2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{b \left(a (a C-b B)+A b^2\right) (b \sin (c)-a \sin (d x))}{a (a-b) (a+b) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}-C (a \cos (c+d x)+b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+C (a \cos (c+d x)+b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{b^2 d (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \left(a^3 (-C)+a A b^2+2 a b^2 C-b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"(2*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-(C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]) + C*(b + a*Cos[c + d*x])*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*(b^3*B + a^3*C - a*b^2*(A + 2*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(3/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b*(A*b^2 + a*(-(b*B) + a*C))*(b*Sin[c] - a*Sin[d*x]))/(a*(a - b)*(a + b)*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(b^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","C",0
912,1,299,138,2.285934,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","\frac{2 (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{2 i (\cos (c)-i \sin (c)) \left(a^3 B-a^2 b (2 A+C)+A b^3\right) (a \cos (c+d x)+b) \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{d \left(a^2-b^2\right)^{3/2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{\left(a (a C-b B)+A b^2\right) (a \sin (d x)-b \sin (c))}{d (a-b) (a+b) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right)}+A x (a \cos (c+d x)+b)\right)}{a^2 (a+b \sec (c+d x))^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{A x}{a^2}-\frac{2 \left(a^3 (-B)+2 a^2 A b+a^2 b C-A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*x*(b + a*Cos[c + d*x]) - ((2*I)*(A*b^3 + a^3*B - a^2*b*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^(3/2)*d*Sqrt[(Cos[c] - I*Sin[c])^2]) + ((A*b^2 + a*(-(b*B) + a*C))*(-(b*Sin[c]) + a*Sin[d*x]))/((a - b)*(a + b)*d*(Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2]))))/(a^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^2)","C",0
913,1,160,202,0.9644078,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{-\frac{2 \left(a^4 C-2 a^3 b B+3 a^2 A b^2+a b^3 B-2 A b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{a b \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a \cos (c+d x)+b)}+(c+d x) (a B-2 A b)+a A \sin (c+d x)}{a^3 d}","-\frac{x (2 A b-a B)}{a^3}-\frac{\sin (c+d x) \left(-\left(a^2 (A-C)\right)-a b B+2 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{2 \left(a^4 C-2 a^3 b B+3 a^2 A b^2+a b^3 B-2 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((-2*A*b + a*B)*(c + d*x) - (2*(3*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + a*b^3*B + a^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + a*A*Sin[c + d*x] - (a*b*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])))/(a^3*d)","A",1
914,1,206,298,1.4448435,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (c+d x) \left(a^2 (A+2 C)-4 a b B+6 A b^2\right)+a^2 A \sin (2 (c+d x))-\frac{8 b \left(-2 a^4 C+3 a^3 b B+a^2 b^2 (C-4 A)-2 a b^3 B+3 A b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{4 a b^2 \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a \cos (c+d x)+b)}+4 a (a B-2 A b) \sin (c+d x)}{4 a^4 d}","-\frac{\sin (c+d x) \cos (c+d x) \left(-\left(a^2 (A-2 C)\right)-2 a b B+3 A b^2\right)}{2 a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \cos (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x \left(a^2 (A+2 C)-4 a b B+6 A b^2\right)}{2 a^4}+\frac{\sin (c+d x) \left(a^3 B-a^2 b (2 A-C)-2 a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{2 b \left(2 a^4 C-3 a^3 b B+4 a^2 A b^2-a^2 b^2 C+2 a b^3 B-3 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*(6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*(c + d*x) - (8*b*(3*A*b^4 + 3*a^3*b*B - 2*a*b^3*B - 2*a^4*C + a^2*b^2*(-4*A + C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 4*a*(-2*A*b + a*B)*Sin[c + d*x] + (4*a*b^2*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*A*Sin[2*(c + d*x)])/(4*a^4*d)","A",1
915,1,255,396,1.7812094,"\int \frac{\cos ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{a^3 A \sin (3 (c+d x))+3 a \sin (c+d x) \left(a^2 (3 A+4 C)-8 a b B+12 A b^2\right)+3 a^2 (a B-2 A b) \sin (2 (c+d x))+6 (c+d x) \left(a^3 B-2 a^2 b (A+2 C)+6 a b^2 B-8 A b^3\right)+\frac{24 b^2 \left(-3 a^4 C+4 a^3 b B+a^2 b^2 (2 C-5 A)-3 a b^3 B+4 A b^4\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{12 a b^3 \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{(a-b) (a+b) (a \cos (c+d x)+b)}}{12 a^5 d}","-\frac{\sin (c+d x) \cos ^2(c+d x) \left(-\left(a^2 (A-3 C)\right)-3 a b B+4 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \cos ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\sin (c+d x) \cos (c+d x) \left(a^3 B-2 a^2 b (A-C)-3 a b^2 B+4 A b^3\right)}{2 a^3 d \left(a^2-b^2\right)}-\frac{x \left(a^3 (-B)+2 a^2 b (A+2 C)-6 a b^2 B+8 A b^3\right)}{2 a^5}-\frac{\sin (c+d x) \left(-\left(a^4 (2 A+3 C)\right)+6 a^3 b B-a^2 b^2 (7 A-6 C)-9 a b^3 B+12 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(3 a^4 C-4 a^3 b B+5 a^2 A b^2-2 a^2 b^2 C+3 a b^3 B-4 A b^4\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(6*(-8*A*b^3 + a^3*B + 6*a*b^2*B - 2*a^2*b*(A + 2*C))*(c + d*x) + (24*b^2*(4*A*b^4 + 4*a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(-5*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 3*a*(12*A*b^2 - 8*a*b*B + a^2*(3*A + 4*C))*Sin[c + d*x] - (12*a*b^3*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + 3*a^2*(-2*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^5*d)","A",1
916,1,1124,465,6.4309602,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\left(-12 C a^2+6 b B a-2 A b^2-b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^3}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\left(12 C a^2-6 b B a+2 A b^2+b^2 C\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^3}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{2 a \left(12 C a^6-6 b B a^5+2 A b^2 a^4-29 b^2 C a^4+15 b^3 B a^3-5 A b^4 a^2+20 b^4 C a^2-12 b^5 B a+6 A b^6\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^3}{b^5 \sqrt{a^2-b^2} \left(b^2-a^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^3(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-24 C \sin (2 (c+d x)) a^7-12 C \sin (4 (c+d x)) a^7-36 b C \sin (c+d x) a^6+12 b B \sin (2 (c+d x)) a^6-36 b C \sin (3 (c+d x)) a^6+6 b B \sin (4 (c+d x)) a^6+18 b^2 B \sin (c+d x) a^5-4 A b^2 \sin (2 (c+d x)) a^5+26 b^2 C \sin (2 (c+d x)) a^5+18 b^2 B \sin (3 (c+d x)) a^5-2 A b^2 \sin (4 (c+d x)) a^5+21 b^2 C \sin (4 (c+d x)) a^5-6 A b^3 \sin (c+d x) a^4+72 b^3 C \sin (c+d x) a^4-14 b^3 B \sin (2 (c+d x)) a^4-6 A b^3 \sin (3 (c+d x)) a^4+64 b^3 C \sin (3 (c+d x)) a^4-11 b^3 B \sin (4 (c+d x)) a^4-32 b^4 B \sin (c+d x) a^3+10 A b^4 \sin (2 (c+d x)) a^3+20 b^4 C \sin (2 (c+d x)) a^3-32 b^4 B \sin (3 (c+d x)) a^3+5 A b^4 \sin (4 (c+d x)) a^3-6 b^4 C \sin (4 (c+d x)) a^3+12 A b^5 \sin (c+d x) a^2-38 b^5 C \sin (c+d x) a^2-12 b^5 B \sin (2 (c+d x)) a^2+12 A b^5 \sin (3 (c+d x)) a^2-22 b^5 C \sin (3 (c+d x)) a^2+2 b^5 B \sin (4 (c+d x)) a^2+8 b^6 B \sin (c+d x) a-16 b^6 C \sin (2 (c+d x)) a+8 b^6 B \sin (3 (c+d x)) a+8 b^7 C \sin (c+d x)+8 b^7 B \sin (2 (c+d x))\right) (b+a \cos (c+d x))}{8 b^4 \left(b^2-a^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","-\frac{\tan (c+d x) \sec ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(12 a^2 C-6 a b B+2 A b^2+b^2 C\right) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(a \left(-4 a^3 C+2 a^2 b B+7 a b^2 C-5 b^3 B\right)+3 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\tan (c+d x) \sec (c+d x) \left(-6 a^4 C+3 a^3 b B-a^2 b^2 (A-10 C)-6 a b^3 B+b^4 (4 A-C)\right)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{\tan (c+d x) \left(-12 a^5 C+6 a^4 b B-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+a b^4 (5 A-6 C)+2 b^5 B\right)}{2 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(12 a^6 C-6 a^5 b B+a^4 b^2 (2 A-29 C)+15 a^3 b^3 B-5 a^2 b^4 (A-4 C)-12 a b^5 B+6 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(2*a*(2*a^4*A*b^2 - 5*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + 12*a^6*C - 29*a^4*b^2*C + 20*a^2*b^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((-2*A*b^2 + 6*a*b*B - 12*a^2*C - b^2*C)*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*(b + a*Cos[c + d*x])^3*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-6*a^4*A*b^3*Sin[c + d*x] + 12*a^2*A*b^5*Sin[c + d*x] + 18*a^5*b^2*B*Sin[c + d*x] - 32*a^3*b^4*B*Sin[c + d*x] + 8*a*b^6*B*Sin[c + d*x] - 36*a^6*b*C*Sin[c + d*x] + 72*a^4*b^3*C*Sin[c + d*x] - 38*a^2*b^5*C*Sin[c + d*x] + 8*b^7*C*Sin[c + d*x] - 4*a^5*A*b^2*Sin[2*(c + d*x)] + 10*a^3*A*b^4*Sin[2*(c + d*x)] + 12*a^6*b*B*Sin[2*(c + d*x)] - 14*a^4*b^3*B*Sin[2*(c + d*x)] - 12*a^2*b^5*B*Sin[2*(c + d*x)] + 8*b^7*B*Sin[2*(c + d*x)] - 24*a^7*C*Sin[2*(c + d*x)] + 26*a^5*b^2*C*Sin[2*(c + d*x)] + 20*a^3*b^4*C*Sin[2*(c + d*x)] - 16*a*b^6*C*Sin[2*(c + d*x)] - 6*a^4*A*b^3*Sin[3*(c + d*x)] + 12*a^2*A*b^5*Sin[3*(c + d*x)] + 18*a^5*b^2*B*Sin[3*(c + d*x)] - 32*a^3*b^4*B*Sin[3*(c + d*x)] + 8*a*b^6*B*Sin[3*(c + d*x)] - 36*a^6*b*C*Sin[3*(c + d*x)] + 64*a^4*b^3*C*Sin[3*(c + d*x)] - 22*a^2*b^5*C*Sin[3*(c + d*x)] - 2*a^5*A*b^2*Sin[4*(c + d*x)] + 5*a^3*A*b^4*Sin[4*(c + d*x)] + 6*a^6*b*B*Sin[4*(c + d*x)] - 11*a^4*b^3*B*Sin[4*(c + d*x)] + 2*a^2*b^5*B*Sin[4*(c + d*x)] - 12*a^7*C*Sin[4*(c + d*x)] + 21*a^5*b^2*C*Sin[4*(c + d*x)] - 6*a^3*b^4*C*Sin[4*(c + d*x)]))/(8*b^4*(-a^2 + b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","B",1
917,1,492,323,2.9267166,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 b \sin (c+d x) \left(6 a^6 C-2 a^5 b B-7 a^4 b^2 C+5 a^3 b^3 B-3 a^2 A b^4-6 a^2 b^4 C+a^2 \cos (2 (c+d x)) \left(6 a^4 C-2 a^3 b B-11 a^2 b^2 C+5 a b^3 B-3 A b^4+2 b^4 C\right)+2 a b \cos (c+d x) \left(9 a^4 C-3 a^3 b B+a^2 b^2 (A-16 C)+6 a b^3 B+4 b^4 (C-A)\right)+4 b^6 C\right)}{\left(a^2-b^2\right)^2}-\frac{8 \cos (c+d x) \left(6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}-8 (b B-3 a C) \cos (c+d x) (a \cos (c+d x)+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 (b B-3 a C) \cos (c+d x) (a \cos (c+d x)+b)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 b^4 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{\tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\tan (c+d x) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a \tan (c+d x) \left(-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(6 a^6 C-2 a^5 b B-15 a^4 b^2 C+5 a^3 b^3 B+a^2 b^4 (A+12 C)-6 a b^5 B+2 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-8*(2*A*b^6 - 2*a^5*b*B + 5*a^3*b^3*B - 6*a*b^5*B + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*Cos[c + d*x]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) - 8*(b*B - 3*a*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*(b*B - 3*a*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*(-3*a^2*A*b^4 - 2*a^5*b*B + 5*a^3*b^3*B + 6*a^6*C - 7*a^4*b^2*C - 6*a^2*b^4*C + 4*b^6*C + 2*a*b*(-3*a^3*b*B + 6*a*b^3*B + a^2*b^2*(A - 16*C) + 9*a^4*C + 4*b^4*(-A + C))*Cos[c + d*x] + a^2*(-3*A*b^4 - 2*a^3*b*B + 5*a*b^3*B + 6*a^4*C - 11*a^2*b^2*C + 2*b^4*C)*Cos[2*(c + d*x)])*Sin[c + d*x])/(a^2 - b^2)^2))/(4*b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","A",0
918,1,514,242,5.8598855,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 (\sin (c)+i \cos (c)) \left(2 a^5 C-5 a^3 b^2 C-a^2 b^3 B+3 a b^4 (A+2 C)-2 b^5 B\right) (a \cos (c+d x)+b)^2 \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{5/2} \sqrt{(\cos (c)-i \sin (c))^2}}+\frac{b \left(\left(a^2+2 b^2\right) \tan (c) \left(2 a^4 C-a^2 b^2 (2 A+5 C)+3 a b^3 B-A b^4\right)+a \sec (c) \left(a \left(\sin (c+2 d x) \left(-2 a^4 C+a^2 b^2 (2 A+5 C)-3 a b^3 B+A b^4\right)+b \sin (2 c+d x) \left(a^3 C+a^2 b B-a b^2 (3 A+4 C)+2 b^3 B\right)\right)+b \sin (d x) \left(-7 a^4 C+a^3 b B+a^2 b^2 (5 A+16 C)-10 a b^3 B+4 A b^4\right)\right)\right)}{a \left(a^2-b^2\right)^2}-4 C (a \cos (c+d x)+b)^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 C (a \cos (c+d x)+b)^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 b^3 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{a \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(-2 a^5 C+5 a^3 b^2 C+a^2 b^3 B-3 a b^4 (A+2 C)+2 b^5 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{\tan (c+d x) \left(-3 a^4 C+a^3 b B+a^2 b^2 (A+6 C)-4 a b^3 B+2 A b^4\right)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-4*C*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*C*(b + a*Cos[c + d*x])^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*(-(a^2*b^3*B) - 2*b^5*B + 2*a^5*C - 5*a^3*b^2*C + 3*a*b^4*(A + 2*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(I*Cos[c] + Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (b*(a*Sec[c]*(b*(4*A*b^4 + a^3*b*B - 10*a*b^3*B - 7*a^4*C + a^2*b^2*(5*A + 16*C))*Sin[d*x] + a*(b*(a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*Sin[2*c + d*x] + (A*b^4 - 3*a*b^3*B - 2*a^4*C + a^2*b^2*(2*A + 5*C))*Sin[c + 2*d*x])) + (a^2 + 2*b^2)*(-(A*b^4) + 3*a*b^3*B + 2*a^4*C - a^2*b^2*(2*A + 5*C))*Tan[c]))/(a*(a^2 - b^2)^2)))/(2*b^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","C",0
919,1,410,202,4.3864273,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{a \sec (c) \left(a \sin (c+2 d x) \left(2 a^3 B-a^2 b (4 A+3 C)+a b^2 B+A b^3\right)+\sin (2 c+d x) \left(a^4 C-3 a^3 b B+a^2 b^2 (5 A+2 C)-2 A b^4\right)+\sin (d x) \left(a^4 C+5 a^3 b B-a^2 b^2 (11 A+10 C)+4 a b^3 B+2 A b^4\right)\right)-\left(a^2+2 b^2\right) \tan (c) \left(2 a^3 B-a^2 b (4 A+3 C)+a b^2 B+A b^3\right)}{\left(a^3-a b^2\right)^2}-\frac{4 i (\cos (c)-i \sin (c)) \left(a^2 (2 A+C)-3 a b B+b^2 (A+2 C)\right) (a \cos (c+d x)+b)^2 \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{5/2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{2 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{\left(-\left(a^2 (2 A+C)\right)+3 a b B-b^2 (A+2 C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\tan (c+d x) \left(a^3 C+a^2 b B-a b^2 (3 A+4 C)+2 b^3 B\right)}{2 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-4*I)*(-3*a*b*B + a^2*(2*A + C) + b^2*(A + 2*C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (a*Sec[c]*((2*A*b^4 + 5*a^3*b*B + 4*a*b^3*B + a^4*C - a^2*b^2*(11*A + 10*C))*Sin[d*x] + (-2*A*b^4 - 3*a^3*b*B + a^4*C + a^2*b^2*(5*A + 2*C))*Sin[2*c + d*x] + a*(A*b^3 + 2*a^3*B + a*b^2*B - a^2*b*(4*A + 3*C))*Sin[c + 2*d*x]) - (a^2 + 2*b^2)*(A*b^3 + 2*a^3*B + a*b^2*B - a^2*b*(4*A + 3*C))*Tan[c])/(a^3 - a*b^2)^2))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","C",0
920,1,793,229,6.2380006,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{\sec (c) \left(a^6 A d x \cos (c+2 d x)+a^6 A d x \cos (3 c+2 d x)+2 a^6 C \sin (c+2 d x)-2 a^6 C \sin (c)+4 a^5 A b d x \cos (2 c+d x)-4 a^5 b B \sin (c+2 d x)+4 a^5 b B \sin (c)-3 a^5 b C \sin (2 c+d x)+5 a^5 b C \sin (d x)+6 a^4 A b^2 \sin (c+2 d x)-2 a^4 A b^2 d x \cos (c+2 d x)-2 a^4 A b^2 d x \cos (3 c+2 d x)-6 a^4 A b^2 \sin (c)+5 a^4 b^2 B \sin (2 c+d x)-11 a^4 b^2 B \sin (d x)+a^4 b^2 C \sin (c+2 d x)-5 a^4 b^2 C \sin (c)-7 a^3 A b^3 \sin (2 c+d x)-8 a^3 A b^3 d x \cos (2 c+d x)+17 a^3 A b^3 \sin (d x)+a^3 b^3 B \sin (c+2 d x)+7 a^3 b^3 B \sin (c)+4 a^3 b^3 C \sin (d x)-3 a^2 A b^4 \sin (c+2 d x)+a^2 A b^4 d x \cos (c+2 d x)+a^2 A b^4 d x \cos (3 c+2 d x)-9 a^2 A b^4 \sin (c)+2 A d x \left(a^2-b^2\right)^2 \left(a^2+2 b^2\right) \cos (c)+4 a A b d x \left(a^2-b^2\right)^2 \cos (d x)-2 a^2 b^4 B \sin (2 c+d x)+2 a^2 b^4 B \sin (d x)-2 a^2 b^4 C \sin (c)+4 a A b^5 \sin (2 c+d x)+4 a A b^5 d x \cos (2 c+d x)-8 a A b^5 \sin (d x)-2 a b^5 B \sin (c)+6 A b^6 \sin (c)\right)}{\left(a^2-b^2\right)^2}-\frac{4 i (\cos (c)-i \sin (c)) \left(2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right) (a \cos (c+d x)+b)^2 \tan ^{-1}\left(\frac{(\sin (c)+i \cos (c)) \left(\tan \left(\frac{d x}{2}\right) (a \cos (c)-b)+a \sin (c)\right)}{\sqrt{a^2-b^2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{\left(a^2-b^2\right)^{5/2} \sqrt{(\cos (c)-i \sin (c))^2}}\right)}{2 a^3 d (a+b \sec (c+d x))^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{A x}{a^3}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\tan (c+d x) \left(a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+2 C)+2 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(2 a^5 B-3 a^4 b (2 A+C)+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-4*I)*(5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B - 3*a^4*b*(2*A + C))*ArcTan[((I*Cos[c] + Sin[c])*(a*Sin[c] + (-b + a*Cos[c])*Tan[(d*x)/2]))/(Sqrt[a^2 - b^2]*Sqrt[(Cos[c] - I*Sin[c])^2])]*(b + a*Cos[c + d*x])^2*(Cos[c] - I*Sin[c]))/((a^2 - b^2)^(5/2)*Sqrt[(Cos[c] - I*Sin[c])^2]) + (Sec[c]*(2*A*(a^2 - b^2)^2*(a^2 + 2*b^2)*d*x*Cos[c] + 4*a*A*b*(a^2 - b^2)^2*d*x*Cos[d*x] + 4*a^5*A*b*d*x*Cos[2*c + d*x] - 8*a^3*A*b^3*d*x*Cos[2*c + d*x] + 4*a*A*b^5*d*x*Cos[2*c + d*x] + a^6*A*d*x*Cos[c + 2*d*x] - 2*a^4*A*b^2*d*x*Cos[c + 2*d*x] + a^2*A*b^4*d*x*Cos[c + 2*d*x] + a^6*A*d*x*Cos[3*c + 2*d*x] - 2*a^4*A*b^2*d*x*Cos[3*c + 2*d*x] + a^2*A*b^4*d*x*Cos[3*c + 2*d*x] - 6*a^4*A*b^2*Sin[c] - 9*a^2*A*b^4*Sin[c] + 6*A*b^6*Sin[c] + 4*a^5*b*B*Sin[c] + 7*a^3*b^3*B*Sin[c] - 2*a*b^5*B*Sin[c] - 2*a^6*C*Sin[c] - 5*a^4*b^2*C*Sin[c] - 2*a^2*b^4*C*Sin[c] + 17*a^3*A*b^3*Sin[d*x] - 8*a*A*b^5*Sin[d*x] - 11*a^4*b^2*B*Sin[d*x] + 2*a^2*b^4*B*Sin[d*x] + 5*a^5*b*C*Sin[d*x] + 4*a^3*b^3*C*Sin[d*x] - 7*a^3*A*b^3*Sin[2*c + d*x] + 4*a*A*b^5*Sin[2*c + d*x] + 5*a^4*b^2*B*Sin[2*c + d*x] - 2*a^2*b^4*B*Sin[2*c + d*x] - 3*a^5*b*C*Sin[2*c + d*x] + 6*a^4*A*b^2*Sin[c + 2*d*x] - 3*a^2*A*b^4*Sin[c + 2*d*x] - 4*a^5*b*B*Sin[c + 2*d*x] + a^3*b^3*B*Sin[c + 2*d*x] + 2*a^6*C*Sin[c + 2*d*x] + a^4*b^2*C*Sin[c + 2*d*x]))/(a^2 - b^2)^2))/(2*a^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a + b*Sec[c + d*x])^3)","C",0
921,1,1015,330,7.1233683,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","-\frac{2 (3 A b-a B) x \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^3}{a^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\left(2 C a^6-6 b B a^5+12 A b^2 a^4+b^2 C a^4+5 b^3 B a^3-15 A b^4 a^2-2 b^5 B a+6 A b^6\right) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 i \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{2 \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^3}{\left(b^2-a^2\right)^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{2 A \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \tan (c+d x) (b+a \cos (c+d x))^3}{a^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec (c) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-6 A \sin (c) b^6+4 a B \sin (c) b^5+5 a A \sin (d x) b^5+9 a^2 A \sin (c) b^4-2 a^2 C \sin (c) b^4-3 a^2 B \sin (d x) b^4-7 a^3 B \sin (c) b^3-8 a^3 A \sin (d x) b^3+a^3 C \sin (d x) b^3+5 a^4 C \sin (c) b^2+6 a^4 B \sin (d x) b^2-4 a^5 C \sin (d x) b\right) (b+a \cos (c+d x))^2}{a^4 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec (c) \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-A \sin (c) b^5+a B \sin (c) b^4+a A \sin (d x) b^4-a^2 C \sin (c) b^3-a^2 B \sin (d x) b^3+a^3 C \sin (d x) b^2\right) (b+a \cos (c+d x))}{a^4 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","-\frac{x (3 A b-a B)}{a^4}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\sin (c+d x) \left(-\left(a^4 (2 A-3 C)\right)-5 a^3 b B+11 a^2 A b^2+2 a b^3 B-6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(-2 a^4 C+4 a^3 b B-a^2 b^2 (6 A+C)-a b^3 B+3 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(-2 a^6 C+6 a^5 b B-a^4 b^2 (12 A+C)-5 a^3 b^3 B+15 a^2 A b^4+2 a b^5 B-6 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}",1,"(-2*(3*A*b - a*B)*x*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((12*a^4*A*b^2 - 15*a^2*A*b^4 + 6*A*b^6 - 6*a^5*b*B + 5*a^3*b^3*B - 2*a*b^5*B + 2*a^6*C + a^4*b^2*C)*(b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-(A*b^5*Sin[c]) + a*b^4*B*Sin[c] - a^2*b^3*C*Sin[c] + a*A*b^4*Sin[d*x] - a^2*b^3*B*Sin[d*x] + a^3*b^2*C*Sin[d*x]))/(a^4*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(9*a^2*A*b^4*Sin[c] - 6*A*b^6*Sin[c] - 7*a^3*b^3*B*Sin[c] + 4*a*b^5*B*Sin[c] + 5*a^4*b^2*C*Sin[c] - 2*a^2*b^4*C*Sin[c] - 8*a^3*A*b^3*Sin[d*x] + 5*a*A*b^5*Sin[d*x] + 6*a^4*b^2*B*Sin[d*x] - 3*a^2*b^4*B*Sin[d*x] - 4*a^5*b*C*Sin[d*x] + a^3*b^3*C*Sin[d*x]))/(a^4*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + (2*A*(b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Tan[c + d*x])/(a^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","C",0
922,1,881,453,4.9161843,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{16 b \left(6 C a^6-12 b B a^5+5 b^2 (4 A-C) a^4+15 b^3 B a^3+b^4 (2 C-29 A) a^2-6 b^5 B a+12 A b^6\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{4 A c a^8+8 c C a^8+4 A d x a^8+8 C d x a^8+4 B \sin (c+d x) a^8+2 A \sin (2 (c+d x)) a^8+4 B \sin (3 (c+d x)) a^8+A \sin (4 (c+d x)) a^8-24 b B c a^7-24 b B d x a^7-8 A b \sin (c+d x) a^7+16 b B \sin (2 (c+d x)) a^7-8 A b \sin (3 (c+d x)) a^7+48 A b^2 c a^6+48 A b^2 d x a^6+8 b^2 B \sin (c+d x) a^6-48 A b^2 \sin (2 (c+d x)) a^6+24 b^2 C \sin (2 (c+d x)) a^6-8 b^2 B \sin (3 (c+d x)) a^6-2 A b^2 \sin (4 (c+d x)) a^6-32 A b^3 \sin (c+d x) a^5+40 b^3 C \sin (c+d x) a^5-64 b^3 B \sin (2 (c+d x)) a^5+16 A b^3 \sin (3 (c+d x)) a^5-12 A b^4 c a^4-24 b^4 c C a^4-12 A b^4 d x a^4-24 b^4 C d x a^4-84 b^4 B \sin (c+d x) a^4+130 A b^4 \sin (2 (c+d x)) a^4-12 b^4 C \sin (2 (c+d x)) a^4+4 b^4 B \sin (3 (c+d x)) a^4+A b^4 \sin (4 (c+d x)) a^4+72 b^5 B c a^3+72 b^5 B d x a^3+160 A b^5 \sin (c+d x) a^3-16 b^5 C \sin (c+d x) a^3+36 b^5 B \sin (2 (c+d x)) a^3-8 A b^5 \sin (3 (c+d x)) a^3-136 A b^6 c a^2+16 b^6 c C a^2-136 A b^6 d x a^2+16 b^6 C d x a^2+48 b^6 B \sin (c+d x) a^2-72 A b^6 \sin (2 (c+d x)) a^2-48 b^7 B c a-48 b^7 B d x a+16 b \left(a^2-b^2\right)^2 \left((A+2 C) a^2-6 b B a+12 A b^2\right) (c+d x) \cos (c+d x) a-96 A b^7 \sin (c+d x) a+96 A b^8 c+96 A b^8 d x+4 \left(a^3-a b^2\right)^2 \left((A+2 C) a^2-6 b B a+12 A b^2\right) (c+d x) \cos (2 (c+d x))}{\left(a^2-b^2\right)^2 (b+a \cos (c+d x))^2}}{16 a^5 d}","\frac{\sin (c+d x) \cos (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{x \left(a^2 (A+2 C)-6 a b B+12 A b^2\right)}{2 a^5}+\frac{\sin (c+d x) \cos (c+d x) \left(a^4 (A-4 C)+6 a^3 b B-a^2 b^2 (10 A-C)-3 a b^3 B+6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \cos (c+d x) \left(3 a^4 C-5 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \left(-2 a^5 B+a^4 b (6 A-5 C)+11 a^3 b^2 B-a^2 b^3 (21 A-2 C)-6 a b^4 B+12 A b^5\right)}{2 a^4 d \left(a^2-b^2\right)^2}-\frac{b \left(6 a^6 C-12 a^5 b B+5 a^4 b^2 (4 A-C)+15 a^3 b^3 B-a^2 b^4 (29 A-2 C)-6 a b^5 B+12 A b^6\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((16*b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B + 5*a^4*b^2*(4*A - C) + 6*a^6*C + a^2*b^4*(-29*A + 2*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^8*A*c + 48*a^6*A*b^2*c - 12*a^4*A*b^4*c - 136*a^2*A*b^6*c + 96*A*b^8*c - 24*a^7*b*B*c + 72*a^3*b^5*B*c - 48*a*b^7*B*c + 8*a^8*c*C - 24*a^4*b^4*c*C + 16*a^2*b^6*c*C + 4*a^8*A*d*x + 48*a^6*A*b^2*d*x - 12*a^4*A*b^4*d*x - 136*a^2*A*b^6*d*x + 96*A*b^8*d*x - 24*a^7*b*B*d*x + 72*a^3*b^5*B*d*x - 48*a*b^7*B*d*x + 8*a^8*C*d*x - 24*a^4*b^4*C*d*x + 16*a^2*b^6*C*d*x + 16*a*b*(a^2 - b^2)^2*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[c + d*x] + 4*(a^3 - a*b^2)^2*(12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*(c + d*x)*Cos[2*(c + d*x)] - 8*a^7*A*b*Sin[c + d*x] - 32*a^5*A*b^3*Sin[c + d*x] + 160*a^3*A*b^5*Sin[c + d*x] - 96*a*A*b^7*Sin[c + d*x] + 4*a^8*B*Sin[c + d*x] + 8*a^6*b^2*B*Sin[c + d*x] - 84*a^4*b^4*B*Sin[c + d*x] + 48*a^2*b^6*B*Sin[c + d*x] + 40*a^5*b^3*C*Sin[c + d*x] - 16*a^3*b^5*C*Sin[c + d*x] + 2*a^8*A*Sin[2*(c + d*x)] - 48*a^6*A*b^2*Sin[2*(c + d*x)] + 130*a^4*A*b^4*Sin[2*(c + d*x)] - 72*a^2*A*b^6*Sin[2*(c + d*x)] + 16*a^7*b*B*Sin[2*(c + d*x)] - 64*a^5*b^3*B*Sin[2*(c + d*x)] + 36*a^3*b^5*B*Sin[2*(c + d*x)] + 24*a^6*b^2*C*Sin[2*(c + d*x)] - 12*a^4*b^4*C*Sin[2*(c + d*x)] - 8*a^7*A*b*Sin[3*(c + d*x)] + 16*a^5*A*b^3*Sin[3*(c + d*x)] - 8*a^3*A*b^5*Sin[3*(c + d*x)] + 4*a^8*B*Sin[3*(c + d*x)] - 8*a^6*b^2*B*Sin[3*(c + d*x)] + 4*a^4*b^4*B*Sin[3*(c + d*x)] + a^8*A*Sin[4*(c + d*x)] - 2*a^6*A*b^2*Sin[4*(c + d*x)] + a^4*A*b^4*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(16*a^5*d)","A",1
923,1,1197,470,6.4912082,"\int \frac{\sec ^4(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\frac{2 (b B-4 a C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^4}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{2 (b B-4 a C) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^4}{b^5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}-\frac{2 \left(-8 C a^8+2 b B a^7+28 b^2 C a^6-7 b^3 B a^5-35 b^4 C a^4+8 b^5 B a^3+3 A b^6 a^2+20 b^6 C a^2-8 b^7 B a+2 A b^8\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^4}{b^5 \sqrt{a^2-b^2} \left(b^2-a^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec ^3(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-48 C \sin (2 (c+d x)) a^9-24 C \sin (4 (c+d x)) a^9-120 b C \sin (c+d x) a^8+12 b B \sin (2 (c+d x)) a^8-120 b C \sin (3 (c+d x)) a^8+6 b B \sin (4 (c+d x)) a^8+30 b^2 B \sin (c+d x) a^7-40 b^2 C \sin (2 (c+d x)) a^7+30 b^2 B \sin (3 (c+d x)) a^7+68 b^2 C \sin (4 (c+d x)) a^7+294 b^3 C \sin (c+d x) a^6+10 b^3 B \sin (2 (c+d x)) a^6+342 b^3 C \sin (3 (c+d x)) a^6-17 b^3 B \sin (4 (c+d x)) a^6-90 b^4 B \sin (c+d x) a^5-16 A b^4 \sin (2 (c+d x)) a^5+370 b^4 C \sin (2 (c+d x)) a^5-90 b^4 B \sin (3 (c+d x)) a^5-4 A b^4 \sin (4 (c+d x)) a^5-65 b^4 C \sin (4 (c+d x)) a^5-6 A b^5 \sin (c+d x) a^4-174 b^5 C \sin (c+d x) a^4-76 b^5 B \sin (2 (c+d x)) a^4-6 A b^5 \sin (3 (c+d x)) a^4-318 b^5 C \sin (3 (c+d x)) a^4+26 b^5 B \sin (4 (c+d x)) a^4+120 b^6 B \sin (c+d x) a^3-2 A b^6 \sin (2 (c+d x)) a^3-444 b^6 C \sin (2 (c+d x)) a^3+120 b^6 B \sin (3 (c+d x)) a^3-11 A b^6 \sin (4 (c+d x)) a^3+6 b^6 C \sin (4 (c+d x)) a^3-54 A b^7 \sin (c+d x) a^2-108 b^7 C \sin (c+d x) a^2+144 b^7 B \sin (2 (c+d x)) a^2-54 A b^7 \sin (3 (c+d x)) a^2+36 b^7 C \sin (3 (c+d x)) a^2-72 A b^8 \sin (2 (c+d x)) a+72 b^8 C \sin (2 (c+d x)) a+48 b^9 C \sin (c+d x)\right) (b+a \cos (c+d x))}{24 b^4 \left(b^2-a^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}","-\frac{\tan (c+d x) \sec ^3(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\tan (c+d x) \left(-12 a^4 C+3 a^3 b B+23 a^2 b^2 C-8 a b^3 B+5 A b^4-6 b^4 C\right)}{6 b^4 d \left(a^2-b^2\right)^2}+\frac{\tan (c+d x) \sec ^2(c+d x) \left(-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{a \tan (c+d x) \left(4 a^6 C-a^5 b B-11 a^4 b^2 C+2 a^3 b^3 B+3 a^2 b^4 (A+4 C)-6 a b^5 B+2 A b^6\right)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\left(-8 a^8 C+2 a^7 b B+28 a^6 b^2 C-7 a^5 b^3 B-35 a^4 b^4 C+8 a^3 b^5 B+a^2 b^6 (3 A+20 C)-8 a b^7 B+2 A b^8\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"(-2*(3*a^2*A*b^6 + 2*A*b^8 + 2*a^7*b*B - 7*a^5*b^3*B + 8*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 28*a^6*b^2*C - 35*a^4*b^4*C + 20*a^2*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b*B - 4*a*C)*(b + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b*B - 4*a*C)*(b + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-6*a^4*A*b^5*Sin[c + d*x] - 54*a^2*A*b^7*Sin[c + d*x] + 30*a^7*b^2*B*Sin[c + d*x] - 90*a^5*b^4*B*Sin[c + d*x] + 120*a^3*b^6*B*Sin[c + d*x] - 120*a^8*b*C*Sin[c + d*x] + 294*a^6*b^3*C*Sin[c + d*x] - 174*a^4*b^5*C*Sin[c + d*x] - 108*a^2*b^7*C*Sin[c + d*x] + 48*b^9*C*Sin[c + d*x] - 16*a^5*A*b^4*Sin[2*(c + d*x)] - 2*a^3*A*b^6*Sin[2*(c + d*x)] - 72*a*A*b^8*Sin[2*(c + d*x)] + 12*a^8*b*B*Sin[2*(c + d*x)] + 10*a^6*b^3*B*Sin[2*(c + d*x)] - 76*a^4*b^5*B*Sin[2*(c + d*x)] + 144*a^2*b^7*B*Sin[2*(c + d*x)] - 48*a^9*C*Sin[2*(c + d*x)] - 40*a^7*b^2*C*Sin[2*(c + d*x)] + 370*a^5*b^4*C*Sin[2*(c + d*x)] - 444*a^3*b^6*C*Sin[2*(c + d*x)] + 72*a*b^8*C*Sin[2*(c + d*x)] - 6*a^4*A*b^5*Sin[3*(c + d*x)] - 54*a^2*A*b^7*Sin[3*(c + d*x)] + 30*a^7*b^2*B*Sin[3*(c + d*x)] - 90*a^5*b^4*B*Sin[3*(c + d*x)] + 120*a^3*b^6*B*Sin[3*(c + d*x)] - 120*a^8*b*C*Sin[3*(c + d*x)] + 342*a^6*b^3*C*Sin[3*(c + d*x)] - 318*a^4*b^5*C*Sin[3*(c + d*x)] + 36*a^2*b^7*C*Sin[3*(c + d*x)] - 4*a^5*A*b^4*Sin[4*(c + d*x)] - 11*a^3*A*b^6*Sin[4*(c + d*x)] + 6*a^8*b*B*Sin[4*(c + d*x)] - 17*a^6*b^3*B*Sin[4*(c + d*x)] + 26*a^4*b^5*B*Sin[4*(c + d*x)] - 24*a^9*C*Sin[4*(c + d*x)] + 68*a^7*b^2*C*Sin[4*(c + d*x)] - 65*a^5*b^4*C*Sin[4*(c + d*x)] + 6*a^3*b^6*C*Sin[4*(c + d*x)]))/(24*b^4*(-a^2 + b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","B",1
924,1,1302,358,7.3341059,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\frac{2 C \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^4}{b^4 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{2 C \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^4}{b^4 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\left(2 C a^7-7 b^2 C a^5-A b^4 a^3+8 b^4 C a^3+3 b^5 B a^2-4 A b^6 a-8 b^6 C a+2 b^7 B\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 i \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{b^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{2 \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{b^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^4}{\left(b^2-a^2\right)^3 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(6 C \sin (d x) a^6-3 b C \sin (c) a^5-17 b^2 C \sin (d x) a^4-3 A b^3 \sin (c) a^3+6 b^3 C \sin (c) a^3-4 b^3 B \sin (d x) a^3+9 b^4 B \sin (c) a^2+13 A b^4 \sin (d x) a^2+26 b^4 C \sin (d x) a^2-12 A b^5 \sin (c) a-18 b^5 C \sin (c) a-11 b^5 B \sin (d x) a+6 b^6 B \sin (c)+2 A b^6 \sin (d x)\right) (b+a \cos (c+d x))^3}{3 b^3 \left(b^2-a^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-3 C \sin (d x) a^4+b C \sin (c) a^3+2 b^2 B \sin (c) a^2+3 A b^2 \sin (d x) a^2+8 b^2 C \sin (d x) a^2-5 A b^3 \sin (c) a-6 b^3 C \sin (c) a-5 b^3 B \sin (d x) a+3 b^4 B \sin (c)+2 A b^4 \sin (d x)\right) (b+a \cos (c+d x))^2}{3 b^2 \left(b^2-a^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}-\frac{2 \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-C \sin (d x) a^3+b C \sin (c) a^2+b B \sin (d x) a^2-b^2 B \sin (c) a-A b^2 \sin (d x) a+A b^3 \sin (c)\right) (b+a \cos (c+d x))}{3 a b \left(b^2-a^2\right) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}","-\frac{\tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{a \tan (c+d x) \left(-3 a^4 C+a^2 b^2 (3 A+8 C)-5 a b^3 B+2 A b^4\right)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{\left(2 a^7 C-7 a^5 b^2 C-a^3 b^4 (A-8 C)+3 a^2 b^5 B-4 a b^6 (A+2 C)+2 b^7 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\tan (c+d x) \left(9 a^6 C-a^4 b^2 (3 A+28 C)+a^3 b^3 B+2 a^2 b^4 (7 A+17 C)-16 a b^5 B+4 A b^6\right)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{C \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(-2*C*(b + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*C*(b + a*Cos[c + d*x])^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-(a^3*A*b^4) - 4*a*A*b^6 + 3*a^2*b^5*B + 2*b^7*B + 2*a^7*C - 7*a^5*b^2*C + 8*a^3*b^4*C - 8*a*b^6*C)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(b^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(b^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*b^3*Sin[c] - a*b^2*B*Sin[c] + a^2*b*C*Sin[c] - a*A*b^2*Sin[d*x] + a^2*b*B*Sin[d*x] - a^3*C*Sin[d*x]))/(3*a*b*(-a^2 + b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-5*a*A*b^3*Sin[c] + 2*a^2*b^2*B*Sin[c] + 3*b^4*B*Sin[c] + a^3*b*C*Sin[c] - 6*a*b^3*C*Sin[c] + 3*a^2*A*b^2*Sin[d*x] + 2*A*b^4*Sin[d*x] - 5*a*b^3*B*Sin[d*x] - 3*a^4*C*Sin[d*x] + 8*a^2*b^2*C*Sin[d*x]))/(3*b^2*(-a^2 + b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-3*a^3*A*b^3*Sin[c] - 12*a*A*b^5*Sin[c] + 9*a^2*b^4*B*Sin[c] + 6*b^6*B*Sin[c] - 3*a^5*b*C*Sin[c] + 6*a^3*b^3*C*Sin[c] - 18*a*b^5*C*Sin[c] + 13*a^2*A*b^4*Sin[d*x] + 2*A*b^6*Sin[d*x] - 4*a^3*b^3*B*Sin[d*x] - 11*a*b^5*B*Sin[d*x] + 6*a^6*C*Sin[d*x] - 17*a^4*b^2*C*Sin[d*x] + 26*a^2*b^4*C*Sin[d*x]))/(3*b^3*(-a^2 + b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","C",0
925,1,299,314,1.5423704,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{24 \left(a^3 B-a^2 b (4 A+3 C)+4 a b^2 B-b^3 (A+2 C)\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-\frac{2 \sin (c+d x) \left(6 a^5 A+8 a^5 C-11 a^4 b B+14 a^3 A b^2+a^3 b^2 C-22 a^2 b^3 B+a \cos (2 (c+d x)) \left(a^4 (6 A+4 C)-13 a^3 b B+a^2 b^2 (10 A+11 C)-2 a b^3 B-A b^4\right)+6 \cos (c+d x) \left(a^5 B+a^4 b (2 A+C)-9 a^3 b^2 B+9 a^2 b^3 (A+C)-2 a b^4 B-A b^5\right)+25 a A b^4+36 a b^4 C-12 b^5 B\right)}{(a \cos (c+d x)+b)^3}}{24 d \left(b^2-a^2\right)^3}","\frac{a \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{\left(a^3 B-a^2 b (4 A+3 C)+4 a b^2 B-b^3 (A+2 C)\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\tan (c+d x) \left(-4 a^4 C+a^3 b B+a^2 b^2 (2 A+9 C)-6 a b^3 B+3 A b^4\right)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\tan (c+d x) \left(2 a^5 C+a^4 b B+a^3 b^2 (2 A-5 C)-10 a^2 b^3 B+a b^4 (13 A+18 C)-6 b^5 B\right)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((24*(a^3*B + 4*a*b^2*B - b^3*(A + 2*C) - a^2*b*(4*A + 3*C))*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (2*(6*a^5*A + 14*a^3*A*b^2 + 25*a*A*b^4 - 11*a^4*b*B - 22*a^2*b^3*B - 12*b^5*B + 8*a^5*C + a^3*b^2*C + 36*a*b^4*C + 6*(-(A*b^5) + a^5*B - 9*a^3*b^2*B - 2*a*b^4*B + 9*a^2*b^3*(A + C) + a^4*b*(2*A + C))*Cos[c + d*x] + a*(-(A*b^4) - 13*a^3*b*B - 2*a*b^3*B + a^4*(6*A + 4*C) + a^2*b^2*(10*A + 11*C))*Cos[2*(c + d*x)])*Sin[c + d*x])/(b + a*Cos[c + d*x])^3)/(24*(-a^2 + b^2)^3*d)","A",1
926,1,1069,299,7.5511343,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{\left(-2 A a^3-C a^3+4 b B a^2-3 A b^2 a-4 b^2 C a+b^3 B\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 i \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{\sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{2 \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{\sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^4}{\left(b^2-a^2\right)^3 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(3 C \sin (c) a^6+6 B \sin (d x) a^6-12 b B \sin (c) a^5-18 A b \sin (d x) a^5-13 b C \sin (d x) a^5+27 A b^2 \sin (c) a^4+12 b^2 C \sin (c) a^4+10 b^2 B \sin (d x) a^4-3 b^3 B \sin (c) a^3+5 A b^3 \sin (d x) a^3-2 b^3 C \sin (d x) a^3-18 A b^4 \sin (c) a^2-b^4 B \sin (d x) a^2-2 A b^5 \sin (d x) a+6 A b^6 \sin (c)\right) (b+a \cos (c+d x))^3}{3 a^3 \left(a^2-b^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(3 C \sin (d x) a^5-5 b C \sin (c) a^4-6 b B \sin (d x) a^4+8 b^2 B \sin (c) a^3+9 A b^2 \sin (d x) a^3+2 b^2 C \sin (d x) a^3-11 A b^3 \sin (c) a^2+b^3 B \sin (d x) a^2-3 b^4 B \sin (c) a-4 A b^4 \sin (d x) a+6 A b^5 \sin (c)\right) (b+a \cos (c+d x))^2}{3 a^3 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{2 \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(A \sin (c) b^4-a B \sin (c) b^3-a A \sin (d x) b^3+a^2 C \sin (c) b^2+a^2 B \sin (d x) b^2-a^3 C \sin (d x) b\right) (b+a \cos (c+d x))}{3 a^3 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}","-\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\left(-\left(a^3 (2 A+C)\right)+4 a^2 b B-a b^2 (3 A+4 C)+b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\tan (c+d x) \left(a^3 C+2 a^2 b B-a b^2 (5 A+6 C)+3 b^3 B\right)}{6 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\tan (c+d x) \left(a^4 C+2 a^3 b B-a^2 b^2 (11 A+10 C)+13 a b^3 B-2 b^4 (2 A+3 C)\right)}{6 b d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((-2*a^3*A - 3*a*A*b^2 + 4*a^2*b*B + b^3*B - a^3*C - 4*a*b^2*C)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*b^4*Sin[c] - a*b^3*B*Sin[c] + a^2*b^2*C*Sin[c] - a*A*b^3*Sin[d*x] + a^2*b^2*B*Sin[d*x] - a^3*b*C*Sin[d*x]))/(3*a^3*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-11*a^2*A*b^3*Sin[c] + 6*A*b^5*Sin[c] + 8*a^3*b^2*B*Sin[c] - 3*a*b^4*B*Sin[c] - 5*a^4*b*C*Sin[c] + 9*a^3*A*b^2*Sin[d*x] - 4*a*A*b^4*Sin[d*x] - 6*a^4*b*B*Sin[d*x] + a^2*b^3*B*Sin[d*x] + 3*a^5*C*Sin[d*x] + 2*a^3*b^2*C*Sin[d*x]))/(3*a^3*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(27*a^4*A*b^2*Sin[c] - 18*a^2*A*b^4*Sin[c] + 6*A*b^6*Sin[c] - 12*a^5*b*B*Sin[c] - 3*a^3*b^3*B*Sin[c] + 3*a^6*C*Sin[c] + 12*a^4*b^2*C*Sin[c] - 18*a^5*A*b*Sin[d*x] + 5*a^3*A*b^3*Sin[d*x] - 2*a*A*b^5*Sin[d*x] + 6*a^6*B*Sin[d*x] + 10*a^4*b^2*B*Sin[d*x] - a^2*b^4*B*Sin[d*x] - 13*a^5*b*C*Sin[d*x] - 2*a^3*b^3*C*Sin[d*x]))/(3*a^3*(a^2 - b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","C",0
927,1,1230,336,7.8724004,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4,x]","\frac{2 A x \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^4}{a^4 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\left(2 B a^7-8 A b a^6-4 b C a^6+3 b^2 B a^5+8 A b^3 a^4-b^3 C a^4-7 A b^5 a^2+2 A b^7\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 i \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}+\frac{2 \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{a^4 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^4}{\left(b^2-a^2\right)^3 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(6 C \sin (d x) a^7-12 b C \sin (c) a^6-18 b B \sin (d x) a^6+27 b^2 B \sin (c) a^5+36 A b^2 \sin (d x) a^5+10 b^2 C \sin (d x) a^5-48 A b^3 \sin (c) a^4-3 b^3 C \sin (c) a^4+5 b^3 B \sin (d x) a^4-18 b^4 B \sin (c) a^3-32 A b^4 \sin (d x) a^3-b^4 C \sin (d x) a^3+51 A b^5 \sin (c) a^2-2 b^5 B \sin (d x) a^2+6 b^6 B \sin (c) a+11 A b^6 \sin (d x) a-18 A b^7 \sin (c)\right) (b+a \cos (c+d x))^3}{3 a^4 \left(a^2-b^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-9 A \sin (c) b^6+6 a B \sin (c) b^5+7 a A \sin (d x) b^5+14 a^2 A \sin (c) b^4-3 a^2 C \sin (c) b^4-4 a^2 B \sin (d x) b^4-11 a^3 B \sin (c) b^3-12 a^3 A \sin (d x) b^3+a^3 C \sin (d x) b^3+8 a^4 C \sin (c) b^2+9 a^4 B \sin (d x) b^2-6 a^5 C \sin (d x) b\right) (b+a \cos (c+d x))^2}{3 a^4 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}-\frac{2 \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(A \sin (c) b^5-a B \sin (c) b^4-a A \sin (d x) b^4+a^2 C \sin (c) b^3+a^2 B \sin (d x) b^3-a^3 C \sin (d x) b^2\right) (b+a \cos (c+d x))}{3 a^4 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}","\frac{A x}{a^4}+\frac{\tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\tan (c+d x) \left(-2 a^4 C+5 a^3 b B-a^2 b^2 (8 A+3 C)+3 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{\left(-2 a^7 B+4 a^6 b (2 A+C)-3 a^5 b^2 B-a^4 b^3 (8 A-C)+7 a^2 A b^5-2 A b^7\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\tan (c+d x) \left(-2 a^6 C+11 a^5 b B-13 a^4 b^2 (2 A+C)+4 a^3 b^3 B+17 a^2 A b^4-6 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"(2*A*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-8*a^6*A*b + 8*a^4*A*b^3 - 7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^2*B - 4*a^6*b*C - a^4*b^3*C)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) + (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^4*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) - (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*b^5*Sin[c] - a*b^4*B*Sin[c] + a^2*b^3*C*Sin[c] - a*A*b^4*Sin[d*x] + a^2*b^3*B*Sin[d*x] - a^3*b^2*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(14*a^2*A*b^4*Sin[c] - 9*A*b^6*Sin[c] - 11*a^3*b^3*B*Sin[c] + 6*a*b^5*B*Sin[c] + 8*a^4*b^2*C*Sin[c] - 3*a^2*b^4*C*Sin[c] - 12*a^3*A*b^3*Sin[d*x] + 7*a*A*b^5*Sin[d*x] + 9*a^4*b^2*B*Sin[d*x] - 4*a^2*b^4*B*Sin[d*x] - 6*a^5*b*C*Sin[d*x] + a^3*b^3*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-48*a^4*A*b^3*Sin[c] + 51*a^2*A*b^5*Sin[c] - 18*A*b^7*Sin[c] + 27*a^5*b^2*B*Sin[c] - 18*a^3*b^4*B*Sin[c] + 6*a*b^6*B*Sin[c] - 12*a^6*b*C*Sin[c] - 3*a^4*b^3*C*Sin[c] + 36*a^5*A*b^2*Sin[d*x] - 32*a^3*A*b^4*Sin[d*x] + 11*a*A*b^6*Sin[d*x] - 18*a^6*b*B*Sin[d*x] + 5*a^4*b^3*B*Sin[d*x] - 2*a^2*b^5*B*Sin[d*x] + 6*a^7*C*Sin[d*x] + 10*a^5*b^2*C*Sin[d*x] - a^3*b^4*C*Sin[d*x]))/(3*a^4*(a^2 - b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","C",0
928,1,1367,471,8.196342,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\frac{2 (4 A b-a B) x \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) (b+a \cos (c+d x))^4}{a^5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\left(-2 C a^8+8 b B a^7-20 A b^2 a^6-3 b^2 C a^6-8 b^3 B a^5+35 A b^4 a^4+7 b^5 B a^3-28 A b^6 a^2-2 b^7 B a+8 A b^8\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 i \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \cos (c)}{a^5 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{2 \tan ^{-1}\left(\sec \left(\frac{d x}{2}\right) \left(\frac{\cos (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}-\frac{i \sin (c)}{\sqrt{a^2-b^2} \sqrt{\cos (2 c)-i \sin (2 c)}}\right) \left(i a \sin \left(c+\frac{d x}{2}\right)-i b \sin \left(\frac{d x}{2}\right)\right)\right) \sin (c)}{a^5 \sqrt{a^2-b^2} d \sqrt{\cos (2 c)-i \sin (2 c)}}\right) (b+a \cos (c+d x))^4}{\left(b^2-a^2\right)^3 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{2 A \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \tan (c+d x) (b+a \cos (c+d x))^4}{a^4 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(36 A \sin (c) b^8-18 a B \sin (c) b^7-26 a A \sin (d x) b^7-96 a^2 A \sin (c) b^6+6 a^2 C \sin (c) b^6+11 a^2 B \sin (d x) b^6+51 a^3 B \sin (c) b^5+71 a^3 A \sin (d x) b^5-2 a^3 C \sin (d x) b^5+75 a^4 A \sin (c) b^4-18 a^4 C \sin (c) b^4-32 a^4 B \sin (d x) b^4-48 a^5 B \sin (c) b^3-60 a^5 A \sin (d x) b^3+5 a^5 C \sin (d x) b^3+27 a^6 C \sin (c) b^2+36 a^6 B \sin (d x) b^2-18 a^7 C \sin (d x) b\right) (b+a \cos (c+d x))^3}{3 a^5 \left(a^2-b^2\right)^3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{\sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(12 A \sin (c) b^7-9 a B \sin (c) b^6-10 a A \sin (d x) b^6-17 a^2 A \sin (c) b^5+6 a^2 C \sin (c) b^5+7 a^2 B \sin (d x) b^5+14 a^3 B \sin (c) b^4+15 a^3 A \sin (d x) b^4-4 a^3 C \sin (d x) b^4-11 a^4 C \sin (c) b^3-12 a^4 B \sin (d x) b^3+9 a^5 C \sin (d x) b^2\right) (b+a \cos (c+d x))^2}{3 a^5 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}+\frac{2 \sec (c) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(A \sin (c) b^6-a B \sin (c) b^5-a A \sin (d x) b^5+a^2 C \sin (c) b^4+a^2 B \sin (d x) b^4-a^3 C \sin (d x) b^3\right) (b+a \cos (c+d x))}{3 a^5 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^4}","-\frac{x (4 A b-a B)}{a^5}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\sin (c+d x) \left(-3 a^4 C+6 a^3 b B-a^2 b^2 (9 A+2 C)-a b^3 B+4 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \left(a^6 (6 A-11 C)+26 a^5 b B-a^4 b^2 (65 A+4 C)-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\sin (c+d x) \left(-2 a^6 C+6 a^5 b B-3 a^4 b^2 (4 A+C)-2 a^3 b^3 B+11 a^2 A b^4+a b^5 B-4 A b^6\right)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\left(-2 a^8 C+8 a^7 b B-a^6 b^2 (20 A+3 C)-8 a^5 b^3 B+35 a^4 A b^4+7 a^3 b^5 B-28 a^2 A b^6-2 a b^7 B+8 A b^8\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"(-2*(4*A*b - a*B)*x*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a^5*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((-20*a^6*A*b^2 + 35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 + 8*a^7*b*B - 8*a^5*b^3*B + 7*a^3*b^5*B - 2*a*b^7*B - 2*a^8*C - 3*a^6*b^2*C)*(b + a*Cos[c + d*x])^4*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-2*I)*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Cos[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (2*ArcTan[Sec[(d*x)/2]*(Cos[c]/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]) - (I*Sin[c])/(Sqrt[a^2 - b^2]*Sqrt[Cos[2*c] - I*Sin[2*c]]))*((-I)*b*Sin[(d*x)/2] + I*a*Sin[c + (d*x)/2])]*Sin[c])/(a^5*Sqrt[a^2 - b^2]*d*Sqrt[Cos[2*c] - I*Sin[2*c]])))/((-a^2 + b^2)^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*(b + a*Cos[c + d*x])*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*b^6*Sin[c] - a*b^5*B*Sin[c] + a^2*b^4*C*Sin[c] - a*A*b^5*Sin[d*x] + a^2*b^4*B*Sin[d*x] - a^3*b^3*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^2*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-17*a^2*A*b^5*Sin[c] + 12*A*b^7*Sin[c] + 14*a^3*b^4*B*Sin[c] - 9*a*b^6*B*Sin[c] - 11*a^4*b^3*C*Sin[c] + 6*a^2*b^5*C*Sin[c] + 15*a^3*A*b^4*Sin[d*x] - 10*a*A*b^6*Sin[d*x] - 12*a^4*b^3*B*Sin[d*x] + 7*a^2*b^5*B*Sin[d*x] + 9*a^5*b^2*C*Sin[d*x] - 4*a^3*b^4*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + ((b + a*Cos[c + d*x])^3*Sec[c]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(75*a^4*A*b^4*Sin[c] - 96*a^2*A*b^6*Sin[c] + 36*A*b^8*Sin[c] - 48*a^5*b^3*B*Sin[c] + 51*a^3*b^5*B*Sin[c] - 18*a*b^7*B*Sin[c] + 27*a^6*b^2*C*Sin[c] - 18*a^4*b^4*C*Sin[c] + 6*a^2*b^6*C*Sin[c] - 60*a^5*A*b^3*Sin[d*x] + 71*a^3*A*b^5*Sin[d*x] - 26*a*A*b^7*Sin[d*x] + 36*a^6*b^2*B*Sin[d*x] - 32*a^4*b^4*B*Sin[d*x] + 11*a^2*b^6*B*Sin[d*x] - 18*a^7*b*C*Sin[d*x] + 5*a^5*b^3*C*Sin[d*x] - 2*a^3*b^5*C*Sin[d*x]))/(3*a^5*(a^2 - b^2)^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4) + (2*A*(b + a*Cos[c + d*x])^4*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Tan[c + d*x])/(a^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^4)","C",0
929,1,658,648,7.0053587,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{(4 A b-a B) \left(-\frac{\sin (c+d x)}{2 a^5}-\frac{i \cos (c+d x)}{2 a^5}\right)}{d}+\frac{(4 A b-a B) \left(-\frac{\sin (c+d x)}{2 a^5}+\frac{i \cos (c+d x)}{2 a^5}\right)}{d}+\frac{A \sin (2 (c+d x))}{4 a^4 d}+\frac{(c+d x) \left(a^2 A+2 a^2 C-8 a b B+20 A b^2\right)}{2 a^6 d}+\frac{a^2 b^4 C \sin (c+d x)-a b^5 B \sin (c+d x)+A b^6 \sin (c+d x)}{3 a^5 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^3}+\frac{-12 a^4 b^3 C \sin (c+d x)+15 a^3 b^4 B \sin (c+d x)-18 a^2 A b^5 \sin (c+d x)+7 a^2 b^5 C \sin (c+d x)-10 a b^6 B \sin (c+d x)+13 A b^7 \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{36 a^6 b^2 C \sin (c+d x)-60 a^5 b^3 B \sin (c+d x)+90 a^4 A b^4 \sin (c+d x)-32 a^4 b^4 C \sin (c+d x)+71 a^3 b^5 B \sin (c+d x)-122 a^2 A b^6 \sin (c+d x)+11 a^2 b^6 C \sin (c+d x)-26 a b^7 B \sin (c+d x)+47 A b^8 \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3 (a \cos (c+d x)+b)}+\frac{b \left(-8 a^8 C+20 a^7 b B-40 a^6 A b^2+8 a^6 b^2 C-35 a^5 b^3 B+84 a^4 A b^4-7 a^4 b^4 C+28 a^3 b^5 B-69 a^2 A b^6+2 a^2 b^6 C-8 a b^7 B+20 A b^8\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{a^6 d \sqrt{a^2-b^2} \left(b^2-a^2\right)^3}","\frac{\sin (c+d x) \cos (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{x \left(a^2 (A+2 C)-8 a b B+20 A b^2\right)}{2 a^6}-\frac{\sin (c+d x) \cos (c+d x) \left(-4 a^4 C+7 a^3 b B-a^2 b^2 (10 A+C)-2 a b^3 B+5 A b^4\right)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{\sin (c+d x) \cos (c+d x) \left(-\left(a^6 (A-6 C)\right)-12 a^5 b B+a^4 b^2 (23 A-2 C)+11 a^3 b^3 B-a^2 b^4 (27 A-C)-4 a b^5 B+10 A b^6\right)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{\sin (c+d x) \cos (c+d x) \left(12 a^6 C-27 a^5 b B+a^4 b^2 (48 A+C)+20 a^3 b^3 B-a^2 b^4 (53 A-2 C)-8 a b^5 B+20 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\sin (c+d x) \left(6 a^7 B-a^6 (24 A b-26 b C)-65 a^5 b^2 B+a^4 b^3 (146 A-17 C)+68 a^3 b^4 B-a^2 b^5 (167 A-6 C)-24 a b^6 B+60 A b^7\right)}{6 a^5 d \left(a^2-b^2\right)^3}+\frac{b \left(-8 a^8 C+20 a^7 b B-8 a^6 b^2 (5 A-C)-35 a^5 b^3 B+7 a^4 b^4 (12 A-C)+28 a^3 b^5 B-a^2 b^6 (69 A-2 C)-8 a b^7 B+20 A b^8\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^6 d \sqrt{a-b} \sqrt{a+b} \left(a^2-b^2\right)^3}",1,"((a^2*A + 20*A*b^2 - 8*a*b*B + 2*a^2*C)*(c + d*x))/(2*a^6*d) + (b*(-40*a^6*A*b^2 + 84*a^4*A*b^4 - 69*a^2*A*b^6 + 20*A*b^8 + 20*a^7*b*B - 35*a^5*b^3*B + 28*a^3*b^5*B - 8*a*b^7*B - 8*a^8*C + 8*a^6*b^2*C - 7*a^4*b^4*C + 2*a^2*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^6*Sqrt[a^2 - b^2]*(-a^2 + b^2)^3*d) + ((4*A*b - a*B)*(((-1/2*I)*Cos[c + d*x])/a^5 - Sin[c + d*x]/(2*a^5)))/d + ((4*A*b - a*B)*(((I/2)*Cos[c + d*x])/a^5 - Sin[c + d*x]/(2*a^5)))/d + (A*b^6*Sin[c + d*x] - a*b^5*B*Sin[c + d*x] + a^2*b^4*C*Sin[c + d*x])/(3*a^5*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^3) + (-18*a^2*A*b^5*Sin[c + d*x] + 13*A*b^7*Sin[c + d*x] + 15*a^3*b^4*B*Sin[c + d*x] - 10*a*b^6*B*Sin[c + d*x] - 12*a^4*b^3*C*Sin[c + d*x] + 7*a^2*b^5*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x])^2) + (90*a^4*A*b^4*Sin[c + d*x] - 122*a^2*A*b^6*Sin[c + d*x] + 47*A*b^8*Sin[c + d*x] - 60*a^5*b^3*B*Sin[c + d*x] + 71*a^3*b^5*B*Sin[c + d*x] - 26*a*b^7*B*Sin[c + d*x] + 36*a^6*b^2*C*Sin[c + d*x] - 32*a^4*b^4*C*Sin[c + d*x] + 11*a^2*b^6*C*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(4*a^4*d)","C",1
930,1,23,24,0.0105059,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","-a C x+b B x+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}","x (b B-a C)+\frac{b C \tanh ^{-1}(\sin (c+d x))}{d}",1,"b*B*x - a*C*x + (b*C*ArcTanh[Sin[c + d*x]])/d","A",1
931,1,76,75,0.2182283,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{2 b (b B-2 a C) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+(c+d x) (b B-a C)}{a d}","\frac{x (b B-a C)}{a}-\frac{2 b (b B-2 a C) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a d \sqrt{a-b} \sqrt{a+b}}",1,"((b*B - a*C)*(c + d*x) + (2*b*(b*B - 2*a*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2])/(a*d)","A",1
932,1,211,140,0.8575471,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) (-a C+b B+b C \sec (c+d x)) \left(-\frac{2 b \left(3 a^3 C-2 a^2 b B-a b^2 C+b^3 B\right) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a b^2 (b B-2 a C) \sin (c+d x)}{(a-b) (a+b)}+(c+d x) (b B-a C) (a \cos (c+d x)+b)\right)}{a^2 d (a+b \sec (c+d x))^2 ((b B-a C) \cos (c+d x)+b C)}","\frac{b^2 (b B-2 a C) \tan (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{x (b B-a C)}{a^2}-\frac{2 b \left(-3 a^3 C+2 a^2 b B+a b^2 C-b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*(b*B - a*C + b*C*Sec[c + d*x])*((b*B - a*C)*(c + d*x)*(b + a*Cos[c + d*x]) - (2*b*(-2*a^2*b*B + b^3*B + 3*a^3*C - a*b^2*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x]))/(a^2 - b^2)^(3/2) + (a*b^2*(b*B - 2*a*C)*Sin[c + d*x])/((a - b)*(a + b))))/(a^2*d*(b*C + (b*B - a*C)*Cos[c + d*x])*(a + b*Sec[c + d*x])^2)","A",1
933,1,302,231,1.8549795,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) (-a C+b B+b C \sec (c+d x)) \left(-\frac{a b^2 \left(10 a^3 C-6 a^2 b B-4 a b^2 C+3 b^3 B\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}+\frac{2 b \left(-8 a^5 C+6 a^4 b B+4 a^3 b^2 C-5 a^2 b^3 B-2 a b^4 C+2 b^5 B\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^3 (2 a C-b B) \sin (c+d x)}{(a-b) (a+b)}+2 (c+d x) (b B-a C) (a \cos (c+d x)+b)^2\right)}{2 a^3 d (a+b \sec (c+d x))^3 ((b B-a C) \cos (c+d x)+b C)}","\frac{x (b B-a C)}{a^3}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{b^2 \left(-8 a^3 C+5 a^2 b B+2 a b^2 C-2 b^3 B\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{b \left(-8 a^5 C+6 a^4 b B+4 a^3 b^2 C-5 a^2 b^3 B-2 a b^4 C+2 b^5 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(b*B - a*C + b*C*Sec[c + d*x])*(2*(b*B - a*C)*(c + d*x)*(b + a*Cos[c + d*x])^2 + (2*b*(6*a^4*b*B - 5*a^2*b^3*B + 2*b^5*B - 8*a^5*C + 4*a^3*b^2*C - 2*a*b^4*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) + (a*b^3*(-(b*B) + 2*a*C)*Sin[c + d*x])/((a - b)*(a + b)) - (a*b^2*(-6*a^2*b*B + 3*b^3*B + 10*a^3*C - 4*a*b^2*C)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2)))/(2*a^3*d*(b*C + (b*B - a*C)*Cos[c + d*x])*(a + b*Sec[c + d*x])^3)","A",1
934,1,1097,336,5.4004315,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^5} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^5,x]","\frac{(b+a \cos (c+d x)) \sec ^3(c+d x) (b B-a C+b C \sec (c+d x)) \left(\frac{24 b \left(-10 C a^7+8 b B a^6+5 b^2 C a^5-8 b^3 B a^4-7 b^4 C a^3+7 b^5 B a^2+2 b^6 C a-2 b^7 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right) (b+a \cos (c+d x))^3}{\left(a^2-b^2\right)^{7/2}}+\frac{-6 c C \cos (3 (c+d x)) a^{10}-6 C d x \cos (3 (c+d x)) a^{10}-36 b c C a^9-36 b C d x a^9+6 b B c \cos (3 (c+d x)) a^9+6 b B d x \cos (3 (c+d x)) a^9+36 b^2 B c a^8+36 b^2 B d x a^8+18 b^2 c C \cos (3 (c+d x)) a^8+18 b^2 C d x \cos (3 (c+d x)) a^8-54 b^2 C \sin (c+d x) a^8-54 b^2 C \sin (3 (c+d x)) a^8+84 b^3 c C a^7+84 b^3 C d x a^7-18 b^3 B c \cos (3 (c+d x)) a^7-18 b^3 B d x \cos (3 (c+d x)) a^7+36 b^3 B \sin (c+d x) a^7-174 b^3 C \sin (2 (c+d x)) a^7+36 b^3 B \sin (3 (c+d x)) a^7-84 b^4 B c a^6-84 b^4 B d x a^6-18 b^4 c C \cos (3 (c+d x)) a^6-18 b^4 C d x \cos (3 (c+d x)) a^6-111 b^4 C \sin (c+d x) a^6+120 b^4 B \sin (2 (c+d x)) a^6+37 b^4 C \sin (3 (c+d x)) a^6-36 b^5 c C a^5-36 b^5 C d x a^5+18 b^5 B c \cos (3 (c+d x)) a^5+18 b^5 B d x \cos (3 (c+d x)) a^5+72 b^5 B \sin (c+d x) a^5+84 b^5 C \sin (2 (c+d x)) a^5-32 b^5 B \sin (3 (c+d x)) a^5+36 b^6 B c a^4+36 b^6 B d x a^4+6 b^6 c C \cos (3 (c+d x)) a^4+6 b^6 C d x \cos (3 (c+d x)) a^4+39 b^6 C \sin (c+d x) a^4-90 b^6 B \sin (2 (c+d x)) a^4-13 b^6 C \sin (3 (c+d x)) a^4-36 b^7 c C a^3-36 b^7 C d x a^3-6 b^7 B c \cos (3 (c+d x)) a^3-6 b^7 B d x \cos (3 (c+d x)) a^3-57 b^7 B \sin (c+d x) a^3-30 b^7 C \sin (2 (c+d x)) a^3+11 b^7 B \sin (3 (c+d x)) a^3+36 b^8 B c a^2+36 b^8 B d x a^2-36 b \left(a^2-b^2\right)^3 (a C-b B) (c+d x) \cos (2 (c+d x)) a^2-24 b^8 C \sin (c+d x) a^2+30 b^8 B \sin (2 (c+d x)) a^2+24 b^9 c C a+24 b^9 C d x a-18 \left(a^2-b^2\right)^3 \left(a^2+4 b^2\right) (a C-b B) (c+d x) \cos (c+d x) a+24 b^9 B \sin (c+d x) a-24 b^{10} B c-24 b^{10} B d x}{\left(a^2-b^2\right)^3}\right)}{24 a^4 d (b C+(b B-a C) \cos (c+d x)) (a+b \sec (c+d x))^4}","\frac{x (b B-a C)}{a^4}+\frac{b^2 (b B-2 a C) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{b^2 \left(-13 a^3 C+8 a^2 b B+3 a b^2 C-3 b^3 B\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b^2 \left(-37 a^5 C+26 a^4 b B+13 a^3 b^2 C-17 a^2 b^3 B-6 a b^4 C+6 b^5 B\right) \tan (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{b \left(-10 a^7 C+8 a^6 b B+5 a^5 b^2 C-8 a^4 b^3 B-7 a^3 b^4 C+7 a^2 b^5 B+2 a b^6 C-2 b^7 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(b*B - a*C + b*C*Sec[c + d*x])*((24*b*(8*a^6*b*B - 8*a^4*b^3*B + 7*a^2*b^5*B - 2*b^7*B - 10*a^7*C + 5*a^5*b^2*C - 7*a^3*b^4*C + 2*a*b^6*C)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (36*a^8*b^2*B*c - 84*a^6*b^4*B*c + 36*a^4*b^6*B*c + 36*a^2*b^8*B*c - 24*b^10*B*c - 36*a^9*b*c*C + 84*a^7*b^3*c*C - 36*a^5*b^5*c*C - 36*a^3*b^7*c*C + 24*a*b^9*c*C + 36*a^8*b^2*B*d*x - 84*a^6*b^4*B*d*x + 36*a^4*b^6*B*d*x + 36*a^2*b^8*B*d*x - 24*b^10*B*d*x - 36*a^9*b*C*d*x + 84*a^7*b^3*C*d*x - 36*a^5*b^5*C*d*x - 36*a^3*b^7*C*d*x + 24*a*b^9*C*d*x - 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-(b*B) + a*C)*(c + d*x)*Cos[c + d*x] - 36*a^2*b*(a^2 - b^2)^3*(-(b*B) + a*C)*(c + d*x)*Cos[2*(c + d*x)] + 6*a^9*b*B*c*Cos[3*(c + d*x)] - 18*a^7*b^3*B*c*Cos[3*(c + d*x)] + 18*a^5*b^5*B*c*Cos[3*(c + d*x)] - 6*a^3*b^7*B*c*Cos[3*(c + d*x)] - 6*a^10*c*C*Cos[3*(c + d*x)] + 18*a^8*b^2*c*C*Cos[3*(c + d*x)] - 18*a^6*b^4*c*C*Cos[3*(c + d*x)] + 6*a^4*b^6*c*C*Cos[3*(c + d*x)] + 6*a^9*b*B*d*x*Cos[3*(c + d*x)] - 18*a^7*b^3*B*d*x*Cos[3*(c + d*x)] + 18*a^5*b^5*B*d*x*Cos[3*(c + d*x)] - 6*a^3*b^7*B*d*x*Cos[3*(c + d*x)] - 6*a^10*C*d*x*Cos[3*(c + d*x)] + 18*a^8*b^2*C*d*x*Cos[3*(c + d*x)] - 18*a^6*b^4*C*d*x*Cos[3*(c + d*x)] + 6*a^4*b^6*C*d*x*Cos[3*(c + d*x)] + 36*a^7*b^3*B*Sin[c + d*x] + 72*a^5*b^5*B*Sin[c + d*x] - 57*a^3*b^7*B*Sin[c + d*x] + 24*a*b^9*B*Sin[c + d*x] - 54*a^8*b^2*C*Sin[c + d*x] - 111*a^6*b^4*C*Sin[c + d*x] + 39*a^4*b^6*C*Sin[c + d*x] - 24*a^2*b^8*C*Sin[c + d*x] + 120*a^6*b^4*B*Sin[2*(c + d*x)] - 90*a^4*b^6*B*Sin[2*(c + d*x)] + 30*a^2*b^8*B*Sin[2*(c + d*x)] - 174*a^7*b^3*C*Sin[2*(c + d*x)] + 84*a^5*b^5*C*Sin[2*(c + d*x)] - 30*a^3*b^7*C*Sin[2*(c + d*x)] + 36*a^7*b^3*B*Sin[3*(c + d*x)] - 32*a^5*b^5*B*Sin[3*(c + d*x)] + 11*a^3*b^7*B*Sin[3*(c + d*x)] - 54*a^8*b^2*C*Sin[3*(c + d*x)] + 37*a^6*b^4*C*Sin[3*(c + d*x)] - 13*a^4*b^6*C*Sin[3*(c + d*x)])/(a^2 - b^2)^3))/(24*a^4*d*(b*C + (b*B - a*C)*Cos[c + d*x])*(a + b*Sec[c + d*x])^4)","B",1
935,1,920,517,20.376309,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 (9 b B \sin (c+d x)+a C \sin (c+d x)) \sec ^3(c+d x)}{63 b}+\frac{4}{9} C \tan (c+d x) \sec ^3(c+d x)+\frac{4 \left(-6 C \sin (c+d x) a^2+9 b B \sin (c+d x) a+63 A b^2 \sin (c+d x)+49 b^2 C \sin (c+d x)\right) \sec ^2(c+d x)}{315 b^2}+\frac{4 \left(8 C \sin (c+d x) a^3-12 b B \sin (c+d x) a^2+21 A b^2 \sin (c+d x) a+13 b^2 C \sin (c+d x) a+75 b^3 B \sin (c+d x)\right) \sec (c+d x)}{315 b^3}+\frac{4 \left(-16 C a^4+24 b B a^3-42 A b^2 a^2-24 b^2 C a^2+57 b^3 B a+189 A b^4+147 b^4 C\right) \sin (c+d x)}{315 b^4}\right) \cos ^2(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 \sqrt{a+b \sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left((a+b) \left(16 C a^4-24 b B a^3+6 b^2 (7 A+4 C) a^2-57 b^3 B a-21 b^4 (9 A+7 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+b (a+b) \left(-16 C a^3+12 b (2 B+C) a^2-6 b^2 (7 A+3 B+6 C) a+3 b^3 (63 A+25 B+49 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(16 C a^4-24 b B a^3+6 b^2 (7 A+4 C) a^2-57 b^3 B a-21 b^4 (9 A+7 C)\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)\right)}{315 b^4 d \sqrt{b+a \cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \tan (c+d x) \sec (c+d x) \left(-6 a^2 C+9 a b B+63 A b^2+49 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 \tan (c+d x) \left(-8 a^3 C+12 a^2 b B-a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-16 a^3 C+12 a^2 b (2 B-C)-6 a b^2 (7 A-3 B+6 C)-3 b^3 (63 A-25 B+49 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-16 a^4 C+24 a^3 b B-6 a^2 b^2 (7 A+4 C)+57 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^5 d}+\frac{2 (a C+9 b B) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}",1,"(4*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-16*a^3*C + 12*a^2*b*(2*B + C) - 6*a*b^2*(7*A + 3*B + 6*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-24*a^3*b*B - 57*a*b^3*B + 16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(315*b^4*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(-42*a^2*A*b^2 + 189*A*b^4 + 24*a^3*b*B + 57*a*b^3*B - 16*a^4*C - 24*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^4) + (4*Sec[c + d*x]^3*(9*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(63*b) + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] + 9*a*b*B*Sin[c + d*x] - 6*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b^2) + (4*Sec[c + d*x]*(21*a*A*b^2*Sin[c + d*x] - 12*a^2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] + 8*a^3*C*Sin[c + d*x] + 13*a*b^2*C*Sin[c + d*x]))/(315*b^3) + (4*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","A",0
936,1,3706,413,26.0282919,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \tan (c+d x) \left(8 a^2 C-14 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{105 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(8 a^2 C-a (14 b B-6 b C)+35 A b^2-b^2 (63 B-25 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-8 a^3 C+14 a^2 b B-a b^2 (35 A+19 C)-63 b^3 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}+\frac{2 (7 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{3/2}}{7 b d}",1,"(Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(35*a*A*b^2 - 14*a^2*b*B + 63*b^3*B + 8*a^3*C + 19*a*b^2*C)*Sin[c + d*x])/(105*b^3) + (4*Sec[c + d*x]^2*(7*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(35*b) + (4*Sec[c + d*x]*(35*A*b^2*Sin[c + d*x] + 7*a*b*B*Sin[c + d*x] - 4*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b^2) + (4*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*((-2*a*A)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^2*B)/(15*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*b*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (38*a*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^3*C)/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*A*Sqrt[Sec[c + d*x]])/(3*b*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (4*a*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*B*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^4*C*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (34*a^2*C*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) + (10*b*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*Sqrt[b + a*Cos[c + d*x]]) - (6*a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (38*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(35*A*b^2 + 8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(35*A*b^2 + 8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(35*A*b^2 + 8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(35*A*b^2 + 8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(35*A*b^2 + 8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(35*A*b^2 + 8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*(2*(a + b)*(-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(35*A*b^2 + 8*a^2*C - 2*a*b*(7*B + 3*C) + b^2*(63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*b*B + 63*b^3*B + 8*a^3*C + a*b^2*(35*A + 19*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
937,1,579,324,20.0393371,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 \sin (c+d x) \left(-2 a^2 C+5 a b B+15 A b^2+9 b^2 C\right)}{15 b^2}+\frac{4 \sec (c+d x) (a C \sin (c+d x)+5 b B \sin (c+d x))}{15 b}+\frac{4}{5} C \tan (c+d x) \sec (c+d x)\right)}{d (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{4 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left((a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left(b (-2 a C+15 A b+5 b B+9 b C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-\left(-2 a^2 C+5 a b B+15 A b^2+9 b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)-\cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(-2 a^2 C+5 a b B+15 A b^2+9 b^2 C\right) (a \cos (c+d x)+b)\right)}{15 b^2 d \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) (2 a C+15 A b-5 b B+9 b C) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(a (5 b B-2 a C)+3 b^2 (5 A+3 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}+\frac{2 (5 b B-2 a C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 b d}",1,"(4*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((a + b)*(-((15*A*b^2 + 5*a*b*B - 2*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]) + b*(15*A*b + 5*b*B - 2*a*C + 9*b*C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] - (15*A*b^2 + 5*a*b*B - 2*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(15*b^2*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(15*A*b^2 + 5*a*b*B - 2*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*b^2) + (4*Sec[c + d*x]*(5*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(15*b) + (4*C*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","A",0
938,1,5287,366,23.7579661,"\int \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sqrt{a+b} \cot (c+d x) ((a-b) (3 B-C)+3 A b) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 (a-b) \sqrt{a+b} (a C+3 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"Result too large to show","B",0
939,1,922,362,16.7081808,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 C \sqrt{a+b \sec (c+d x)} \sin (c+d x)}{d}+\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a A \tan ^5\left(\frac{1}{2} (c+d x)\right)-A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a C \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 a C \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a A \tan \left(\frac{1}{2} (c+d x)\right)+A b \tan \left(\frac{1}{2} (c+d x)\right)-2 a C \tan \left(\frac{1}{2} (c+d x)\right)-2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (A-2 C) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 (A b-(B+C) b+a (B-C)) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 a B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \cot (c+d x) (2 a C+A b+2 b B-2 b C) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}-\frac{\sqrt{a+b} (2 a B+A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}+\frac{(a-b) \sqrt{a+b} (A-2 C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}+\frac{A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}",1,"(2*C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*A*Tan[(c + d*x)/2] + A*b*Tan[(c + d*x)/2] - 2*a*C*Tan[(c + d*x)/2] - 2*b*C*Tan[(c + d*x)/2] - 2*a*A*Tan[(c + d*x)/2]^3 + 4*a*C*Tan[(c + d*x)/2]^3 + a*A*Tan[(c + d*x)/2]^5 - A*b*Tan[(c + d*x)/2]^5 - 2*a*C*Tan[(c + d*x)/2]^5 + 2*b*C*Tan[(c + d*x)/2]^5 + 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(A - 2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(A*b + a*(B - C) - b*(B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
940,1,1842,435,19.9830814,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{A \sqrt{a+b \sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{\sqrt{a+b \sec (c+d x)} \left(-4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-4 a b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i (a-b) (A b+4 a B) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i (a-b) (A b+2 a (A+2 C)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a \sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \cot (c+d x) \left(-4 a^2 (A+2 C)-4 a b B+A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}+\frac{\sqrt{a+b} \cot (c+d x) (2 a (A+2 B+4 C)+A b) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{(4 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a d}+\frac{(a-b) \sqrt{a+b} (4 a B+A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a b d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"(A*Sqrt[a + b*Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (Sqrt[a + b*Sec[c + d*x]]*(-(a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]) - A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 2*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 8*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(A*b + 4*a*B)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(a - b)*(A*b + 2*a*(A + 2*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a*Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
941,1,542,538,14.2657961,"\int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{(6 a B+A b) \sin (2 (c+d x))}{24 a}+\frac{1}{12} A \sin (c+d x)+\frac{1}{12} A \sin (3 (c+d x))\right)}{d}-\frac{\cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(-a \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(8 a^2 (2 A+3 C)+6 a b B-3 A b^2\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)+b (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(4 a^2 (4 A+3 B+6 C)-6 a b (A+B)+3 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-a (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(8 a^2 (2 A+3 C)+6 a b B-3 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-3 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(8 a^3 B+4 a^2 b (A+2 C)-2 a b^2 B+A b^3\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left((b-a) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{24 a^3 d \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)}","-\frac{\sin (c+d x) \left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)}}{24 a^2 d}-\frac{\sqrt{a+b} \cot (c+d x) \left(-4 a^2 (4 A+3 B+6 C)-2 a b (A+3 B)+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(-8 a^2 (2 A+3 C)-6 a b B+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(8 a^3 B+4 a^2 b (A+2 C)-2 a b^2 B+A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a^3 d}+\frac{(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((A*Sin[c + d*x])/12 + ((A*b + 6*a*B)*Sin[2*(c + d*x)])/(24*a) + (A*Sin[3*(c + d*x)])/12))/d - (Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(-(a*(a + b)*(-3*A*b^2 + 6*a*b*B + 8*a^2*(2*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(3*A*b^2 - 6*a*b*(A + B) + 4*a^2*(4*A + 3*B + 6*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(A*b^3 + 8*a^3*B - 2*a*b^2*B + 4*a^2*b*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*(-3*A*b^2 + 6*a*b*B + 8*a^2*(2*A + 3*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(24*a^3*d*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2))","A",1
942,1,1087,628,21.2851343,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4}{99} (11 b B \sin (c+d x)+12 a C \sin (c+d x)) \sec ^4(c+d x)+\frac{4}{11} b C \tan (c+d x) \sec ^4(c+d x)+\frac{4 \left(3 C \sin (c+d x) a^2+110 b B \sin (c+d x) a+99 A b^2 \sin (c+d x)+81 b^2 C \sin (c+d x)\right) \sec ^3(c+d x)}{693 b}+\frac{4 \left(-18 C \sin (c+d x) a^3+33 b B \sin (c+d x) a^2+792 A b^2 \sin (c+d x) a+606 b^2 C \sin (c+d x) a+539 b^3 B \sin (c+d x)\right) \sec ^2(c+d x)}{3465 b^2}+\frac{4 \left(24 C \sin (c+d x) a^4-44 b B \sin (c+d x) a^3+99 A b^2 \sin (c+d x) a^2+57 b^2 C \sin (c+d x) a^2+968 b^3 B \sin (c+d x) a+825 A b^4 \sin (c+d x)+675 b^4 C \sin (c+d x)\right) \sec (c+d x)}{3465 b^3}-\frac{4 \left(48 C a^5-88 b B a^4+198 A b^2 a^3+108 b^2 C a^3-363 b^3 B a^2-2706 A b^4 a-2088 b^4 C a-1617 b^5 B\right) \sin (c+d x)}{3465 b^4}\right) \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 (a+b \sec (c+d x))^{3/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left((a+b) \left(48 C a^5-88 b B a^4+18 b^2 (11 A+6 C) a^3-363 b^3 B a^2-6 b^4 (451 A+348 C) a-1617 b^5 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+b (a+b) \left(-48 C a^4+4 b (22 B+9 C) a^3-6 b^2 (33 A+11 B+24 C) a^2+3 b^3 (627 A+143 B+471 C) a+3 b^4 (275 A+539 B+225 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(48 C a^5-88 b B a^4+18 b^2 (11 A+6 C) a^3-363 b^3 B a^2-6 b^4 (451 A+348 C) a-1617 b^5 B\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)\right)}{3465 b^4 d (b+a \cos (c+d x))^{3/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \tan (c+d x) \sec ^2(c+d x) \left(3 a^2 C+110 a b B+99 A b^2+81 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{693 b d}+\frac{2 \tan (c+d x) \sec (c+d x) \left(-18 a^3 C+33 a^2 b B+6 a b^2 (132 A+101 C)+539 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{3465 b^2 d}-\frac{2 \tan (c+d x) \left(-24 a^4 C+44 a^3 b B-3 a^2 b^2 (33 A+19 C)-968 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{a+b \sec (c+d x)}}{3465 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-48 a^4 C+4 a^3 b (22 B-9 C)-6 a^2 b^2 (33 A-11 B+24 C)-3 a b^3 (627 A-143 B+471 C)+3 b^4 (275 A-539 B+225 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-48 a^5 C+88 a^4 b B-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+6 a b^4 (451 A+348 C)+1617 b^5 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^5 d}+\frac{2 (3 a C+11 b B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{99 d}+\frac{2 C \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}",1,"(4*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-48*a^4*C + 4*a^3*b*(22*B + 9*C) - 6*a^2*b^2*(33*A + 11*B + 24*C) + 3*b^4*(275*A + 539*B + 225*C) + 3*a*b^3*(627*A + 143*B + 471*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 18*a^3*b^2*(11*A + 6*C) - 6*a*b^4*(451*A + 348*C))*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(3465*b^4*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(198*a^3*A*b^2 - 2706*a*A*b^4 - 88*a^4*b*B - 363*a^2*b^3*B - 1617*b^5*B + 48*a^5*C + 108*a^3*b^2*C - 2088*a*b^4*C)*Sin[c + d*x])/(3465*b^4) + (4*Sec[c + d*x]^4*(11*b*B*Sin[c + d*x] + 12*a*C*Sin[c + d*x]))/99 + (4*Sec[c + d*x]^3*(99*A*b^2*Sin[c + d*x] + 110*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 81*b^2*C*Sin[c + d*x]))/(693*b) + (4*Sec[c + d*x]^2*(792*a*A*b^2*Sin[c + d*x] + 33*a^2*b*B*Sin[c + d*x] + 539*b^3*B*Sin[c + d*x] - 18*a^3*C*Sin[c + d*x] + 606*a*b^2*C*Sin[c + d*x]))/(3465*b^2) + (4*Sec[c + d*x]*(99*a^2*A*b^2*Sin[c + d*x] + 825*A*b^4*Sin[c + d*x] - 44*a^3*b*B*Sin[c + d*x] + 968*a*b^3*B*Sin[c + d*x] + 24*a^4*C*Sin[c + d*x] + 57*a^2*b^2*C*Sin[c + d*x] + 675*b^4*C*Sin[c + d*x]))/(3465*b^3) + (4*b*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","A",0
943,1,4186,505,26.1504491,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \tan (c+d x) \left(8 a^2 C-18 a b B+63 A b^2+49 b^2 C\right) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \tan (c+d x) \left(-8 a^3 C+18 a^2 b B-3 a b^2 (21 A+13 C)-75 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-8 a^3 C+6 a^2 b (3 B-C)-3 a b^2 (21 A-57 B+13 C)+3 b^3 (63 A-25 B+49 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-8 a^4 C+18 a^3 b B-3 a^2 b^2 (21 A+11 C)-246 a b^3 B-21 b^4 (9 A+7 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^4 d}+\frac{2 (9 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(63*a^2*A*b^2 + 189*A*b^4 - 18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 33*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^3) + (4*Sec[c + d*x]^3*(9*b*B*Sin[c + d*x] + 10*a*C*Sin[c + d*x]))/63 + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] + 72*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/(315*b) + (4*Sec[c + d*x]*(126*a*A*b^2*Sin[c + d*x] + 9*a^2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] - 4*a^3*C*Sin[c + d*x] + 88*a*b^2*C*Sin[c + d*x]))/(315*b^2) + (4*b*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*((-2*a^2*A)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*A*b^2)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^3*B)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (164*a*b*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (22*a^2*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^4*C)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b^2*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^3*A*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (2*a*A*b*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (62*a^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*B*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (10*b^2*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*C*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (62*a^3*C*Sqrt[Sec[c + d*x]])/(315*b*Sqrt[b + a*Cos[c + d*x]]) + (26*a*b*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (2*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) - (6*a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (164*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (22*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) - (14*a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(7/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^3*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) - (2*((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - a*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*(a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a + b)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-1/2*(b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])) + (a + b)*((-18*a^3*b*B + 246*a*b^3*B + 8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(8*a^3*C - 6*a^2*b*(3*B + C) + 3*a*b^2*(21*A + 57*B + 13*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))","B",0
944,1,3724,406,26.2377367,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \tan (c+d x) \left(-6 a^2 C+21 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{105 b d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(6 a^2 C+3 a b (35 A-7 B+19 C)-b^2 (35 A-63 B+25 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-6 a^3 C+21 a^2 b B+2 a b^2 (70 A+41 C)+63 b^3 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^3 d}+\frac{2 (7 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(-140*a*A*b^2 - 21*a^2*b*B - 63*b^3*B + 6*a^3*C - 82*a*b^2*C)*Sin[c + d*x])/(105*b^2) + (4*Sec[c + d*x]^2*(7*b*B*Sin[c + d*x] + 8*a*C*Sin[c + d*x]))/35 + (4*Sec[c + d*x]*(35*A*b^2*Sin[c + d*x] + 42*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b) + (4*b*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*((-8*a*A*b)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*b^2*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^3*C)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (164*a*b*C)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*A*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^2*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (2*a^3*B*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (2*a*b*B*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (62*a^2*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*C*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (10*b^2*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (8*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (2*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) - (6*a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (164*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*C + 3*a*b*(35*A + 7*B + 19*C) + b^2*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(7/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*C + 3*a*b*(35*A + 7*B + 19*C) + b^2*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*C + 3*a*b*(35*A + 7*B + 19*C) + b^2*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-6*a^2*C + 3*a*b*(35*A + 7*B + 19*C) + b^2*(35*A + 63*B + 25*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-6*a^2*C + 3*a*b*(35*A + 7*B + 19*C) + b^2*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-6*a^2*C + 3*a*b*(35*A + 7*B + 19*C) + b^2*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(2*(a + b)*(-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*C + 3*a*b*(35*A + 7*B + 19*C) + b^2*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*b*B - 63*b^3*B + 6*a^3*C - 2*a*b^2*(70*A + 41*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
945,1,6946,443,26.3264301,"\int (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sqrt{a+b} \cot (c+d x) \left(3 a^2 (5 B-C)+2 a b (15 A-10 B+6 C)-b^2 (15 A-5 B+9 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(3 a^2 C+20 a b B+15 A b^2+9 b^2 C\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^2 d}-\frac{2 a A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 (3 a C+5 b B) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"Result too large to show","B",0
946,1,7670,426,26.2152293,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \cot (c+d x) \left(6 a^2 C+a b (3 A+12 B-8 C)+2 b^2 (3 A-3 B+C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (3 a A-8 a C-6 b B) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b d}-\frac{\sqrt{a+b} (2 a B+3 A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{b (3 A-2 C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}",1,"Result too large to show","B",0
947,1,538,442,20.2794202,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{1}{2} \left(\frac{\cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(\frac{1}{2} a A \sin (2 (c+d x))+4 b C \sin (c+d x)\right)}{d (a \cos (c+d x)+b)}-\frac{\sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left(\sec ^2\left(\frac{1}{2} (c+d x)\right) \left(4 a^2 (A+2 C)+12 a b B+3 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)-a \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 a B+5 A b-8 b C) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)+b (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) (2 a (A+2 B-4 C)+3 A b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-a (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) (4 a B+5 A b-8 b C) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{2 a d \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}\right)","-\frac{\sqrt{a+b} \cot (c+d x) \left(4 a^2 (A+2 C)+12 a b B+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a d}+\frac{\sqrt{a+b} \cot (c+d x) (2 a (A+2 B+8 C)+b (5 A+8 B-8 C)) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) (4 a B+5 A b-8 b C) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 b d}+\frac{(4 a B+3 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}",1,"((Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(4*b*C*Sin[c + d*x] + (a*A*Sin[2*(c + d*x)])/2))/(d*(b + a*Cos[c + d*x])) - (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(-(a*(a + b)*(5*A*b + 4*a*B - 8*b*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(3*A*b + 2*a*(A + 2*B - 4*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (3*A*b^2 + 12*a*b*B + 4*a^2*(A + 2*C))*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*(5*A*b + 4*a*B - 8*b*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(2*a*d*(b + a*Cos[c + d*x])^2*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(3/2)))/2","A",0
948,1,5040,540,24.1456222,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)}}{24 a d}+\frac{\sqrt{a+b} \cot (c+d x) \left(4 a^2 (4 A+3 B+6 C)+2 a b (7 A+15 B+24 C)+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(8 a^2 (2 A+3 C)+30 a b B+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a b d}+\frac{\sqrt{a+b} \cot (c+d x) \left(-8 a^3 B-12 a^2 b (A+2 C)-6 a b^2 B+A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a^2 d}+\frac{(2 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}",1,"Result too large to show","B",0
949,1,759,650,17.2881716,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{\sin (2 (c+d x)) \left(48 a^2 A+48 a^2 C+56 a b B+3 A b^2\right)}{96 a}+\frac{1}{48} (8 a B+9 A b) \sin (c+d x)+\frac{1}{48} (8 a B+9 A b) \sin (3 (c+d x))+\frac{1}{16} a A \sin (4 (c+d x))\right)}{d (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{\cos ^5(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-a \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(128 a^3 B+12 a^2 b (13 A+20 C)+24 a b^2 B-9 A b^3\right) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)+b (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(8 a^3 (9 A+16 B+12 C)+12 a^2 b (7 A+4 (B+3 C))-6 a b^2 (3 A+4 B)+9 A b^3\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-a (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(128 a^3 B+12 a^2 b (13 A+20 C)+24 a b^2 B-9 A b^3\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-3 \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left((b-a) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{96 a^3 d \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{\sin (c+d x) \cos (c+d x) \left(12 a^2 (3 A+4 C)+56 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)}}{96 a d}-\frac{\sin (c+d x) \left(-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right) \sqrt{a+b \sec (c+d x)}}{192 a^2 d}-\frac{\sqrt{a+b} \cot (c+d x) \left(-8 a^3 (9 A+16 B+12 C)-4 a^2 b (39 A+28 B+60 C)-6 a b^2 (A+4 B)+9 A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(-128 a^3 B-12 a^2 b (13 A+20 C)-24 a b^2 B+9 A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(16 a^4 (3 A+4 C)+96 a^3 b B+24 a^2 b^2 (A+2 C)-8 a b^3 B+3 A b^4\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^3 d}+\frac{(8 a B+3 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((9*A*b + 8*a*B)*Sin[c + d*x])/48 + ((48*a^2*A + 3*A*b^2 + 56*a*b*B + 48*a^2*C)*Sin[2*(c + d*x)])/(96*a) + ((9*A*b + 8*a*B)*Sin[3*(c + d*x)])/48 + (a*A*Sin[4*(c + d*x)])/16))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-(a*(a + b)*(-9*A*b^3 + 128*a^3*B + 24*a*b^2*B + 12*a^2*b*(13*A + 20*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(9*A*b^3 - 6*a*b^2*(3*A + 4*B) + 8*a^3*(9*A + 16*B + 12*C) + 12*a^2*b*(7*A + 4*(B + 3*C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(3*A*b^4 + 96*a^3*b*B - 8*a*b^3*B + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*(-9*A*b^3 + 128*a^3*B + 24*a*b^2*B + 12*a^2*b*(13*A + 20*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(96*a^3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2))","A",0
950,1,1090,610,21.772348,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4}{99} \left(11 B \sin (c+d x) b^2+23 a C \sin (c+d x) b\right) \sec ^4(c+d x)+\frac{4}{11} b^2 C \tan (c+d x) \sec ^4(c+d x)+\frac{4}{693} \left(113 C \sin (c+d x) a^2+209 b B \sin (c+d x) a+99 A b^2 \sin (c+d x)+81 b^2 C \sin (c+d x)\right) \sec ^3(c+d x)+\frac{4 \left(15 C \sin (c+d x) a^3+825 b B \sin (c+d x) a^2+1485 A b^2 \sin (c+d x) a+1145 b^2 C \sin (c+d x) a+539 b^3 B \sin (c+d x)\right) \sec ^2(c+d x)}{3465 b}+\frac{4 \left(-20 C \sin (c+d x) a^4+55 b B \sin (c+d x) a^3+1485 A b^2 \sin (c+d x) a^2+1025 b^2 C \sin (c+d x) a^2+1793 b^3 B \sin (c+d x) a+825 A b^4 \sin (c+d x)+675 b^4 C \sin (c+d x)\right) \sec (c+d x)}{3465 b^2}+\frac{4 \left(40 C a^5-110 b B a^4+495 A b^2 a^3+255 b^2 C a^3+3069 b^3 B a^2+4785 A b^4 a+3705 b^4 C a+1617 b^5 B\right) \sin (c+d x)}{3465 b^3}\right)}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 (a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left((a+b) \left(40 C a^5-110 b B a^4+15 b^2 (33 A+17 C) a^3+3069 b^3 B a^2+15 b^4 (319 A+247 C) a+1617 b^5 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)-b (a+b) \left(40 C a^4-10 b (11 B+3 C) a^3+15 b^2 (33 A+121 B+19 C) a^2+6 b^3 (660 A+209 B+505 C) a+3 b^4 (275 A+539 B+225 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(40 C a^5-110 b B a^4+15 b^2 (33 A+17 C) a^3+3069 b^3 B a^2+15 b^4 (319 A+247 C) a+1617 b^5 B\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)\right)}{3465 b^3 d (b+a \cos (c+d x))^{5/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \tan (c+d x) \left(8 a^2 C-22 a b B+99 A b^2+81 b^2 C\right) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \tan (c+d x) \left(-40 a^3 C+110 a^2 b B-5 a b^2 (99 A+67 C)-539 b^3 B\right) (a+b \sec (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \tan (c+d x) \left(-40 a^4 C+110 a^3 b B-15 a^2 b^2 (33 A+19 C)-1254 a b^3 B-75 b^4 (11 A+9 C)\right) \sqrt{a+b \sec (c+d x)}}{3465 b^2 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-40 a^4 C+10 a^3 b (11 B-3 C)-15 a^2 b^2 (33 A-121 B+19 C)+6 a b^3 (660 A-209 B+505 C)-3 b^4 (275 A-539 B+225 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-40 a^5 C+110 a^4 b B-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B-15 a b^4 (319 A+247 C)-1617 b^5 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3465 b^4 d}+\frac{2 (11 b B-4 a C) \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}",1,"(-4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 15*a^3*b^2*(33*A + 17*C) + 15*a*b^4*(319*A + 247*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(40*a^4*C - 10*a^3*b*(11*B + 3*C) + 15*a^2*b^2*(33*A + 121*B + 19*C) + 3*b^4*(275*A + 539*B + 225*C) + 6*a*b^3*(660*A + 209*B + 505*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 15*a^3*b^2*(33*A + 17*C) + 15*a*b^4*(319*A + 247*C))*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(3465*b^3*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(495*a^3*A*b^2 + 4785*a*A*b^4 - 110*a^4*b*B + 3069*a^2*b^3*B + 1617*b^5*B + 40*a^5*C + 255*a^3*b^2*C + 3705*a*b^4*C)*Sin[c + d*x])/(3465*b^3) + (4*Sec[c + d*x]^4*(11*b^2*B*Sin[c + d*x] + 23*a*b*C*Sin[c + d*x]))/99 + (4*Sec[c + d*x]^3*(99*A*b^2*Sin[c + d*x] + 209*a*b*B*Sin[c + d*x] + 113*a^2*C*Sin[c + d*x] + 81*b^2*C*Sin[c + d*x]))/693 + (4*Sec[c + d*x]^2*(1485*a*A*b^2*Sin[c + d*x] + 825*a^2*b*B*Sin[c + d*x] + 539*b^3*B*Sin[c + d*x] + 15*a^3*C*Sin[c + d*x] + 1145*a*b^2*C*Sin[c + d*x]))/(3465*b) + (4*Sec[c + d*x]*(1485*a^2*A*b^2*Sin[c + d*x] + 825*A*b^4*Sin[c + d*x] + 55*a^3*b*B*Sin[c + d*x] + 1793*a*b^3*B*Sin[c + d*x] - 20*a^4*C*Sin[c + d*x] + 1025*a^2*b^2*C*Sin[c + d*x] + 675*b^4*C*Sin[c + d*x]))/(3465*b^2) + (4*b^2*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","A",0
951,1,4220,502,26.6968933,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \tan (c+d x) \left(-10 a^2 C+45 a b B+63 A b^2+49 b^2 C\right) (a+b \sec (c+d x))^{3/2}}{315 b d}+\frac{2 \tan (c+d x) \left(-10 a^3 C+45 a^2 b B+6 a b^2 (28 A+19 C)+75 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{315 b d}+\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(10 a^3 C+15 a^2 b (21 A-3 B+11 C)-6 a b^2 (28 A-60 B+19 C)+3 b^3 (63 A-25 B+49 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-10 a^4 C+45 a^3 b B+3 a^2 b^2 (161 A+93 C)+435 a b^3 B+21 b^4 (9 A+7 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^3 d}+\frac{2 (9 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}",1,"(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(483*a^2*A*b^2 + 189*A*b^4 + 45*a^3*b*B + 435*a*b^3*B - 10*a^4*C + 279*a^2*b^2*C + 147*b^4*C)*Sin[c + d*x])/(315*b^2) + (4*Sec[c + d*x]^3*(9*b^2*B*Sin[c + d*x] + 19*a*b*C*Sin[c + d*x]))/63 + (4*Sec[c + d*x]^2*(63*A*b^2*Sin[c + d*x] + 135*a*b*B*Sin[c + d*x] + 75*a^2*C*Sin[c + d*x] + 49*b^2*C*Sin[c + d*x]))/315 + (4*Sec[c + d*x]*(231*a*A*b^2*Sin[c + d*x] + 135*a^2*b*B*Sin[c + d*x] + 75*b^3*B*Sin[c + d*x] + 5*a^3*C*Sin[c + d*x] + 163*a*b^2*C*Sin[c + d*x]))/(315*b) + (4*b^2*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*((-46*a^2*A*b)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*A*b^3)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^3*B)/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (58*a*b^2*B)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^4*C)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (62*a^2*b*C)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*b^3*C)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^3*A*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (16*a*A*b^2*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (2*a^4*B*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (4*a^2*b*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (10*b^3*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (248*a^3*C*Sqrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x]]) + (4*a^5*C*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) + (76*a*b^2*C*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (46*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (6*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (2*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (58*a^2*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (62*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (4*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (14*a*b^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((a + b)*((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^2*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(9/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(315*b^2*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (2*((a + b)*((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^6)/2 - a*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] - (-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Sin[c + d*x]*Tan[(c + d*x)/2] + 2*(-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]^2 + (3*(a + b)*((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (a + b)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*((b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])) + (a + b)*((-45*a^3*b*B - 435*a*b^3*B + 10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-10*a^3*C + 15*a^2*b*(21*A + 3*B + 11*C) + 6*a*b^2*(28*A + 60*B + 19*C) + 3*b^3*(63*A + 25*B + 49*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x]*Tan[c + d*x]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))","B",0
952,1,1401,521,20.9000635,"\int (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4}{35} \left(7 B \sin (c+d x) b^2+15 a C \sin (c+d x) b\right) \sec ^2(c+d x)+\frac{4}{7} b^2 C \tan (c+d x) \sec ^2(c+d x)+\frac{4}{105} \left(45 C \sin (c+d x) a^2+77 b B \sin (c+d x) a+35 A b^2 \sin (c+d x)+25 b^2 C \sin (c+d x)\right) \sec (c+d x)+\frac{4 \left(15 C a^3+161 b B a^2+245 A b^2 a+145 b^2 C a+63 b^3 B\right) \sin (c+d x)}{105 b}\right)}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 (a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-245 a A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+245 a^2 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)-63 b^4 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+63 a b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-161 a^2 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+161 a^3 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a^4 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-145 a b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+145 a^2 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a^3 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-490 a^2 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-126 a b^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-322 a^3 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)-30 a^4 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-290 a^2 b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right)-210 a^3 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+245 a A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+245 a^2 A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+63 b^4 B \tan \left(\frac{1}{2} (c+d x)\right)+63 a b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+161 a^2 b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+161 a^3 b B \tan \left(\frac{1}{2} (c+d x)\right)+15 a^4 C \tan \left(\frac{1}{2} (c+d x)\right)+145 a b^3 C \tan \left(\frac{1}{2} (c+d x)\right)+145 a^2 b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+15 a^3 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(15 C a^3+161 b B a^2+5 b^2 (49 A+29 C) a+63 b^3 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-b \left(-15 (7 A-7 B-C) a^3+b (315 A+161 B+135 C) a^2+b^2 (245 A+119 B+145 C) a+b^3 (35 A+63 B+25 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-210 a^3 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{105 b d (b+a \cos (c+d x))^{5/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \tan (c+d x) \left(15 a^2 C+56 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{105 d}-\frac{2 a^2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 \sqrt{a+b} \cot (c+d x) \left(15 a^3 (7 B-C)+a^2 b (315 A-161 B+135 C)-a b^2 (245 A-119 B+145 C)+b^3 (35 A-63 B+25 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(15 a^3 C+161 a^2 b B+5 a b^2 (49 A+29 C)+63 b^3 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^2 d}+\frac{2 (5 a C+7 b B) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 d}+\frac{2 C \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}",1,"(-4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(245*a^2*A*b^2*Tan[(c + d*x)/2] + 245*a*A*b^3*Tan[(c + d*x)/2] + 161*a^3*b*B*Tan[(c + d*x)/2] + 161*a^2*b^2*B*Tan[(c + d*x)/2] + 63*a*b^3*B*Tan[(c + d*x)/2] + 63*b^4*B*Tan[(c + d*x)/2] + 15*a^4*C*Tan[(c + d*x)/2] + 15*a^3*b*C*Tan[(c + d*x)/2] + 145*a^2*b^2*C*Tan[(c + d*x)/2] + 145*a*b^3*C*Tan[(c + d*x)/2] - 490*a^2*A*b^2*Tan[(c + d*x)/2]^3 - 322*a^3*b*B*Tan[(c + d*x)/2]^3 - 126*a*b^3*B*Tan[(c + d*x)/2]^3 - 30*a^4*C*Tan[(c + d*x)/2]^3 - 290*a^2*b^2*C*Tan[(c + d*x)/2]^3 + 245*a^2*A*b^2*Tan[(c + d*x)/2]^5 - 245*a*A*b^3*Tan[(c + d*x)/2]^5 + 161*a^3*b*B*Tan[(c + d*x)/2]^5 - 161*a^2*b^2*B*Tan[(c + d*x)/2]^5 + 63*a*b^3*B*Tan[(c + d*x)/2]^5 - 63*b^4*B*Tan[(c + d*x)/2]^5 + 15*a^4*C*Tan[(c + d*x)/2]^5 - 15*a^3*b*C*Tan[(c + d*x)/2]^5 + 145*a^2*b^2*C*Tan[(c + d*x)/2]^5 - 145*a*b^3*C*Tan[(c + d*x)/2]^5 - 210*a^3*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 210*a^3*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - b*(-15*a^3*(7*A - 7*B - C) + b^3*(35*A + 63*B + 25*C) + a^2*b*(315*A + 161*B + 135*C) + a*b^2*(245*A + 119*B + 145*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(105*b*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(245*a*A*b^2 + 161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Sin[c + d*x])/(105*b) + (4*Sec[c + d*x]^2*(7*b^2*B*Sin[c + d*x] + 15*a*b*C*Sin[c + d*x]))/35 + (4*Sec[c + d*x]*(35*A*b^2*Sin[c + d*x] + 77*a*b*B*Sin[c + d*x] + 45*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/105 + (4*b^2*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","B",0
953,1,1490,505,20.8780738,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4}{5} C \sec (c+d x) \tan (c+d x) b^2+\frac{4}{15} \left(23 C a^2+35 b B a+15 A b^2+9 b^2 C\right) \sin (c+d x)+\frac{4}{15} \sec (c+d x) \left(5 B \sin (c+d x) b^2+11 a C \sin (c+d x) b\right)\right) \cos ^4(c+d x)}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 (a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(30 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-15 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+70 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-70 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-46 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+18 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)-18 a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+46 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+60 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-30 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)+140 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+92 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+36 a b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+150 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+60 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-30 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-30 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+15 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+15 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)-70 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-70 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-46 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-18 b^3 C \tan \left(\frac{1}{2} (c+d x)\right)-18 a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)-46 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left((15 A-46 C) a^2-70 b B a-6 b^2 (5 A+3 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(15 (B-C) a^3+b (45 A-45 B-23 C) a^2-b^2 (45 A+35 B+17 C) a-b^3 (15 A+5 B+9 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+150 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+60 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{15 d (b+a \cos (c+d x))^{5/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(-\left(a^2 (15 A-46 C)\right)+70 a b B+6 b^2 (5 A+3 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}+\frac{\sqrt{a+b} \cot (c+d x) \left(30 a^3 C+a^2 b (15 A+90 B-46 C)+2 a b^2 (45 A-35 B+17 C)-2 b^3 (15 A-5 B+9 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b d}-\frac{b \tan (c+d x) (15 a A-16 a C-10 b B) \sqrt{a+b \sec (c+d x)}}{15 d}-\frac{a \sqrt{a+b} (2 a B+5 A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{b (5 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{d}",1,"(2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(15*a^3*A*Tan[(c + d*x)/2] + 15*a^2*A*b*Tan[(c + d*x)/2] - 30*a*A*b^2*Tan[(c + d*x)/2] - 30*A*b^3*Tan[(c + d*x)/2] - 70*a^2*b*B*Tan[(c + d*x)/2] - 70*a*b^2*B*Tan[(c + d*x)/2] - 46*a^3*C*Tan[(c + d*x)/2] - 46*a^2*b*C*Tan[(c + d*x)/2] - 18*a*b^2*C*Tan[(c + d*x)/2] - 18*b^3*C*Tan[(c + d*x)/2] - 30*a^3*A*Tan[(c + d*x)/2]^3 + 60*a*A*b^2*Tan[(c + d*x)/2]^3 + 140*a^2*b*B*Tan[(c + d*x)/2]^3 + 92*a^3*C*Tan[(c + d*x)/2]^3 + 36*a*b^2*C*Tan[(c + d*x)/2]^3 + 15*a^3*A*Tan[(c + d*x)/2]^5 - 15*a^2*A*b*Tan[(c + d*x)/2]^5 - 30*a*A*b^2*Tan[(c + d*x)/2]^5 + 30*A*b^3*Tan[(c + d*x)/2]^5 - 70*a^2*b*B*Tan[(c + d*x)/2]^5 + 70*a*b^2*B*Tan[(c + d*x)/2]^5 - 46*a^3*C*Tan[(c + d*x)/2]^5 + 46*a^2*b*C*Tan[(c + d*x)/2]^5 - 18*a*b^2*C*Tan[(c + d*x)/2]^5 + 18*b^3*C*Tan[(c + d*x)/2]^5 + 150*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 150*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-70*a*b*B + a^2*(15*A - 46*C) - 6*b^2*(5*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(a^2*b*(45*A - 45*B - 23*C) + 15*a^3*(B - C) - b^3*(15*A + 5*B + 9*C) - a*b^2*(45*A + 35*B + 17*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(15*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(15*A*b^2 + 35*a*b*B + 23*a^2*C + 9*b^2*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]*(5*b^2*B*Sin[c + d*x] + 11*a*b*C*Sin[c + d*x]))/15 + (4*b^2*C*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","B",0
954,1,4887,507,25.3477147,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \cot (c+d x) \left(6 a^2 (A+2 (B+6 C))+a b (27 A+72 B-56 C)+8 b^2 (3 A-3 B+C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(12 a^2 B+a b (27 A-56 C)-24 b^2 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{12 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(4 a^2 (A+2 C)+20 a b B+15 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 d}-\frac{b \tan (c+d x) (12 a B+21 A b-8 b C) \sqrt{a+b \sec (c+d x)}}{12 d}+\frac{(4 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}+\frac{A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{5/2}}{2 d}",1,"((Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((4*b*(3*b*B + 7*a*C)*Sin[c + d*x])/3 + (a^2*A*Sin[2*(c + d*x)])/2 + (4*b^2*C*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*((a^3*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a*A*b^2)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^2*b*B)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^3*B)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^3*C)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (14*a*b^2*C)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (11*a^2*A*b*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^3*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (a^3*B*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (4*a*b^2*B*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] + (4*a^2*b*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (9*a^2*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(4*Sqrt[b + a*Cos[c + d*x]]) + (a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] - (2*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]] - (14*a^2*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(6*d*(b + a*Cos[c + d*x])^3*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(5/2)*((a*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(12*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(4*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(12*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(12*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x])/2 + (a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + (3*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 + ((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*((-a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(21*A*b + 6*a*(A + 2*B - 8*C) - 8*b*(3*B + C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]) + 3*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*(((-a + b)*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + (a*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])) - a*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]*Tan[c + d*x] + (12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]*Tan[c + d*x]))/(6*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)))))/2","B",0
955,1,5347,549,26.0040324,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(8 a^2 (2 A+3 C)+42 a b B+15 A b^2\right) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{\sqrt{a+b} \cot (c+d x) \left(4 a^2 (4 A+3 B+6 C)+2 a b (13 A+27 B+72 C)+3 b^2 (11 A+16 (B-C))\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(8 a^2 (2 A+3 C)+54 a b B+3 b^2 (11 A-16 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(8 a^3 B+20 a^2 b (A+2 C)+30 a b^2 B+5 A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a d}+\frac{(6 a B+5 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2}}{3 d}",1,"Result too large to show","B",0
956,1,5667,652,26.5193715,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \cos (c+d x) \left(4 a^2 (3 A+4 C)+24 a b B+5 A b^2\right) \sqrt{a+b \sec (c+d x)}}{32 d}+\frac{\sin (c+d x) \left(128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\sqrt{a+b} \cot (c+d x) \left(8 a^3 (9 A+16 B+12 C)+4 a^2 b (71 A+52 B+108 C)+2 a b^2 (59 A+132 B+192 C)+15 A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a d}+\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(128 a^3 B+4 a^2 b (71 A+108 C)+264 a b^2 B+15 A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a b d}+\frac{\sqrt{a+b} \cot (c+d x) \left(-16 a^4 (3 A+4 C)-160 a^3 b B-120 a^2 b^2 (A+2 C)-40 a b^3 B+5 A b^4\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{64 a^2 d}+\frac{(8 a B+5 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{24 d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}",1,"Result too large to show","B",0
957,1,800,774,20.8867937,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{1}{40} A \sin (5 (c+d x)) a^2+\frac{1}{160} (21 A b+10 a B) \sin (4 (c+d x)) a+\frac{1}{480} \left(88 A a^2+80 C a^2+170 b B a+93 A b^2\right) \sin (c+d x)+\frac{1}{480} \left(100 A a^2+80 C a^2+170 b B a+93 A b^2\right) \sin (3 (c+d x))+\frac{\left(480 B a^3+1024 A b a^2+1040 b C a^2+590 b^2 B a+15 A b^3\right) \sin (2 (c+d x))}{960 a}\right)}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{\cos ^5(c+d x) (a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{i \left((a-b) \left(256 (4 A+5 C) a^4+2840 b B a^3+12 b^2 (141 A+220 C) a^2+150 b^3 B a-45 A b^4\right) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)-2 (a-b) \left(720 B a^4+8 b (161 A+45 B+220 C) a^3+4 b^2 (129 A+185 B+180 C) a^2-30 b^3 (A-5 B) a-45 A b^4\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)+30 \left(96 B a^5+80 b (3 A+4 C) a^4+240 b^2 B a^3+40 b^3 (A+2 C) a^2-10 b^4 B a+3 A b^5\right) \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right)\right) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}}}{\sqrt{\frac{b-a}{a+b}} (b+a \cos (c+d x)) \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)}}-\left(256 (4 A+5 C) a^4+2840 b B a^3+12 b^2 (141 A+220 C) a^2+150 b^3 B a-45 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right)\right)}{960 a^2 d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}","\frac{\sin (c+d x) \cos ^2(c+d x) \left(16 a^2 (4 A+5 C)+110 a b B+15 A b^2\right) \sqrt{a+b \sec (c+d x)}}{240 d}+\frac{\sin (c+d x) \cos (c+d x) \left(360 a^3 B+4 a^2 b (193 A+260 C)+590 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{960 a d}-\frac{\sin (c+d x) \left(-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right) \sqrt{a+b \sec (c+d x)}}{1920 a^2 d}-\frac{\sqrt{a+b} \cot (c+d x) \left(-16 a^4 (64 A+45 B+80 C)-8 a^3 b (193 A+355 B+260 C)-4 a^2 b^2 (423 A+295 B+660 C)-30 a b^3 (A+5 B)+45 A b^4\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{1920 a^2 d}-\frac{(a-b) \sqrt{a+b} \cot (c+d x) \left(-256 a^4 (4 A+5 C)-2840 a^3 b B-12 a^2 b^2 (141 A+220 C)-150 a b^3 B+45 A b^4\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{1920 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(96 a^5 B+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+40 a^2 b^3 (A+2 C)-10 a b^4 B+3 A b^5\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{128 a^3 d}+\frac{(2 a B+A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{8 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}",1,"(Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((88*a^2*A + 93*A*b^2 + 170*a*b*B + 80*a^2*C)*Sin[c + d*x])/480 + ((1024*a^2*A*b + 15*A*b^3 + 480*a^3*B + 590*a*b^2*B + 1040*a^2*b*C)*Sin[2*(c + d*x)])/(960*a) + ((100*a^2*A + 93*A*b^2 + 170*a*b*B + 80*a^2*C)*Sin[3*(c + d*x)])/480 + (a*(21*A*b + 10*a*B)*Sin[4*(c + d*x)])/160 + (a^2*A*Sin[5*(c + d*x)])/40))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (Cos[c + d*x]^5*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((I*((a - b)*(-45*A*b^4 + 2840*a^3*b*B + 150*a*b^3*B + 256*a^4*(4*A + 5*C) + 12*a^2*b^2*(141*A + 220*C))*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] - 2*(a - b)*(-45*A*b^4 - 30*a*b^3*(A - 5*B) + 720*a^4*B + 4*a^2*b^2*(129*A + 185*B + 180*C) + 8*a^3*b*(161*A + 45*B + 220*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)] + 30*(3*A*b^5 + 96*a^5*B + 240*a^3*b^2*B - 10*a*b^4*B + 40*a^2*b^3*(A + 2*C) + 80*a^4*b*(3*A + 4*C))*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)])*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(Sqrt[(-a + b)/(a + b)]*(b + a*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]) - (-45*A*b^4 + 2840*a^3*b*B + 150*a*b^3*B + 256*a^4*(4*A + 5*C) + 12*a^2*b^2*(141*A + 220*C))*Tan[(c + d*x)/2]))/(960*a^2*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
958,1,3811,429,26.186906,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{2 \tan (c+d x) \left(24 a^2 C-28 a b B+35 A b^2+25 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{105 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(-48 a^3 C+56 a^2 b B-2 a b^2 (35 A+22 C)+63 b^3 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^5 d}+\frac{2 \sqrt{a+b} \cot (c+d x) \left(48 a^3 C-4 a^2 b (14 B+3 C)+2 a b^2 (35 A+7 B+22 C)+b^3 (35 A-63 B+25 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}+\frac{2 (7 b B-6 a C) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b^2 d}+\frac{2 C \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 b d}",1,"(Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(-70*a*A*b^2 + 56*a^2*b*B + 63*b^3*B - 48*a^3*C - 44*a*b^2*C)*Sin[c + d*x])/(105*b^4) + (4*Sec[c + d*x]^2*(7*b*B*Sin[c + d*x] - 6*a*C*Sin[c + d*x]))/(35*b^2) + (4*Sec[c + d*x]*(35*A*b^2*Sin[c + d*x] - 28*a*b*B*Sin[c + d*x] + 24*a^2*C*Sin[c + d*x] + 25*b^2*C*Sin[c + d*x]))/(105*b^3) + (4*C*Sec[c + d*x]^2*Tan[c + d*x])/(7*b)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]]) + (4*((4*a*A)/(3*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^2*B)/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (32*a^3*C)/(35*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (88*a*C)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (4*a^2*A*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^3*B*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (14*a*B*Sqrt[Sec[c + d*x]])/(15*b*Sqrt[b + a*Cos[c + d*x]]) + (10*C*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (32*a^4*C*Sqrt[Sec[c + d*x]])/(35*b^4*Sqrt[b + a*Cos[c + d*x]]) + (64*a^2*C*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]) + (4*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (6*a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (32*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^4*Sqrt[b + a*Cos[c + d*x]]) + (88*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(4*a^2*b*(14*B - 3*C) - 48*a^3*C - 2*a*b^2*(35*A - 7*B + 22*C) + b^3*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(4*a^2*b*(14*B - 3*C) - 48*a^3*C - 2*a*b^2*(35*A - 7*B + 22*C) + b^3*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(4*a^2*b*(14*B - 3*C) - 48*a^3*C - 2*a*b^2*(35*A - 7*B + 22*C) + b^3*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(4*a^2*b*(14*B - 3*C) - 48*a^3*C - 2*a*b^2*(35*A - 7*B + 22*C) + b^3*(35*A + 63*B + 25*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(4*a^2*b*(14*B - 3*C) - 48*a^3*C - 2*a*b^2*(35*A - 7*B + 22*C) + b^3*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(4*a^2*b*(14*B - 3*C) - 48*a^3*C - 2*a*b^2*(35*A - 7*B + 22*C) + b^3*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(2*(a + b)*(-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(4*a^2*b*(14*B - 3*C) - 48*a^3*C - 2*a*b^2*(35*A - 7*B + 22*C) + b^3*(35*A + 63*B + 25*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-56*a^2*b*B - 63*b^3*B + 48*a^3*C + 2*a*b^2*(35*A + 22*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
959,1,3332,342,24.755634,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \cot (c+d x) \left(8 a^2 C-10 a b B+15 A b^2+9 b^2 C\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d}-\frac{2 \sqrt{a+b} \cot (c+d x) \left(8 a^2 C-2 a b (5 B+C)+15 A b^2-b^2 (5 B-9 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^3 d}+\frac{2 (5 b B-4 a C) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 C \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}",1,"(Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*b^3) + (4*Sec[c + d*x]*(5*b*B*Sin[c + d*x] - 4*a*C*Sin[c + d*x]))/(15*b^2) + (4*C*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]]) + (4*((-2*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a*B)/(3*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (6*C)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^2*C)/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*A*Sqrt[Sec[c + d*x]])/(b*Sqrt[b + a*Cos[c + d*x]]) + (2*B*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (4*a^2*B*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^3*C*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (14*a*C*Sqrt[Sec[c + d*x]])/(15*b*Sqrt[b + a*Cos[c + d*x]]) - (2*a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*Sqrt[b + a*Cos[c + d*x]]) + (4*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (6*a*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-2*(a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(15*A*b^2 + 8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-2*(a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(15*A*b^2 + 8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-2*(a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(15*A*b^2 + 8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*((15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4) - ((a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(15*A*b^2 + 8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - ((a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(15*A*b^2 + 8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + a*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(15*A*b^2 + 8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(-2*(a + b)*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(15*A*b^2 + 8*a^2*C + 2*a*b*(-5*B + C) + b^2*(5*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
960,1,470,267,17.2880815,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\cos (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 (3 b B-2 a C) \sin (c+d x)}{3 b^2}+\frac{4 C \tan (c+d x)}{3 b}\right)}{d \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{4 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 b \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} (-2 a C+3 A b+b (3 B+C)) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-\left((3 b B-2 a C) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)+2 (a+b) (2 a C-3 b B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sqrt{a+b} \cot (c+d x) (2 a C+3 A b-b (3 B-C)) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d}-\frac{2 (a-b) \sqrt{a+b} (3 b B-2 a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}",1,"(4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(a + b)*(-3*b*B + 2*a*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(3*A*b - 2*a*C + b*(3*B + C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (3*b*B - 2*a*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(3*b*B - 2*a*C)*Sin[c + d*x])/(3*b^2) + (4*C*Tan[c + d*x])/(3*b)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]])","A",0
961,1,758,317,16.9529024,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{4 C \sin (c+d x) \cos (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{b d \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{4 \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{a \cos (c+d x)+b} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(b (A-B-C) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 A b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 A b \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+C (a+b) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{a+b}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+a C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a C \tan ^3\left(\frac{1}{2} (c+d x)\right)+a C \tan \left(\frac{1}{2} (c+d x)\right)-b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+b C \tan \left(\frac{1}{2} (c+d x)\right)\right)}{b d \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}-\frac{2 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d}+\frac{2 \sqrt{a+b} (B-C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b d}",1,"(4*C*Cos[c + d*x]*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c + d*x])/(b*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[a + b*Sec[c + d*x]]) - (4*Sqrt[b + a*Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*C*Tan[(c + d*x)/2] + b*C*Tan[(c + d*x)/2] - 2*a*C*Tan[(c + d*x)/2]^3 + a*C*Tan[(c + d*x)/2]^5 - b*C*Tan[(c + d*x)/2]^5 - 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*C*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(A - B - C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(b*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
962,1,853,358,16.6297082,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{b+a \cos (c+d x)} (B+A \cos (c+d x)+C \sec (c+d x)) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a A \tan ^5\left(\frac{1}{2} (c+d x)\right)-A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a A \tan \left(\frac{1}{2} (c+d x)\right)+A b \tan \left(\frac{1}{2} (c+d x)\right)+A (a+b) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a (B-C) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 a B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} (A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{\sqrt{a+b} (2 a C+A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}+\frac{A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{a d}+\frac{A (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d}",1,"(2*Sqrt[b + a*Cos[c + d*x]]*(B + A*Cos[c + d*x] + C*Sec[c + d*x])*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*A*Tan[(c + d*x)/2] + A*b*Tan[(c + d*x)/2] - 2*a*A*Tan[(c + d*x)/2]^3 + a*A*Tan[(c + d*x)/2]^5 - A*b*Tan[(c + d*x)/2]^5 - 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + A*(a + b)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(B - C)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
963,1,1905,439,15.6796148,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{A (b+a \cos (c+d x)) \sec (c+d x) \sin (2 (c+d x))}{4 a d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+4 a b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-3 A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-3 a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) (4 a B-3 A b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i \left(2 (A+2 C) a^2-b (A+4 B) a+3 A b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-16 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a^2 \sqrt{\frac{b-a}{a+b}} d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{(3 A b-4 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}-\frac{\sqrt{a+b} (3 A b-2 a (A+2 B)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{(a-b) \sqrt{a+b} (3 A b-4 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 b d}-\frac{\sqrt{a+b} \cot (c+d x) \left(4 a^2 (A+2 C)-4 a b B+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"(A*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[2*(c + d*x)])/(4*a*d*Sqrt[a + b*Sec[c + d*x]]) + (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-3*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 3*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 6*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 8*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - 3*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 3*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(-3*A*b + 4*a*B)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(3*A*b^2 - a*b*(A + 4*B) + 2*a^2*(A + 2*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a^2*Sqrt[(-a + b)/(a + b)]*d*Sqrt[a + b*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
964,1,874,510,20.8741841,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \left(-48 C a^4+40 b B a^3-30 A b^2 a^2+24 b^2 C a^2-25 b^3 B a+15 A b^4+9 b^4 C\right) \sin (c+d x)}{15 b^4 \left(b^2-a^2\right)}+\frac{4 \sec (c+d x) (5 b B \sin (c+d x)-9 a C \sin (c+d x))}{15 b^3}+\frac{4 \left(C \sin (c+d x) a^4-b B \sin (c+d x) a^3+A b^2 \sin (c+d x) a^2\right)}{b^3 \left(b^2-a^2\right) (b+a \cos (c+d x))}+\frac{4 C \sec (c+d x) \tan (c+d x)}{5 b^2}\right) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{3/2}}+\frac{4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left((a+b) \left(48 C a^4-40 b B a^3+6 b^2 (5 A-4 C) a^2+25 b^3 B a-3 b^4 (5 A+3 C)\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+b (a+b) \left(-48 C a^3+4 b (10 B+9 C) a^2-6 b^2 (5 A+5 B+2 C) a+b^3 (15 A+5 B+9 C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(48 C a^4-40 b B a^3+6 b^2 (5 A-4 C) a^2+25 b^3 B a-3 b^4 (5 A+3 C)\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)\right) (b+a \cos (c+d x))^{3/2}}{15 b^4 \left(b^2-a^2\right) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 \tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \tan (c+d x) \sec (c+d x) \left(6 a^2 C-5 a b B+5 A b^2-b^2 C\right) \sqrt{a+b \sec (c+d x)}}{5 b^2 d \left(a^2-b^2\right)}+\frac{2 \tan (c+d x) \left(-24 a^3 C+20 a^2 b B-3 a b^2 (5 A-3 C)-5 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{15 b^3 d \left(a^2-b^2\right)}+\frac{2 \cot (c+d x) \left(-48 a^3 C+a^2 b (40 B-36 C)-6 a b^2 (5 A-5 B+2 C)-b^3 (15 A-5 B+9 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^4 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \left(-48 a^4 C+40 a^3 b B-6 a^2 b^2 (5 A-4 C)-25 a b^3 B+3 b^4 (5 A+3 C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 b^5 d \sqrt{a+b}}",1,"(4*(b + a*Cos[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-40*a^3*b*B + 25*a*b^3*B + 6*a^2*b^2*(5*A - 4*C) + 48*a^4*C - 3*b^4*(5*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-48*a^3*C - 6*a*b^2*(5*A + 5*B + 2*C) + b^3*(15*A + 5*B + 9*C) + 4*a^2*b*(10*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-40*a^3*b*B + 25*a*b^3*B + 6*a^2*b^2*(5*A - 4*C) + 48*a^4*C - 3*b^4*(5*A + 3*C))*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(15*b^4*(-a^2 + b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + ((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(-30*a^2*A*b^2 + 15*A*b^4 + 40*a^3*b*B - 25*a*b^3*B - 48*a^4*C + 24*a^2*b^2*C + 9*b^4*C)*Sin[c + d*x])/(15*b^4*(-a^2 + b^2)) + (4*Sec[c + d*x]*(5*b*B*Sin[c + d*x] - 9*a*C*Sin[c + d*x]))/(15*b^3) + (4*(a^2*A*b^2*Sin[c + d*x] - a^3*b*B*Sin[c + d*x] + a^4*C*Sin[c + d*x]))/(b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (4*C*Sec[c + d*x]*Tan[c + d*x])/(5*b^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2))","A",0
965,1,3856,352,26.3697152,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 a \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \cot (c+d x) \left(-8 a^3 C+6 a^2 b B-a b^2 (3 A-5 C)-3 b^3 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b}}+\frac{2 \cot (c+d x) \left(3 A b^2-(2 a+b) (b (3 B-C)-4 a C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b}}+\frac{2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(3*a*A*b^2 - 6*a^2*b*B + 3*b^3*B + 8*a^3*C - 5*a*b^2*C)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)) - (4*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (4*C*Tan[c + d*x])/(3*b^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) - (4*(b + a*Cos[c + d*x])*((-2*a*A)/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^2*B)/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b*B)/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (10*a*C)/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^3*C)/(3*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*A*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (4*a*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*B*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (16*a^4*C*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (14*a^2*C*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*b*C*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (16*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (10*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(a + b)*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*A*b^2 + (2*a - b)*(4*a*C - b*(3*B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*A*b^2 + (2*a - b)*(4*a*C - b*(3*B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*A*b^2 + (2*a - b)*(4*a*C - b*(3*B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(3*A*b^2 + (2*a - b)*(4*a*C - b*(3*B + C)))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(3*A*b^2 + (2*a - b)*(4*a*C - b*(3*B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(3*A*b^2 + (2*a - b)*(4*a*C - b*(3*B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*(2*(a + b)*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*A*b^2 + (2*a - b)*(4*a*C - b*(3*B + C)))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 5*C) + 8*a^3*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
966,1,603,293,20.2308245,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a \cos (c+d x)+b)^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)+A b^2 \sin (c+d x)\right)}{b \left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{4 \sin (c+d x) \left(2 a^2 C-a b B+A b^2-b^2 C\right)}{b^2 \left(b^2-a^2\right)}\right)}{d (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{4 \sqrt{2} \sqrt{\frac{\cos (c+d x)}{(\cos (c+d x)+1)^2}} \sqrt{\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^4\left(\frac{1}{2} (c+d x)\right) \left(2 a^2 C-a b B+A b^2-b^2 C\right) (a \cos (c+d x)+b)+(a+b) \sec (c+d x) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} \left(\left(2 a^2 C-a b B+A b^2-b^2 C\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+b (-2 a C-A b+b (B+C)) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)\right)}{b^2 d \left(b^2-a^2\right) \sqrt{\frac{1}{\cos (c+d x)+1}} \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{2 \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \cot (c+d x) \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^3 d \sqrt{a+b}}+\frac{2 \cot (c+d x) (-2 a C+A b+b (B-C)) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) + (4*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) + (4*Sqrt[2]*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])^2]*(b + a*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((a + b)*((A*b^2 - a*b*B + 2*a^2*C - b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + b*(-(A*b) - 2*a*C + b*(B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sec[c + d*x] + (A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4*Tan[(c + d*x)/2]))/(b^2*(-a^2 + b^2)*d*Sqrt[(1 + Cos[c + d*x])^(-1)]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2))","B",0
967,1,7666,395,25.4927373,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","\frac{2 \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}+\frac{2 \cot (c+d x) \left(A b^2-a (b B-a C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b^2 d \sqrt{a+b}}-\frac{2 \cot (c+d x) (A b-a (B+C)) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}",1,"Result too large to show","B",0
968,1,1798,451,20.2770344,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \left(C a^2-b B a+A b^2\right) \sin (c+d x)}{a^2 \left(a^2-b^2\right)}-\frac{4 \left(A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right)}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{3/2}}-\frac{2 (b+a \cos (c+d x))^{3/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-4 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 a^3 C \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+2 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+2 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)-2 a^3 C \tan \left(\frac{1}{2} (c+d x)\right)-2 a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left((A-2 C) a^2+2 b B a-3 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a (a+b) (a (B-C)-A b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a^2 \left(a^2-b^2\right) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\sqrt{a+b} (3 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}-\frac{b \tan (c+d x) \left(-\left(a^2 (A-2 C)\right)-2 a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\cot (c+d x) \left(2 a^2 C+a b (A-2 B)+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}-\frac{\cot (c+d x) \left(-\left(a^2 (A-2 C)\right)-2 a b B+3 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 b d \sqrt{a+b}}+\frac{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) - (4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a^3*A*Tan[(c + d*x)/2] + a^2*A*b*Tan[(c + d*x)/2] - 3*a*A*b^2*Tan[(c + d*x)/2] - 3*A*b^3*Tan[(c + d*x)/2] + 2*a^2*b*B*Tan[(c + d*x)/2] + 2*a*b^2*B*Tan[(c + d*x)/2] - 2*a^3*C*Tan[(c + d*x)/2] - 2*a^2*b*C*Tan[(c + d*x)/2] - 2*a^3*A*Tan[(c + d*x)/2]^3 + 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 4*a^2*b*B*Tan[(c + d*x)/2]^3 + 4*a^3*C*Tan[(c + d*x)/2]^3 + a^3*A*Tan[(c + d*x)/2]^5 - a^2*A*b*Tan[(c + d*x)/2]^5 - 3*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*A*b^3*Tan[(c + d*x)/2]^5 + 2*a^2*b*B*Tan[(c + d*x)/2]^5 - 2*a*b^2*B*Tan[(c + d*x)/2]^5 - 2*a^3*C*Tan[(c + d*x)/2]^5 + 2*a^2*b*C*Tan[(c + d*x)/2]^5 - 6*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-3*A*b^2 + 2*a*b*B + a^2*(A - 2*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(a + b)*(-(A*b) + a*(B - C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a^2*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
969,1,3404,552,25.1152756,"\int \frac{\cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(5 A b-4 a B) \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b} \cot (c+d x) \left(4 a^2 (A+2 C)-12 a b B+15 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^4 d}-\frac{\cot (c+d x) \left(-2 a^2 (A+2 B-4 C)+a b (5 A-12 B)+15 A b^2\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d \sqrt{a+b}}+\frac{b \tan (c+d x) \left(4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\cot (c+d x) \left(4 a^3 B-a^2 (7 A b-8 b C)-12 a b^2 B+15 A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 b d \sqrt{a+b}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"(((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((4*b*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a^3*(-a^2 + b^2)) + (4*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(2*a^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 4*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 4*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 12*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 12*a*b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 8*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + 8*a^2*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + 14*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 30*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 8*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 24*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - 16*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 - 7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 15*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 4*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 12*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 12*a*b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 8*a^3*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - 8*a^2*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - (8*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (24*I)*a^3*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b^3*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^4*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*b^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (24*I)*a^3*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b^3*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (16*I)*a^4*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (16*I)*a^2*b^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(15*A*b^3 + 4*a^3*B - 12*a*b^2*B + a^2*(-7*A*b + 8*b*C))*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(15*A*b^3 + 2*a*b^2*(5*A - 6*B) + 2*a^3*(A + 2*C) + a^2*b*(A - 8*B + 8*C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(2*a^3*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2))))/2","C",0
970,1,989,549,21.6658995,"\int \frac{\sec ^3(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \left(16 C a^5-8 b B a^4+2 A b^2 a^3-28 b^2 C a^3+15 b^3 B a^2-6 A b^4 a+8 b^4 C a-3 b^5 B\right) \sin (c+d x)}{3 b^4 \left(a^2-b^2\right)^2}-\frac{4 \left(C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a\right)}{3 b^2 \left(b^2-a^2\right) (b+a \cos (c+d x))^2}-\frac{4 \left(-7 C \sin (c+d x) a^5+4 b B \sin (c+d x) a^4-A b^2 \sin (c+d x) a^3+11 b^2 C \sin (c+d x) a^3-8 b^3 B \sin (c+d x) a^2+5 A b^4 \sin (c+d x) a\right)}{3 b^3 \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}+\frac{4 C \tan (c+d x)}{3 b^3}\right) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}}+\frac{4 \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left((a+b) \left(16 C a^5-8 b B a^4+2 b^2 (A-14 C) a^3+15 b^3 B a^2+2 b^4 (4 C-3 A) a-3 b^5 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+b (a+b) \left(-16 C a^4+4 b (2 B+3 C) a^3-2 b^2 (A+3 B-8 C) a^2+3 b^3 (A-3 (B+C)) a+b^4 (3 A+3 B+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)+\left(16 C a^5-8 b B a^4+2 b^2 (A-14 C) a^3+15 b^3 B a^2+2 b^4 (4 C-3 A) a-3 b^5 B\right) \tan \left(\frac{1}{2} (c+d x)\right) \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)\right) (b+a \cos (c+d x))^{5/2}}{3 b^4 \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 \tan (c+d x) \sec ^2(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \tan (c+d x) \left(2 a^2 C-a b B+A b^2-b^2 C\right) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 a \tan (c+d x) \left(a \left(-6 a^3 C+3 a^2 b B+10 a b^2 C-7 b^3 B\right)+4 A b^4\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \cot (c+d x) \left(-16 a^4 C+a^3 b (8 B-12 C)-2 a^2 b^2 (A-3 B-8 C)-3 a b^3 (A+3 B-3 C)+b^4 (3 A-3 B+C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \cot (c+d x) \left(-16 a^5 C+8 a^4 b B-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+2 a b^4 (3 A-4 C)+3 b^5 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^5 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"(4*(b + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*((a + b)*(-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 2*a^3*b^2*(A - 14*C) + 16*a^5*C + 2*a*b^4*(-3*A + 4*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(-2*a^2*b^2*(A + 3*B - 8*C) - 16*a^4*C + b^4*(3*A + 3*B + C) + 4*a^3*b*(2*B + 3*C) + 3*a*b^3*(A - 3*(B + C)))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (-8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 2*a^3*b^2*(A - 14*C) + 16*a^5*C + 2*a*b^4*(-3*A + 4*C))*Tan[(c + d*x)/2]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2)))/(3*b^4*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(2*a^3*A*b^2 - 6*a*A*b^4 - 8*a^4*b*B + 15*a^2*b^3*B - 3*b^5*B + 16*a^5*C - 28*a^3*b^2*C + 8*a*b^4*C)*Sin[c + d*x])/(3*b^4*(a^2 - b^2)^2) - (4*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (4*(-(a^3*A*b^2*Sin[c + d*x]) + 5*a*A*b^4*Sin[c + d*x] + 4*a^4*b*B*Sin[c + d*x] - 8*a^2*b^3*B*Sin[c + d*x] - 7*a^5*C*Sin[c + d*x] + 11*a^3*b^2*C*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (4*C*Tan[c + d*x])/(3*b^3)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2))","A",0
971,1,4504,449,27.6628843,"\int \frac{\sec ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 a \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \cot (c+d x) \left(-8 a^3 C+2 a^2 b (B-3 C)+a b^2 (A+3 B+9 C)-3 b^3 (A+B-C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \tan (c+d x) \left(-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \cot (c+d x) \left(-8 a^4 C+2 a^3 b B+a^2 b^2 (A+15 C)-6 a b^3 B+3 b^4 (A-C)\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d \sqrt{a+b} \left(a^2-b^2\right)}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(a^2*A*b^2 + 3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 8*a^4*C + 15*a^2*b^2*C - 3*b^4*C)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)^2) + (4*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (4*(2*a^2*A*b^2*Sin[c + d*x] + 2*A*b^4*Sin[c + d*x] + a^3*b*B*Sin[c + d*x] - 5*a*b^3*B*Sin[c + d*x] - 4*a^4*C*Sin[c + d*x] + 8*a^2*b^2*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (4*(b + a*Cos[c + d*x])^2*((2*a^2*A)/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*A*b^2)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^3*B)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*b*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (10*a^2*C)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a^4*C)/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^2*C)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^3*A*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a*A*b*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (10*a^2*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*B*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*C*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (34*a^3*C*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (6*a*b*C*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (16*a^5*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (10*a^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*b^3*(A - B - C) - 8*a^3*C + 2*a^2*b*(B + 3*C) + a*b^2*(A - 3*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*((-2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*b^3*(A - B - C) - 8*a^3*C + 2*a^2*b*(B + 3*C) + a*b^2*(A - 3*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*b^3*(A - B - C) - 8*a^3*C + 2*a^2*b*(B + 3*C) + a*b^2*(A - 3*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(3*b^3*(A - B - C) - 8*a^3*C + 2*a^2*b*(B + 3*C) + a*b^2*(A - 3*B + 9*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(3*b^3*(A - B - C) - 8*a^3*C + 2*a^2*b*(B + 3*C) + a*b^2*(A - 3*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(3*b^3*(A - B - C) - 8*a^3*C + 2*a^2*b*(B + 3*C) + a*b^2*(A - 3*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*(2*(a + b)*(-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*b^3*(A - B - C) - 8*a^3*C + 2*a^2*b*(B + 3*C) + a*b^2*(A - 3*B + 9*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*b*B + 6*a*b^3*B + 8*a^4*C + 3*b^4*(-A + C) - a^2*b^2*(A + 15*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
972,1,3980,416,26.3381806,"\int \frac{\sec (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \cot (c+d x) \left(2 a^2 C+a b (3 A+B+3 C)-b^2 (A+3 (B+C))\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \tan (c+d x) \left(2 a^3 C+a^2 b B-2 a b^2 (2 A+3 C)+3 b^3 B\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \cot (c+d x) \left(-2 a^3 C-a^2 b B+4 a A b^2+6 a b^2 C-3 b^3 B\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(-4*a*A*b^2 + a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2) + (4*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(-5*a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] + 2*a^3*b*B*Sin[c + d*x] + 2*a*b^3*B*Sin[c + d*x] + a^4*C*Sin[c + d*x] - 5*a^2*b^2*C*Sin[c + d*x]))/(3*a*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) + (4*(b + a*Cos[c + d*x])^2*((-8*a*A*b)/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*B)/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^3*C)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*b*C)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*A*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^2*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*B*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a*b*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (10*a^2*C*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*C*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*a^2*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*a^4*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-2*a^2*C + b^2*(A - 3*B + 3*C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*((2*a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-2*a^2*C + b^2*(A - 3*B + 3*C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-2*a^2*C + b^2*(A - 3*B + 3*C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (4*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-2*a^2*C + b^2*(A - 3*B + 3*C) + a*b*(3*A - B + 3*C))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-2*a^2*C + b^2*(A - 3*B + 3*C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-2*a^2*C + b^2*(A - 3*B + 3*C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*(2*(a + b)*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-2*a^2*C + b^2*(A - 3*B + 3*C) + a*b*(3*A - B + 3*C))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
973,1,1907,541,21.7705394,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(b+a \cos (c+d x))^3 \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \left(-C a^4+4 b B a^3-7 A b^2 a^2-3 b^2 C a^2+3 A b^4\right) \sin (c+d x)}{3 a^2 b \left(b^2-a^2\right)^2}-\frac{4 \left(A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right)}{3 a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{4 \left(2 C \sin (c+d x) a^4-5 b B \sin (c+d x) a^3+8 A b^2 \sin (c+d x) a^2+2 b^2 C \sin (c+d x) a^2+b^3 B \sin (c+d x) a-4 A b^4 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}}-\frac{4 (b+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^5-2 C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^5+C \tan \left(\frac{1}{2} (c+d x)\right) a^5-4 b B \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4-b C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^4+8 b B \tan ^3\left(\frac{1}{2} (c+d x)\right) a^4-4 b B \tan \left(\frac{1}{2} (c+d x)\right) a^4+b C \tan \left(\frac{1}{2} (c+d x)\right) a^4+6 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4+6 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^4+7 A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3+4 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3+3 b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^3-14 A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right) a^3-6 b^2 C \tan ^3\left(\frac{1}{2} (c+d x)\right) a^3+7 A b^2 \tan \left(\frac{1}{2} (c+d x)\right) a^3-4 b^2 B \tan \left(\frac{1}{2} (c+d x)\right) a^3+3 b^2 C \tan \left(\frac{1}{2} (c+d x)\right) a^3-7 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2-3 b^3 C \tan ^5\left(\frac{1}{2} (c+d x)\right) a^2+7 A b^3 \tan \left(\frac{1}{2} (c+d x)\right) a^2+3 b^3 C \tan \left(\frac{1}{2} (c+d x)\right) a^2-12 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2-12 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a^2-3 A b^4 \tan ^5\left(\frac{1}{2} (c+d x)\right) a+6 A b^4 \tan ^3\left(\frac{1}{2} (c+d x)\right) a-3 A b^4 \tan \left(\frac{1}{2} (c+d x)\right) a-b (a+b) \left((3 A-3 B+C) a^2+b (3 A-B+3 C) a-2 A b^2\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} a+3 A b^5 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 A b^5 \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(C a^4-4 b B a^3+b^2 (7 A+3 C) a^2-3 A b^4\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \tan ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^5 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{3 a^2 b \left(a^2-b^2\right)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{2 \tan (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \cot (c+d x) \left(a^3 (3 B+C)-a^2 b (6 A+B+3 C)+a A b^2+3 A b^3\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \tan (c+d x) \left(a^4 (-C)+4 a^3 b B-a^2 b^2 (7 A+3 C)+3 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \cot (c+d x) \left(a^4 C-4 a^3 b B+7 a^2 A b^2+3 a^2 b^2 C-3 A b^4\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b^2 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(-7*a^2*A*b^2 + 3*A*b^4 + 4*a^3*b*B - a^4*C - 3*a^2*b^2*C)*Sin[c + d*x])/(3*a^2*b*(-a^2 + b^2)^2) - (4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(8*a^2*A*b^2*Sin[c + d*x] - 4*A*b^4*Sin[c + d*x] - 5*a^3*b*B*Sin[c + d*x] + a*b^3*B*Sin[c + d*x] + 2*a^4*C*Sin[c + d*x] + 2*a^2*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (4*(b + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(7*a^3*A*b^2*Tan[(c + d*x)/2] + 7*a^2*A*b^3*Tan[(c + d*x)/2] - 3*a*A*b^4*Tan[(c + d*x)/2] - 3*A*b^5*Tan[(c + d*x)/2] - 4*a^4*b*B*Tan[(c + d*x)/2] - 4*a^3*b^2*B*Tan[(c + d*x)/2] + a^5*C*Tan[(c + d*x)/2] + a^4*b*C*Tan[(c + d*x)/2] + 3*a^3*b^2*C*Tan[(c + d*x)/2] + 3*a^2*b^3*C*Tan[(c + d*x)/2] - 14*a^3*A*b^2*Tan[(c + d*x)/2]^3 + 6*a*A*b^4*Tan[(c + d*x)/2]^3 + 8*a^4*b*B*Tan[(c + d*x)/2]^3 - 2*a^5*C*Tan[(c + d*x)/2]^3 - 6*a^3*b^2*C*Tan[(c + d*x)/2]^3 + 7*a^3*A*b^2*Tan[(c + d*x)/2]^5 - 7*a^2*A*b^3*Tan[(c + d*x)/2]^5 - 3*a*A*b^4*Tan[(c + d*x)/2]^5 + 3*A*b^5*Tan[(c + d*x)/2]^5 - 4*a^4*b*B*Tan[(c + d*x)/2]^5 + 4*a^3*b^2*B*Tan[(c + d*x)/2]^5 + a^5*C*Tan[(c + d*x)/2]^5 - a^4*b*C*Tan[(c + d*x)/2]^5 + 3*a^3*b^2*C*Tan[(c + d*x)/2]^5 - 3*a^2*b^3*C*Tan[(c + d*x)/2]^5 + 6*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(-3*A*b^4 - 4*a^3*b*B + a^4*C + a^2*b^2*(7*A + 3*C))*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - a*b*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a^2*b*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
974,1,20027,618,29.8416674,"\int \frac{\cos (c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} (5 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}-\frac{b \tan (c+d x) \left(-\left(a^2 (3 A-2 C)\right)-2 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{b \tan (c+d x) \left(-\left(a^4 (3 A-8 C)\right)-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{\cot (c+d x) \left(-6 a^4 C-a^3 b (3 A-2 (6 B+C))-a^2 b^2 (21 A+2 B)+a b^3 (5 A-6 B)+15 A b^4\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 b d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{\cot (c+d x) \left(-\left(a^4 (3 A-8 C)\right)-14 a^3 b B+26 a^2 A b^2+6 a b^3 B-15 A b^4\right) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^3 b d (a-b) (a+b)^{3/2}}+\frac{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"Result too large to show","B",0
975,1,4764,448,24.5109286,"\int (a+b \sec (c+d x))^{3/2} \left(a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \left(-12 a^2 C+35 a b B+9 b^2 C\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}-\frac{2 a^2 \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 \sqrt{a+b} \left(30 a^3 C-3 a^2 b (15 B+4 C)+a b^2 (35 B-12 C)-b^3 (5 B-9 C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{15 d}+\frac{2 b^2 (3 a C+5 b B) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 b^2 C \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(b*B - a*C + b*C*Sec[c + d*x])*((2*b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*Sin[c + d*x])/15 + (2*Sec[c + d*x]*(5*b^3*B*Sin[c + d*x] + 6*a*b^2*C*Sin[c + d*x]))/15 + (2*b^3*C*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])^2*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])) + (2*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*((a^3*b*B)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*a*b^3*B)/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^4*C)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a^2*b^2*C)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*b^4*C)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*b^2*B*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (b^4*B*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (6*a^3*b*C*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (a*b^3*C*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (7*a^2*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*b*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (3*a*b^3*C*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(b*B - a*C + b*C*Sec[c + d*x])*(-(b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 15*a^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(15*d*(b + a*Cos[c + d*x])^3*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]^(7/2)*((a*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(-(b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 15*a^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(15*(b + a*Cos[c + d*x])^(3/2)*(Sec[(c + d*x)/2]^2)^(3/2)) - (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(-(b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 15*a^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(5*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 15*a^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(15*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)) + (Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*(-(b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 15*a^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (2*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(-1/2*(b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]) - b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - 15*a^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - (3*b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*Sqrt[Cos[c + d*x]*Sec[(c + d*x)/2]^2]*Sec[c + d*x]*Tan[(c + d*x)/2]*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/2 - (b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) - (15*a^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/(2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + (b*(a + b)*(-15*a^2*C + 3*a*b*(10*B + C) + b^2*(5*B + 9*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (b*(a + b)*(35*a*b*B - 12*a^2*C + 9*b^2*C)*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]) - 15*a^2*(b*B - a*C)*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*(((a - b)*Sec[(c + d*x)/2]^2)/(2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) - (a*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])) + a*b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]*Tan[c + d*x] - b*(35*a*b*B - 12*a^2*C + 9*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]*Tan[c + d*x]))/(15*Sqrt[b + a*Cos[c + d*x]]*(Sec[(c + d*x)/2]^2)^(3/2))))","B",0
976,1,1131,382,19.206432,"\int \sqrt{a+b \sec (c+d x)} \left(a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^{3/2} (b B-a C+b C \sec (c+d x)) \left(\frac{2}{3} C \tan (c+d x) b^2+\frac{2}{3} (3 b B+a C) \sin (c+d x) b\right) \cos ^2(c+d x)}{d (b+a \cos (c+d x)) (b C-a \cos (c+d x) C+b B \cos (c+d x))}+\frac{2 (a+b \sec (c+d x))^{3/2} (b B-a C+b C \sec (c+d x)) \left(-3 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-a b^2 C \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^2 b C \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^2 b C \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)+3 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+a b^2 C \tan \left(\frac{1}{2} (c+d x)\right)+a^2 b C \tan \left(\frac{1}{2} (c+d x)\right)+b (a+b) (3 b B+a C) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-\left(3 C a^3-3 b (B+C) a^2+b^2 (6 B+C) a+b^3 (3 B+C)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a^3 C \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{3 d (b+a \cos (c+d x))^{3/2} (b C-a \cos (c+d x) C+b B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{2 \sqrt{a+b} \left(3 a^2 C-a b (6 B-C)+b^2 (3 B-C)\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}+\frac{2 b^2 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}-\frac{2 (a-b) \sqrt{a+b} (a C+3 b B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 d}-\frac{2 a \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(2*(a + b*Sec[c + d*x])^(3/2)*(b*B - a*C + b*C*Sec[c + d*x])*(3*a*b^2*B*Tan[(c + d*x)/2] + 3*b^3*B*Tan[(c + d*x)/2] + a^2*b*C*Tan[(c + d*x)/2] + a*b^2*C*Tan[(c + d*x)/2] - 6*a*b^2*B*Tan[(c + d*x)/2]^3 - 2*a^2*b*C*Tan[(c + d*x)/2]^3 + 3*a*b^2*B*Tan[(c + d*x)/2]^5 - 3*b^3*B*Tan[(c + d*x)/2]^5 + a^2*b*C*Tan[(c + d*x)/2]^5 - a*b^2*C*Tan[(c + d*x)/2]^5 - 6*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a^3*C*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + b*(a + b)*(3*b*B + a*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (3*a^3*C - 3*a^2*b*(B + C) + b^3*(3*B + C) + a*b^2*(6*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*d*(b + a*Cos[c + d*x])^(3/2)*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(b*B - a*C + b*C*Sec[c + d*x])*((2*b*(3*b*B + a*C)*Sin[c + d*x])/3 + (2*b^2*C*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x]))","B",0
977,1,1232,316,18.5935719,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 b C \cos (c+d x) \sqrt{a+b \sec (c+d x)} (b B-a C+b C \sec (c+d x)) \sin (c+d x)}{d (b C-a \cos (c+d x) C+b B \cos (c+d x))}+\frac{2 \sqrt{a+b \sec (c+d x)} (b B-a C+b C \sec (c+d x)) \left(-b^2 \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} C \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b \sqrt{\frac{b-a}{a+b}} C \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-2 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+b^2 \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} C \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) b C E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i (a-b) (a C+b (C-B)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i a^2 C \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{\sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} (b C-a \cos (c+d x) C+b B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 b \sqrt{a+b} (B-C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 C (a-b) \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}",1,"(2*b*C*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(b*B - a*C + b*C*Sec[c + d*x])*Sin[c + d*x])/(d*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])) + (2*Sqrt[a + b*Sec[c + d*x]]*(b*B - a*C + b*C*Sec[c + d*x])*(a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] - 2*a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 + a*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + (2*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*C*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(a*C + b*(-B + C))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",1
978,1,158,212,3.0274955,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec (c+d x) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \left((a C+b (C-B)) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 (b B-a C) \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 C \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}-\frac{2 \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((a*C + b*(-B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*(b*B - a*C)*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
979,1,2090,379,15.6690486,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^2 d}-\frac{2 (b B-2 a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}+\frac{2 (b B-2 a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]*(b*B - a*C + b*C*Sec[c + d*x])*((2*b*(b*B - 2*a*C)*Sin[c + d*x])/(a*(-a^2 + b^2)) - (2*(-(b^3*B*Sin[c + d*x]) + 2*a*b^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*(b*B - a*C + b*C*Sec[c + d*x])*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-(a*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]) - b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 2*a^2*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + 2*a*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2] + 2*a*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - 4*a^2*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^3 - a*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 2*a^2*b*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 - 2*a*b^2*Sqrt[(-a + b)/(a + b)]*C*Tan[(c + d*x)/2]^5 + (2*I)*a^2*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*b^3*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^3*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*a*b^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*a^2*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*b^3*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^3*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*a*b^2*C*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*b*(-(b*B) + 2*a*C)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(-2*b^2*B - a*b*(B - 3*C) + a^2*C)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
980,1,812,519,15.7247859,"\int \frac{a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx","Integrate[(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]","\frac{\sec ^2(c+d x) (b B-a C+b C \sec (c+d x)) \left(\frac{2 b \left(11 C a^3-7 b B a^2-3 b^2 C a+3 b^3 B\right) \sin (c+d x)}{3 a^2 \left(b^2-a^2\right)^2}-\frac{2 \left(b^4 B \sin (c+d x)-2 a b^3 C \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}-\frac{2 \left(4 B \sin (c+d x) b^5-5 a C \sin (c+d x) b^4-8 a^2 B \sin (c+d x) b^3+13 a^3 C \sin (c+d x) b^2\right)}{3 a^2 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^3}{d (b C-a \cos (c+d x) C+b B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}}+\frac{2 (b B-a C+b C \sec (c+d x)) \left(-a b (a+b) \left(11 C a^3-7 b B a^2-3 b^2 C a+3 b^3 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}} \sec ^2\left(\frac{1}{2} (c+d x)\right)-b (a+b) \left(-12 C a^4+b (9 B+C) a^3-2 b^2 (B-3 C) a^2-3 b^3 (2 B+C) a+3 b^4 B\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}} \sec ^2\left(\frac{1}{2} (c+d x)\right)-3 (a-b)^2 (a+b)^2 (b B-a C) \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}} \sec ^2\left(\frac{1}{2} (c+d x)\right)-a b \left(11 C a^3-7 b B a^2-3 b^2 C a+3 b^3 B\right) (b+a \cos (c+d x)) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sec (c+d x) \tan \left(\frac{1}{2} (c+d x)\right)\right) (b+a \cos (c+d x))^2}{3 a^3 \left(a^2-b^2\right)^2 d (b C-a \cos (c+d x) C+b B \cos (c+d x)) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a+b \sec (c+d x))^{5/2}}","-\frac{2 \sqrt{a+b} (b B-a C) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{2 b^2 (b B-2 a C) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 b^2 \left(-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^3 C-2 a^2 b (3 B+C)+a b^2 (B-3 C)+3 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-11 a^3 C+7 a^2 b B+3 a b^2 C-3 b^3 B\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^2*(b*B - a*C + b*C*Sec[c + d*x])*((2*b*(-7*a^2*b*B + 3*b^3*B + 11*a^3*C - 3*a*b^2*C)*Sin[c + d*x])/(3*a^2*(-a^2 + b^2)^2) - (2*(b^4*B*Sin[c + d*x] - 2*a*b^3*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-8*a^2*b^3*B*Sin[c + d*x] + 4*b^5*B*Sin[c + d*x] + 13*a^3*b^2*C*Sin[c + d*x] - 5*a*b^4*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*(b*B - a*C + b*C*Sec[c + d*x])*(-(a*b*(a + b)*(-7*a^2*b*B + 3*b^3*B + 11*a^3*C - 3*a*b^2*C)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) - b*(a + b)*(3*b^4*B - 2*a^2*b^2*(B - 3*C) - 12*a^4*C - 3*a*b^3*(2*B + C) + a^3*b*(9*B + C))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - 3*(a - b)^2*(a + b)^2*(b*B - a*C)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*b*(-7*a^2*b*B + 3*b^3*B + 11*a^3*C - 3*a*b^2*C)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)^2*d*(b*C + b*B*Cos[c + d*x] - a*C*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*(a + b*Sec[c + d*x])^(5/2))","A",1
981,1,1262,266,7.4512412,"\int \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 a A F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 b B F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 a C F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} A b e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} a B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{14 \sqrt{2} b C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{45 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 b C \sec (c) \sin (d x) \sec ^4(c+d x)}{9 d}+\frac{4 \sec (c) (7 b C \sin (c)+9 b B \sin (d x)+9 a C \sin (d x)) \sec ^3(c+d x)}{63 d}+\frac{4 \sec (c) (45 b B \sin (c)+45 a C \sin (c)+63 A b \sin (d x)+63 a B \sin (d x)+49 b C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac{4 \sec (c) (63 A b \sin (c)+63 a B \sin (c)+49 b C \sin (c)+105 a A \sin (d x)+75 b B \sin (d x)+75 a C \sin (d x)) \sec (c+d x)}{315 d}+\frac{4 (9 A b+7 C b+9 a B) \cos (d x) \csc (c)}{15 d}+\frac{4 (7 a A+5 b B+5 a C) \tan (c)}{21 d}\right)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (9 a B+9 A b+7 b C)}{45 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (7 a A+5 a C+5 b B)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (9 a B+9 A b+7 b C)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+5 a C+5 b B)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a B+9 A b+7 b C)}{15 d}+\frac{2 (a C+b B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{9 d}",1,"(2*Sqrt[2]*A*b*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*Sqrt[2]*a*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (14*Sqrt[2]*b*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*a*A*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*b*B*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*a*C*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(9*A*b + 9*a*B + 7*b*C)*Cos[d*x]*Csc[c])/(15*d) + (4*b*C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(9*d) + (4*Sec[c]*Sec[c + d*x]^3*(7*b*C*Sin[c] + 9*b*B*Sin[d*x] + 9*a*C*Sin[d*x]))/(63*d) + (4*Sec[c]*Sec[c + d*x]*(63*A*b*Sin[c] + 63*a*B*Sin[c] + 49*b*C*Sin[c] + 105*a*A*Sin[d*x] + 75*b*B*Sin[d*x] + 75*a*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]^2*(45*b*B*Sin[c] + 45*a*C*Sin[c] + 63*A*b*Sin[d*x] + 63*a*B*Sin[d*x] + 49*b*C*Sin[d*x]))/(315*d) + (4*(7*a*A + 5*b*B + 5*a*C)*Tan[c])/(21*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))","C",0
982,1,1202,230,7.2049222,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 A b F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 a B F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 b C F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} a A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} b B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} a C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 b C \sec (c) \sin (d x) \sec ^3(c+d x)}{7 d}+\frac{4 \sec (c) (5 b C \sin (c)+7 b B \sin (d x)+7 a C \sin (d x)) \sec ^2(c+d x)}{35 d}+\frac{4 \sec (c) (21 b B \sin (c)+21 a C \sin (c)+35 A b \sin (d x)+35 a B \sin (d x)+25 b C \sin (d x)) \sec (c+d x)}{105 d}+\frac{4 (5 a A+3 b B+3 a C) \cos (d x) \csc (c)}{5 d}+\frac{4 (7 A b+5 C b+7 a B) \tan (c)}{21 d}\right)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (7 a B+7 A b+5 b C)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (5 a A+3 a C+3 b B)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(2*Sqrt[2]*a*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*Sqrt[2]*b*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*Sqrt[2]*a*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*A*b*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*a*B*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*b*C*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(5*a*A + 3*b*B + 3*a*C)*Cos[d*x]*Csc[c])/(5*d) + (4*b*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(7*d) + (4*Sec[c]*Sec[c + d*x]^2*(5*b*C*Sin[c] + 7*b*B*Sin[d*x] + 7*a*C*Sin[d*x]))/(35*d) + (4*Sec[c]*Sec[c + d*x]*(21*b*B*Sin[c] + 21*a*C*Sin[c] + 35*A*b*Sin[d*x] + 35*a*B*Sin[d*x] + 25*b*C*Sin[d*x]))/(105*d) + (4*(7*A*b + 7*a*B + 5*b*C)*Tan[c])/(21*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))","C",0
983,1,1140,192,7.0665074,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 a A F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 b B F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 a C F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} A b e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} a B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{2 \sqrt{2} b C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 b C \sec (c) \sin (d x) \sec ^2(c+d x)}{5 d}+\frac{4 \sec (c) (3 b C \sin (c)+5 b B \sin (d x)+5 a C \sin (d x)) \sec (c+d x)}{15 d}+\frac{4 (5 A b+3 C b+5 a B) \cos (d x) \csc (c)}{5 d}+\frac{4 (b B+a C) \tan (c)}{3 d}\right)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} (5 a B+5 A b+3 b C)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 (a C+b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(2*Sqrt[2]*A*b*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*Sqrt[2]*a*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*Sqrt[2]*b*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*a*A*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*b*B*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*a*C*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(5*A*b + 5*a*B + 3*b*C)*Cos[d*x]*Csc[c])/(5*d) + (4*b*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]*(3*b*C*Sin[c] + 5*b*B*Sin[d*x] + 5*a*C*Sin[d*x]))/(15*d) + (4*(b*B + a*C)*Tan[c])/(3*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))","C",0
984,1,223,152,2.7515201,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(i \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right) (a (C-A)+b B)+2 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)+3 i a A \cos (2 (c+d x))+3 i a A+3 a C \sin (2 (c+d x))-3 i a C \cos (2 (c+d x))-3 i a C+3 b B \sin (2 (c+d x))-3 i b B \cos (2 (c+d x))-3 i b B+2 b C \sin (c+d x)\right)}{3 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 (a C+b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 b C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((3*I)*a*A - (3*I)*b*B - (3*I)*a*C + (3*I)*a*A*Cos[2*(c + d*x)] - (3*I)*b*B*Cos[2*(c + d*x)] - (3*I)*a*C*Cos[2*(c + d*x)] + 2*(3*A*b + 3*a*B + b*C)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + I*(b*B + a*(-A + C))*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*b*C*Sin[c + d*x] + 3*b*B*Sin[2*(c + d*x)] + 3*a*C*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
985,1,197,146,2.8252469,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2 i e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right) (a B+A b-b C)+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)+a A \sin (2 (c+d x))+6 i a B \cos (c+d x)+6 i A b \cos (c+d x)+6 b C \sin (c+d x)-6 i b C \cos (c+d x)\right)}{3 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b C \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((6*I)*A*b*Cos[c + d*x] + (6*I)*a*B*Cos[c + d*x] - (6*I)*b*C*Cos[c + d*x] + 2*(3*b*B + a*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2*I)*(A*b + a*B - b*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 6*b*C*Sin[c + d*x] + a*A*Sin[2*(c + d*x)]))/(3*d*E^(I*d*x))","C",1
986,1,194,156,2.7295977,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-4 i e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right) (3 a A+5 a C+5 b B)+2 \cos (c+d x) (10 (a B+A b) \sin (c+d x)+6 i (3 a A+5 a C+5 b B)+3 a A \sin (2 (c+d x)))+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)\right)}{30 d}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(20*(A*b + a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4*I)*(3*a*A + 5*b*B + 5*a*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((6*I)*(3*a*A + 5*b*B + 5*a*C) + 10*(A*b + a*B)*Sin[c + d*x] + 3*a*A*Sin[2*(c + d*x)])))/(30*d*E^(I*d*x))","C",1
987,1,219,194,3.4469357,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right) (3 a B+3 A b+5 b C)+2 \cos (c+d x) (5 \sin (c+d x) (23 a A+28 a C+28 b B)+42 (a B+A b) \sin (2 (c+d x))+84 i (3 a B+3 A b+5 b C)+15 a A \sin (3 (c+d x)))+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)\right)}{420 d}","\frac{2 \sin (c+d x) (5 a A+7 a C+7 b B)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 (a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(40*(5*a*A + 7*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(3*A*b + 3*a*B + 5*b*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((84*I)*(3*A*b + 3*a*B + 5*b*C) + 5*(23*a*A + 28*b*B + 28*a*C)*Sin[c + d*x] + 42*(A*b + a*B)*Sin[2*(c + d*x)] + 15*a*A*Sin[3*(c + d*x)])))/(420*d*E^(I*d*x))","C",1
988,1,249,230,4.6415748,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right) (7 a A+9 a C+9 b B)+\cos (c+d x) (30 \sin (c+d x) (23 a B+23 A b+28 b C)+14 \sin (2 (c+d x)) (19 a A+18 a C+18 b B)+35 a A \sin (4 (c+d x))+1176 i a A+90 a B \sin (3 (c+d x))+1512 i a C+90 A b \sin (3 (c+d x))+1512 i b B)+120 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+7 b C)\right)}{1260 d}","\frac{2 \sin (c+d x) (7 a A+9 a C+9 b B)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (5 a B+5 A b+7 b C)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+7 b C)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+9 a C+9 b B)}{15 d}+\frac{2 (a B+A b) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(120*(5*A*b + 5*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(7*a*A + 9*b*B + 9*a*C)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((1176*I)*a*A + (1512*I)*b*B + (1512*I)*a*C + 30*(23*A*b + 23*a*B + 28*b*C)*Sin[c + d*x] + 14*(19*a*A + 18*b*B + 18*a*C)*Sin[2*(c + d*x)] + 90*A*b*Sin[3*(c + d*x)] + 90*a*B*Sin[3*(c + d*x)] + 35*a*A*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
989,1,1371,266,6.9621327,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{60 a A F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{77 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 b B F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{20 a C F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^{\frac{7}{2}}(c+d x)}{21 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{14 \sqrt{2} A b e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{45 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{14 \sqrt{2} a B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{45 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 \sqrt{2} b C e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc (c) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{(a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{(149 A b+198 C b+187 A \cos (2 c) b+234 C \cos (2 c) b+149 a B+187 a B \cos (2 c)) \cos (d x) \csc (c)}{180 d}+\frac{(1041 a A+1144 b B+1144 a C) \cos (2 d x) \sin (2 c)}{1848 d}+\frac{(43 A b+36 C b+43 a B) \cos (3 d x) \sin (3 c)}{180 d}+\frac{(16 a A+11 b B+11 a C) \cos (4 d x) \sin (4 c)}{154 d}+\frac{(A b+a B) \cos (5 d x) \sin (5 c)}{36 d}+\frac{a A \cos (6 d x) \sin (6 c)}{88 d}+\frac{(187 A b+234 C b+187 a B) \cos (c) \sin (d x)}{90 d}+\frac{(1041 a A+1144 b B+1144 a C) \cos (2 c) \sin (2 d x)}{1848 d}+\frac{(43 A b+36 C b+43 a B) \cos (3 c) \sin (3 d x)}{180 d}+\frac{(16 a A+11 b B+11 a C) \cos (4 c) \sin (4 d x)}{154 d}+\frac{(A b+a B) \cos (5 c) \sin (5 d x)}{36 d}+\frac{a A \cos (6 c) \sin (6 d x)}{88 d}\right)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{5}{2}}(c+d x)}","\frac{2 \sin (c+d x) (7 a B+7 A b+9 b C)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (9 a A+11 a C+11 b B)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{10 \sin (c+d x) (9 a A+11 a C+11 b B)}{231 d \sqrt{\sec (c+d x)}}+\frac{10 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (9 a A+11 a C+11 b B)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+9 b C)}{15 d}+\frac{2 (a B+A b) \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x)}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(-14*Sqrt[2]*A*b*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (14*Sqrt[2]*a*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(45*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (2*Sqrt[2]*b*C*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^3*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(5*d*E^(I*d*x)*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (60*a*A*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(77*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*b*B*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (20*a*C*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(21*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/180*((149*A*b + 149*a*B + 198*b*C + 187*A*b*Cos[2*c] + 187*a*B*Cos[2*c] + 234*b*C*Cos[2*c])*Cos[d*x]*Csc[c])/d + ((1041*a*A + 1144*b*B + 1144*a*C)*Cos[2*d*x]*Sin[2*c])/(1848*d) + ((43*A*b + 43*a*B + 36*b*C)*Cos[3*d*x]*Sin[3*c])/(180*d) + ((16*a*A + 11*b*B + 11*a*C)*Cos[4*d*x]*Sin[4*c])/(154*d) + ((A*b + a*B)*Cos[5*d*x]*Sin[5*c])/(36*d) + (a*A*Cos[6*d*x]*Sin[6*c])/(88*d) + ((187*A*b + 187*a*B + 234*b*C)*Cos[c]*Sin[d*x])/(90*d) + ((1041*a*A + 1144*b*B + 1144*a*C)*Cos[2*c]*Sin[2*d*x])/(1848*d) + ((43*A*b + 43*a*B + 36*b*C)*Cos[3*c]*Sin[3*d*x])/(180*d) + ((16*a*A + 11*b*B + 11*a*C)*Cos[4*c]*Sin[4*d*x])/(154*d) + ((A*b + a*B)*Cos[5*c]*Sin[5*d*x])/(36*d) + (a*A*Cos[6*c]*Sin[6*d*x])/(88*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))","C",0
990,1,507,343,6.7864678,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4}{15} \sin (c+d x) \left(15 a^2 A+9 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)+\frac{4}{45} \sec ^2(c+d x) \left(9 a^2 C \sin (c+d x)+18 a b B \sin (c+d x)+9 A b^2 \sin (c+d x)+7 b^2 C \sin (c+d x)\right)+\frac{4}{21} \sec (c+d x) \left(7 a^2 B \sin (c+d x)+14 a A b \sin (c+d x)+10 a b C \sin (c+d x)+5 b^2 B \sin (c+d x)\right)+\frac{4}{7} \sec ^3(c+d x) \left(2 a b C \sin (c+d x)+b^2 B \sin (c+d x)\right)+\frac{4}{9} b^2 C \tan (c+d x) \sec ^3(c+d x)\right)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{2 \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-35 a^2 B-70 a A b-50 a b C-25 b^2 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(105 a^2 A+63 a^2 C+126 a b B+63 A b^2+49 b^2 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{105 d (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(4 a^2 C+18 a b B+9 A b^2+7 b^2 C\right)}{45 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 B+14 a A b+10 a b C+5 b^2 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 (5 A+3 C)+18 a b B+b^2 (9 A+7 C)\right)}{15 d}+\frac{2 b (4 a C+9 b B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}",1,"(-2*Cos[c + d*x]^4*((2*(105*a^2*A + 63*A*b^2 + 126*a*b*B + 63*a^2*C + 49*b^2*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(-70*a*A*b - 35*a^2*B - 25*b^2*B - 50*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(15*a^2*A + 9*A*b^2 + 18*a*b*B + 9*a^2*C + 7*b^2*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^3*(b^2*B*Sin[c + d*x] + 2*a*b*C*Sin[c + d*x]))/7 + (4*Sec[c + d*x]*(14*a*A*b*Sin[c + d*x] + 7*a^2*B*Sin[c + d*x] + 5*b^2*B*Sin[c + d*x] + 10*a*b*C*Sin[c + d*x]))/21 + (4*Sec[c + d*x]^2*(9*A*b^2*Sin[c + d*x] + 18*a*b*B*Sin[c + d*x] + 9*a^2*C*Sin[c + d*x] + 7*b^2*C*Sin[c + d*x]))/45 + (4*b^2*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","A",1
991,1,333,289,2.5147102,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{4 (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)-21 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+2 a b (5 A+3 C)+3 b^2 B\right)+105 a^2 B \sin (c+d x)+35 a^2 C \tan (c+d x)+210 a A b \sin (c+d x)+70 a b B \tan (c+d x)+126 a b C \sin (c+d x)+42 a b C \tan (c+d x) \sec (c+d x)+35 A b^2 \tan (c+d x)+63 b^2 B \sin (c+d x)+21 b^2 B \tan (c+d x) \sec (c+d x)+25 b^2 C \tan (c+d x)+15 b^2 C \tan (c+d x) \sec ^2(c+d x)\right)}{105 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 b (4 a C+7 b B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}",1,"(4*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-21*(5*a^2*B + 3*b^2*B + 2*a*b*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 210*a*A*b*Sin[c + d*x] + 105*a^2*B*Sin[c + d*x] + 63*b^2*B*Sin[c + d*x] + 126*a*b*C*Sin[c + d*x] + 35*A*b^2*Tan[c + d*x] + 70*a*b*B*Tan[c + d*x] + 35*a^2*C*Tan[c + d*x] + 25*b^2*C*Tan[c + d*x] + 21*b^2*B*Sec[c + d*x]*Tan[c + d*x] + 42*a*b*C*Sec[c + d*x]*Tan[c + d*x] + 15*b^2*C*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))","A",1
992,1,271,241,2.142182,"\int \frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{4 (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)+3 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 (A-C)-10 a b B-b^2 (5 A+3 C)\right)+15 a^2 C \sin (c+d x)+30 a b B \sin (c+d x)+10 a b C \tan (c+d x)+15 A b^2 \sin (c+d x)+5 b^2 B \tan (c+d x)+9 b^2 C \sin (c+d x)+3 b^2 C \tan (c+d x) \sec (c+d x)\right)}{15 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(4 a^2 C+10 a b B+5 A b^2+3 b^2 C\right)}{5 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}+\frac{2 b (4 a C+5 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}",1,"(4*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(3*(-10*a*b*B + 5*a^2*(A - C) - b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 15*A*b^2*Sin[c + d*x] + 30*a*b*B*Sin[c + d*x] + 15*a^2*C*Sin[c + d*x] + 9*b^2*C*Sin[c + d*x] + 5*b^2*B*Tan[c + d*x] + 10*a*b*C*Tan[c + d*x] + 3*b^2*C*Sec[c + d*x]*Tan[c + d*x]))/(15*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))","A",1
993,1,227,224,2.98806,"\int \frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)+6 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)+a^2 A \sin (2 (c+d x))+12 a b C \sin (c+d x)+6 b^2 B \sin (c+d x)+2 b^2 C \tan (c+d x)\right)}{3 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} (3 b B-2 a (A-3 C))}{3 d}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt{\sec (c+d x)}}-\frac{2 b^2 (A-C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*(a^2*B - b^2*B + 2*a*b*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 6*b^2*B*Sin[c + d*x] + 12*a*b*C*Sin[c + d*x] + a^2*A*Sin[2*(c + d*x)] + 2*b^2*C*Tan[c + d*x]))/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))","A",1
994,1,234,225,4.7892223,"\int \frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{2 (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sin (c+d x) \left(3 \left(a^2 A \cos (2 (c+d x))+a^2 A+10 b^2 C\right)+10 a (a B+2 A b) \cos (c+d x)\right)+10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)+6 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)\right)}{15 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 a (5 a B+4 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 (A-5 C) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}",1,"(2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (10*a*(2*A*b + a*B)*Cos[c + d*x] + 3*(a^2*A + 10*b^2*C + a^2*A*Cos[2*(c + d*x)]))*Sin[c + d*x]))/(15*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))","A",1
995,1,251,242,6.5107267,"\int \frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sin (2 (c+d x)) \left(5 \left(3 a^2 A \cos (2 (c+d x))+a^2 (13 A+14 C)+28 a b B+14 A b^2\right)+42 a (a B+2 A b) \cos (c+d x)\right)+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)+84 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+5 C)+5 b^2 B\right)\right)}{105 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 a (7 a B+4 A b) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(84*(3*a^2*B + 5*b^2*B + 2*a*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (42*a*(2*A*b + a*B)*Cos[c + d*x] + 5*(14*A*b^2 + 28*a*b*B + a^2*(13*A + 14*C) + 3*a^2*A*Cos[2*(c + d*x)]))*Sin[2*(c + d*x)]))/(105*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))","A",1
996,1,286,290,3.6316509,"\int \frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\sin (2 (c+d x)) \left(7 \cos (c+d x) \left(a^2 (43 A+36 C)+72 a b B+36 A b^2\right)+5 \left(7 a^2 A \cos (3 (c+d x))+78 a^2 B+18 a (a B+2 A b) \cos (2 (c+d x))+156 a A b+168 a b C+84 b^2 B\right)\right)+120 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+2 a b (5 A+7 C)+7 b^2 B\right)+168 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)\right)}{630 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(168*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*(5*a^2*B + 7*b^2*B + 2*a*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(36*A*b^2 + 72*a*b*B + a^2*(43*A + 36*C))*Cos[c + d*x] + 5*(156*a*A*b + 78*a^2*B + 84*b^2*B + 168*a*b*C + 18*a*(2*A*b + a*B)*Cos[2*(c + d*x)] + 7*a^2*A*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(630*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(7/2))","A",1
997,1,566,397,7.135075,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(105 a^3 A+35 a^3 C+105 a^2 b B+105 a A b^2+75 a b^2 C+25 b^3 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-105 a^3 B-315 a^2 A b-189 a^2 b C-189 a b^2 B-63 A b^3-49 b^3 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{105 d (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4}{45} \sec ^2(c+d x) \left(27 a^2 b C \sin (c+d x)+27 a b^2 B \sin (c+d x)+9 A b^3 \sin (c+d x)+7 b^3 C \sin (c+d x)\right)+\frac{4}{15} \sin (c+d x) \left(15 a^3 B+45 a^2 A b+27 a^2 b C+27 a b^2 B+9 A b^3+7 b^3 C\right)+\frac{4}{21} \sec (c+d x) \left(7 a^3 C \sin (c+d x)+21 a^2 b B \sin (c+d x)+21 a A b^2 \sin (c+d x)+15 a b^2 C \sin (c+d x)+5 b^3 B \sin (c+d x)\right)+\frac{4}{7} \sec ^3(c+d x) \left(3 a b^2 C \sin (c+d x)+b^3 B \sin (c+d x)\right)+\frac{4}{9} b^3 C \tan (c+d x) \sec ^3(c+d x)\right)}{d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 (2 a C+3 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{21 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3}{9 d}",1,"(2*Cos[c + d*x]^5*((2*(-315*a^2*A*b - 63*A*b^3 - 105*a^3*B - 189*a*b^2*B - 189*a^2*b*C - 49*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(105*a^3*A + 105*a*A*b^2 + 105*a^2*b*B + 25*b^3*B + 35*a^3*C + 75*a*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(45*a^2*A*b + 9*A*b^3 + 15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^3*(b^3*B*Sin[c + d*x] + 3*a*b^2*C*Sin[c + d*x]))/7 + (4*Sec[c + d*x]*(21*a*A*b^2*Sin[c + d*x] + 21*a^2*b*B*Sin[c + d*x] + 5*b^3*B*Sin[c + d*x] + 7*a^3*C*Sin[c + d*x] + 15*a*b^2*C*Sin[c + d*x]))/21 + (4*Sec[c + d*x]^2*(9*A*b^3*Sin[c + d*x] + 27*a*b^2*B*Sin[c + d*x] + 27*a^2*b*C*Sin[c + d*x] + 7*b^3*C*Sin[c + d*x]))/45 + (4*b^3*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","A",1
998,1,377,334,3.9196633,"\int \frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{4 (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(105 a^3 C \sin (c+d x)+315 a^2 b B \sin (c+d x)+105 a^2 b C \tan (c+d x)+5 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right)+21 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 (A-C)-15 a^2 b B-3 a b^2 (5 A+3 C)-3 b^3 B\right)+315 a A b^2 \sin (c+d x)+105 a b^2 B \tan (c+d x)+189 a b^2 C \sin (c+d x)+63 a b^2 C \tan (c+d x) \sec (c+d x)+35 A b^3 \tan (c+d x)+63 b^3 B \sin (c+d x)+21 b^3 B \tan (c+d x) \sec (c+d x)+25 b^3 C \tan (c+d x)+15 b^3 C \tan (c+d x) \sec ^2(c+d x)\right)}{105 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right)}{105 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right)}{35 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}+\frac{2 (6 a C+7 b B) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{35 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3}{7 d}",1,"(4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(21*(-15*a^2*b*B - 3*b^3*B + 5*a^3*(A - C) - 3*a*b^2*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 315*a*A*b^2*Sin[c + d*x] + 315*a^2*b*B*Sin[c + d*x] + 63*b^3*B*Sin[c + d*x] + 105*a^3*C*Sin[c + d*x] + 189*a*b^2*C*Sin[c + d*x] + 35*A*b^3*Tan[c + d*x] + 105*a*b^2*B*Tan[c + d*x] + 105*a^2*b*C*Tan[c + d*x] + 25*b^3*C*Tan[c + d*x] + 21*b^3*B*Sec[c + d*x]*Tan[c + d*x] + 63*a*b^2*C*Sec[c + d*x]*Tan[c + d*x] + 15*b^3*C*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(9/2))","A",1
999,1,311,319,3.5395031,"\int \frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(5 a^3 A \sin (2 (c+d x))+90 a^2 b C \sin (c+d x)+10 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)+6 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+15 a^2 b (A-C)-15 a b^2 B-b^3 (5 A+3 C)\right)+90 a b^2 B \sin (c+d x)+30 a b^2 C \tan (c+d x)+30 A b^3 \sin (c+d x)+10 b^3 B \tan (c+d x)+18 b^3 C \sin (c+d x)+6 b^3 C \tan (c+d x) \sec (c+d x)\right)}{15 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} \left(-\left(a^2 (10 A-42 C)\right)+45 a b B+3 b^2 (5 A+3 C)\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+15 a^2 b (A-C)-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (5 a A-9 a C-5 b B)}{15 d}-\frac{2 b (5 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{15 d}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^3}{3 d \sqrt{\sec (c+d x)}}",1,"(2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(6*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 30*A*b^3*Sin[c + d*x] + 90*a*b^2*B*Sin[c + d*x] + 90*a^2*b*C*Sin[c + d*x] + 18*b^3*C*Sin[c + d*x] + 5*a^3*A*Sin[2*(c + d*x)] + 10*b^3*B*Tan[c + d*x] + 30*a*b^2*C*Tan[c + d*x] + 6*b^3*C*Sec[c + d*x]*Tan[c + d*x]))/(15*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(9/2))","A",1
1000,1,295,313,2.7836211,"\int \frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(3 a^3 A \sin (c+d x)+3 a^3 A \sin (3 (c+d x))+10 a^3 B \sin (2 (c+d x))+30 a^2 A b \sin (2 (c+d x))+20 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 B+3 a^2 b (A+3 C)+9 a b^2 B+b^3 (3 A+C)\right)+12 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)+180 a b^2 C \sin (c+d x)+60 b^3 B \sin (c+d x)+20 b^3 C \tan (c+d x)\right)}{15 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} \left(10 a^2 B+3 a b (7 A-15 C)-15 b^2 B\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 B+3 a^2 b (A+3 C)+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (5 a B+9 A b-5 b C)}{15 d}+\frac{2 (5 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{15 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^3}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(12*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*a^3*A*Sin[c + d*x] + 60*b^3*B*Sin[c + d*x] + 180*a*b^2*C*Sin[c + d*x] + 30*a^2*A*b*Sin[2*(c + d*x)] + 10*a^3*B*Sin[2*(c + d*x)] + 3*a^3*A*Sin[3*(c + d*x)] + 20*b^3*C*Tan[c + d*x]))/(15*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(9/2))","A",1
1001,1,234,317,5.1030501,"\int \frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (c+d x) \left(15 a^3 A \cos (3 (c+d x))+5 a \cos (c+d x) \left(a^2 (29 A+28 C)+84 a b B+84 A b^2\right)+42 a^2 (a B+3 A b) \cos (2 (c+d x))+42 \left(a^3 B+3 a^2 A b+10 b^3 C\right)\right)+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)+168 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right)\right)}{420 d}","\frac{2 a \sin (c+d x) \left(5 a^2 (5 A+7 C)+63 a b B+24 A b^2\right)}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}-\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} (7 a B+11 A b-35 b C)}{35 d}+\frac{2 (7 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^3}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(42*(3*a^2*A*b + a^3*B + 10*b^3*C) + 5*a*(84*A*b^2 + 84*a*b*B + a^2*(29*A + 28*C))*Cos[c + d*x] + 42*a^2*(3*A*b + a*B)*Cos[2*(c + d*x)] + 15*a^3*A*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
1002,1,323,336,6.2041447,"\int \frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sin (2 (c+d x)) \left(7 a \cos (c+d x) \left(a^2 (43 A+36 C)+108 a b B+108 A b^2\right)+5 \left(7 a^3 A \cos (3 (c+d x))+78 a^3 B+18 a^2 (a B+3 A b) \cos (2 (c+d x))+6 a^2 (39 A b+42 b C)+252 a b^2 B+84 A b^3\right)\right)+240 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right)+336 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)\right)}{1260 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 a \sin (c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right)}{63 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 (3 a B+2 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^3}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(336*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(7*a*(108*A*b^2 + 108*a*b*B + a^2*(43*A + 36*C))*Cos[c + d*x] + 5*(84*A*b^3 + 78*a^3*B + 252*a*b^2*B + 6*a^2*(39*A*b + 42*b*C) + 18*a^2*(3*A*b + a*B)*Cos[2*(c + d*x)] + 7*a^3*A*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(9/2))","A",1
1003,1,538,401,6.8745281,"\int \frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{1}{88} a^3 A \sin (6 (c+d x))+\frac{1}{154} a \sin (4 (c+d x)) \left(16 a^2 A+11 a^2 C+33 a b B+33 A b^2\right)+\frac{1}{36} a^2 (a B+3 A b) \sin (5 (c+d x))+\frac{1}{90} \sin (c+d x) \left(19 a^3 B+57 a^2 A b+54 a^2 b C+54 a b^2 B+18 A b^3\right)+\frac{\sin (2 (c+d x)) \left(1041 a^3 A+1144 a^3 C+3432 a^2 b B+3432 a A b^2+3696 a b^2 C+1232 b^3 B\right)}{1848}+\frac{1}{180} \sin (3 (c+d x)) \left(43 a^3 B+129 a^2 A b+108 a^2 b C+108 a b^2 B+36 A b^3\right)\right)}{d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{2 \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(225 a^3 A+275 a^3 C+825 a^2 b B+825 a A b^2+1155 a b^2 C+385 b^3 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(539 a^3 B+1617 a^2 A b+2079 a^2 b C+2079 a b^2 B+693 A b^3+1155 b^3 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{1155 d (a \cos (c+d x)+b)^3 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 a \sin (c+d x) \left(9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right)}{15 d}+\frac{2 (11 a B+6 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^3}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(2*Cos[c + d*x]^5*((2*(1617*a^2*A*b + 693*A*b^3 + 539*a^3*B + 2079*a*b^2*B + 2079*a^2*b*C + 1155*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(225*a^3*A + 825*a*A*b^2 + 825*a^2*b*B + 385*b^3*B + 275*a^3*C + 1155*a*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(1155*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((57*a^2*A*b + 18*A*b^3 + 19*a^3*B + 54*a*b^2*B + 54*a^2*b*C)*Sin[c + d*x])/90 + ((1041*a^3*A + 3432*a*A*b^2 + 3432*a^2*b*B + 1232*b^3*B + 1144*a^3*C + 3696*a*b^2*C)*Sin[2*(c + d*x)])/1848 + ((129*a^2*A*b + 36*A*b^3 + 43*a^3*B + 108*a*b^2*B + 108*a^2*b*C)*Sin[3*(c + d*x)])/180 + (a*(16*a^2*A + 33*A*b^2 + 33*a*b*B + 11*a^2*C)*Sin[4*(c + d*x)])/154 + (a^2*(3*A*b + a*B)*Sin[5*(c + d*x)])/36 + (a^3*A*Sin[6*(c + d*x)])/88))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","A",1
1004,1,713,515,7.5533261,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(1155 a^4 A+385 a^4 C+1540 a^3 b B+2310 a^2 A b^2+1650 a^2 b^2 C+1100 a b^3 B+275 A b^4+225 b^4 C\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-1155 a^4 B-4620 a^3 A b-2772 a^3 b C-4158 a^2 b^2 B-2772 a A b^3-2156 a b^3 C-539 b^4 B\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{1155 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4}{77} \sec ^3(c+d x) \left(66 a^2 b^2 C \sin (c+d x)+44 a b^3 B \sin (c+d x)+11 A b^4 \sin (c+d x)+9 b^4 C \sin (c+d x)\right)+\frac{4}{45} \sec ^2(c+d x) \left(36 a^3 b C \sin (c+d x)+54 a^2 b^2 B \sin (c+d x)+36 a A b^3 \sin (c+d x)+28 a b^3 C \sin (c+d x)+7 b^4 B \sin (c+d x)\right)+\frac{4}{15} \sin (c+d x) \left(15 a^4 B+60 a^3 A b+36 a^3 b C+54 a^2 b^2 B+36 a A b^3+28 a b^3 C+7 b^4 B\right)+\frac{4}{231} \sec (c+d x) \left(77 a^4 C \sin (c+d x)+308 a^3 b B \sin (c+d x)+462 a^2 A b^2 \sin (c+d x)+330 a^2 b^2 C \sin (c+d x)+220 a b^3 B \sin (c+d x)+55 A b^4 \sin (c+d x)+45 b^4 C \sin (c+d x)\right)+\frac{4}{9} \sec ^4(c+d x) \left(4 a b^3 C \sin (c+d x)+b^4 B \sin (c+d x)\right)+\frac{4}{11} b^4 C \tan (c+d x) \sec ^4(c+d x)\right)}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right) (a+b \sec (c+d x))^2}{231 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right)}{3465 d}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right)}{693 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right)}{231 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right)}{15 d}+\frac{2 (8 a C+11 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3}{99 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4}{11 d}",1,"(2*Cos[c + d*x]^6*((2*(-4620*a^3*A*b - 2772*a*A*b^3 - 1155*a^4*B - 4158*a^2*b^2*B - 539*b^4*B - 2772*a^3*b*C - 2156*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(1155*a^4*A + 2310*a^2*A*b^2 + 275*A*b^4 + 1540*a^3*b*B + 1100*a*b^3*B + 385*a^4*C + 1650*a^2*b^2*C + 225*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(1155*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(60*a^3*A*b + 36*a*A*b^3 + 15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 36*a^3*b*C + 28*a*b^3*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^4*(b^4*B*Sin[c + d*x] + 4*a*b^3*C*Sin[c + d*x]))/9 + (4*Sec[c + d*x]^2*(36*a*A*b^3*Sin[c + d*x] + 54*a^2*b^2*B*Sin[c + d*x] + 7*b^4*B*Sin[c + d*x] + 36*a^3*b*C*Sin[c + d*x] + 28*a*b^3*C*Sin[c + d*x]))/45 + (4*Sec[c + d*x]^3*(11*A*b^4*Sin[c + d*x] + 44*a*b^3*B*Sin[c + d*x] + 66*a^2*b^2*C*Sin[c + d*x] + 9*b^4*C*Sin[c + d*x]))/77 + (4*Sec[c + d*x]*(462*a^2*A*b^2*Sin[c + d*x] + 55*A*b^4*Sin[c + d*x] + 308*a^3*b*B*Sin[c + d*x] + 220*a*b^3*B*Sin[c + d*x] + 77*a^4*C*Sin[c + d*x] + 330*a^2*b^2*C*Sin[c + d*x] + 45*b^4*C*Sin[c + d*x]))/231 + (4*b^4*C*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))","A",1
1005,1,609,441,7.4172008,"\int \frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(105 a^4 B+420 a^3 A b+140 a^3 b C+210 a^2 b^2 B+140 a A b^3+100 a b^3 C+25 b^4 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(105 a^4 A-105 a^4 C-420 a^3 b B-630 a^2 A b^2-378 a^2 b^2 C-252 a b^3 B-63 A b^4-49 b^4 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{105 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4}{45} \sec ^2(c+d x) \left(54 a^2 b^2 C \sin (c+d x)+36 a b^3 B \sin (c+d x)+9 A b^4 \sin (c+d x)+7 b^4 C \sin (c+d x)\right)+\frac{4}{21} \sec (c+d x) \left(28 a^3 b C \sin (c+d x)+42 a^2 b^2 B \sin (c+d x)+28 a A b^3 \sin (c+d x)+20 a b^3 C \sin (c+d x)+5 b^4 B \sin (c+d x)\right)+\frac{4}{15} \sin (c+d x) \left(15 a^4 C+60 a^3 b B+90 a^2 A b^2+54 a^2 b^2 C+36 a b^3 B+9 A b^4+7 b^4 C\right)+\frac{4}{7} \sec ^3(c+d x) \left(4 a b^3 C \sin (c+d x)+b^4 B \sin (c+d x)\right)+\frac{4}{9} b^4 C \tan (c+d x) \sec ^3(c+d x)\right)}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right) (a+b \sec (c+d x))^2}{315 d}+\frac{2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right)}{315 d}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right)}{315 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}+\frac{2 (8 a C+9 b B) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3}{63 d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^4}{9 d}",1,"(2*Cos[c + d*x]^6*((2*(105*a^4*A - 630*a^2*A*b^2 - 63*A*b^4 - 420*a^3*b*B - 252*a*b^3*B - 105*a^4*C - 378*a^2*b^2*C - 49*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(420*a^3*A*b + 140*a*A*b^3 + 105*a^4*B + 210*a^2*b^2*B + 25*b^4*B + 140*a^3*b*C + 100*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(90*a^2*A*b^2 + 9*A*b^4 + 60*a^3*b*B + 36*a*b^3*B + 15*a^4*C + 54*a^2*b^2*C + 7*b^4*C)*Sin[c + d*x])/15 + (4*Sec[c + d*x]^3*(b^4*B*Sin[c + d*x] + 4*a*b^3*C*Sin[c + d*x]))/7 + (4*Sec[c + d*x]*(28*a*A*b^3*Sin[c + d*x] + 42*a^2*b^2*B*Sin[c + d*x] + 5*b^4*B*Sin[c + d*x] + 28*a^3*b*C*Sin[c + d*x] + 20*a*b^3*C*Sin[c + d*x]))/21 + (4*Sec[c + d*x]^2*(9*A*b^4*Sin[c + d*x] + 36*a*b^3*B*Sin[c + d*x] + 54*a^2*b^2*C*Sin[c + d*x] + 7*b^4*C*Sin[c + d*x]))/45 + (4*b^4*C*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))","A",1
1006,1,530,419,7.4198175,"\int \frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(35 a^4 A+105 a^4 C+420 a^3 b B+630 a^2 A b^2+210 a^2 b^2 C+140 a b^3 B+35 A b^4+25 b^4 C\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(105 a^4 B+420 a^3 A b-420 a^3 b C-630 a^2 b^2 B-420 a A b^3-252 a b^3 C-63 b^4 B\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{105 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2}{3} a^4 A \sin (2 (c+d x))+\frac{4}{21} \sec (c+d x) \left(42 a^2 b^2 C \sin (c+d x)+28 a b^3 B \sin (c+d x)+7 A b^4 \sin (c+d x)+5 b^4 C \sin (c+d x)\right)+\frac{4}{5} b \sin (c+d x) \left(20 a^3 C+30 a^2 b B+20 a A b^2+12 a b^2 C+3 b^3 B\right)+\frac{4}{5} \sec ^2(c+d x) \left(4 a b^3 C \sin (c+d x)+b^4 B \sin (c+d x)\right)+\frac{4}{7} b^4 C \tan (c+d x) \sec ^2(c+d x)\right)}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-\left(a^2 (35 A-87 C)\right)+98 a b B+5 b^2 (7 A+5 C)\right)}{105 d}+\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} \left(-\left(a^3 (70 A-366 C)\right)+609 a^2 b B+84 a b^2 (5 A+3 C)+63 b^3 B\right)}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} (35 a A-39 a C-21 b B) (a+b \sec (c+d x))^2}{105 d}-\frac{2 b (7 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3}{21 d}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{3 d \sqrt{\sec (c+d x)}}",1,"(2*Cos[c + d*x]^6*((2*(420*a^3*A*b - 420*a*A*b^3 + 105*a^4*B - 630*a^2*b^2*B - 63*b^4*B - 420*a^3*b*C - 252*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(35*a^4*A + 630*a^2*A*b^2 + 35*A*b^4 + 420*a^3*b*B + 140*a*b^3*B + 105*a^4*C + 210*a^2*b^2*C + 25*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*b*(20*a*A*b^2 + 30*a^2*b*B + 3*b^3*B + 20*a^3*C + 12*a*b^2*C)*Sin[c + d*x])/5 + (4*Sec[c + d*x]^2*(b^4*B*Sin[c + d*x] + 4*a*b^3*C*Sin[c + d*x]))/5 + (4*Sec[c + d*x]*(7*A*b^4*Sin[c + d*x] + 28*a*b^3*B*Sin[c + d*x] + 42*a^2*b^2*C*Sin[c + d*x] + 5*b^4*C*Sin[c + d*x]))/21 + (2*a^4*A*Sin[2*(c + d*x)])/3 + (4*b^4*C*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))","A",1
1007,1,485,409,7.408779,"\int \frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 A b+60 a^3 b C+90 a^2 b^2 B+60 a A b^3+20 a b^3 C+5 b^4 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(9 a^4 A+15 a^4 C+60 a^3 b B+90 a^2 A b^2-90 a^2 b^2 C-60 a b^3 B-15 A b^4-9 b^4 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{15 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{1}{5} a^4 A \sin (3 (c+d x))+\frac{2}{3} a^3 (a B+4 A b) \sin (2 (c+d x))+\frac{1}{5} \sin (c+d x) \left(a^4 A+120 a^2 b^2 C+80 a b^3 B+20 A b^4+12 b^4 C\right)+\frac{4}{3} \sec (c+d x) \left(4 a b^3 C \sin (c+d x)+b^4 B \sin (c+d x)\right)+\frac{4}{5} b^4 C \tan (c+d x) \sec (c+d x)\right)}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 B+14 a b (A-C)-5 b^2 B\right)}{15 d}-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} \left(10 a^3 B+a^2 b (31 A-87 C)-60 a b^2 B-3 b^3 (5 A+3 C)\right)}{15 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 B+4 a^3 b (A+3 C)+18 a^2 b^2 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (3 A+5 C)+20 a^3 b B+30 a^2 b^2 (A-C)-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} (5 a B+11 A b-3 b C) (a+b \sec (c+d x))^2}{15 d}+\frac{2 (5 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{15 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*Cos[c + d*x]^6*((2*(9*a^4*A + 90*a^2*A*b^2 - 15*A*b^4 + 60*a^3*b*B - 60*a*b^3*B + 15*a^4*C - 90*a^2*b^2*C - 9*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(20*a^3*A*b + 60*a*A*b^3 + 5*a^4*B + 90*a^2*b^2*B + 5*b^4*B + 60*a^3*b*C + 20*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((a^4*A + 20*A*b^4 + 80*a*b^3*B + 120*a^2*b^2*C + 12*b^4*C)*Sin[c + d*x])/5 + (4*Sec[c + d*x]*(b^4*B*Sin[c + d*x] + 4*a*b^3*C*Sin[c + d*x]))/3 + (2*a^3*(4*A*b + a*B)*Sin[2*(c + d*x)])/3 + (a^4*A*Sin[3*(c + d*x)])/5 + (4*b^4*C*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))","A",1
1008,1,394,429,6.7126548,"\int \frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(130 a^4 A \sin (2 (c+d x))+15 a^4 A \sin (4 (c+d x))+42 a^4 B \sin (c+d x)+42 a^4 B \sin (3 (c+d x))+140 a^4 C \sin (2 (c+d x))+168 a^3 A b \sin (c+d x)+168 a^3 A b \sin (3 (c+d x))+560 a^3 b B \sin (2 (c+d x))+840 a^2 A b^2 \sin (2 (c+d x))+40 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right)+168 \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right)+3360 a b^3 C \sin (c+d x)+840 b^4 B \sin (c+d x)+280 b^4 C \tan (c+d x)\right)}{210 d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 (5 A+7 C)+98 a b B+b^2 (87 A-35 C)\right)}{105 d}+\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+77 a b B+48 A b^2\right) (a+b \sec (c+d x))^2}{105 d \sqrt{\sec (c+d x)}}-\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} \left(10 a^3 (5 A+7 C)+217 a^2 b B+12 a b^2 (19 A-35 C)-105 b^3 B\right)}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 (7 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(168*(3*a^4*B + 30*a^2*b^2*B - 5*b^4*B + 20*a*b^3*(A - C) + 4*a^3*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 40*(28*a^3*b*B + 84*a*b^3*B + 7*b^4*(3*A + C) + 42*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 168*a^3*A*b*Sin[c + d*x] + 42*a^4*B*Sin[c + d*x] + 840*b^4*B*Sin[c + d*x] + 3360*a*b^3*C*Sin[c + d*x] + 130*a^4*A*Sin[2*(c + d*x)] + 840*a^2*A*b^2*Sin[2*(c + d*x)] + 560*a^3*b*B*Sin[2*(c + d*x)] + 140*a^4*C*Sin[2*(c + d*x)] + 168*a^3*A*b*Sin[3*(c + d*x)] + 42*a^4*B*Sin[3*(c + d*x)] + 15*a^4*A*Sin[4*(c + d*x)] + 280*b^4*C*Tan[c + d*x]))/(210*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(11/2))","A",1
1009,1,517,426,7.3418542,"\int \frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{1}{36} a^4 A \sin (5 (c+d x))+\frac{1}{14} a^3 (a B+4 A b) \sin (4 (c+d x))+\frac{1}{180} a^2 \sin (3 (c+d x)) \left(43 a^2 A+36 a^2 C+144 a b B+216 A b^2\right)+\frac{1}{21} a \sin (2 (c+d x)) \left(13 a^3 B+52 a^2 A b+56 a^2 b C+84 a b^2 B+56 A b^3\right)+\frac{1}{90} \sin (c+d x) \left(19 a^4 A+18 a^4 C+72 a^3 b B+108 a^2 A b^2+360 b^4 C\right)\right)}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(25 a^4 B+100 a^3 A b+140 a^3 b C+210 a^2 b^2 B+140 a A b^3+420 a b^3 C+105 b^4 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(49 a^4 A+63 a^4 C+252 a^3 b B+378 a^2 A b^2+630 a^2 b^2 C+420 a b^3 B+105 A b^4-105 b^4 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{105 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+117 a b B+48 A b^2\right) (a+b \sec (c+d x))^2}{315 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 b^2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right)}{315 d}+\frac{2 a \sin (c+d x) \left(75 a^3 B+a^2 (202 A b+294 b C)+261 a b^2 B+64 A b^3\right)}{315 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 (9 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*Cos[c + d*x]^6*((2*(49*a^4*A + 378*a^2*A*b^2 + 105*A*b^4 + 252*a^3*b*B + 420*a*b^3*B + 63*a^4*C + 630*a^2*b^2*C - 105*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(100*a^3*A*b + 140*a*A*b^3 + 25*a^4*B + 210*a^2*b^2*B + 105*b^4*B + 140*a^3*b*C + 420*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(105*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((19*a^4*A + 108*a^2*A*b^2 + 72*a^3*b*B + 18*a^4*C + 360*b^4*C)*Sin[c + d*x])/90 + (a*(52*a^2*A*b + 56*A*b^3 + 13*a^3*B + 84*a*b^2*B + 56*a^2*b*C)*Sin[2*(c + d*x)])/21 + (a^2*(43*a^2*A + 216*A*b^2 + 144*a*b*B + 36*a^2*C)*Sin[3*(c + d*x)])/180 + (a^3*(4*A*b + a*B)*Sin[4*(c + d*x)])/14 + (a^4*A*Sin[5*(c + d*x)])/36))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))","A",1
1010,1,580,444,7.0941776,"\int \frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{1}{88} a^4 A \sin (6 (c+d x))+\frac{1}{36} a^3 (a B+4 A b) \sin (5 (c+d x))+\frac{1}{154} a^2 \sin (4 (c+d x)) \left(16 a^2 A+11 a^2 C+44 a b B+66 A b^2\right)+\frac{1}{90} a \sin (c+d x) \left(19 a^3 B+76 a^2 A b+72 a^2 b C+108 a b^2 B+72 A b^3\right)+\frac{1}{180} a \sin (3 (c+d x)) \left(43 a^3 B+172 a^2 A b+144 a^2 b C+216 a b^2 B+144 A b^3\right)+\frac{\sin (2 (c+d x)) \left(1041 a^4 A+1144 a^4 C+4576 a^3 b B+6864 a^2 A b^2+7392 a^2 b^2 C+4928 a b^3 B+1232 A b^4\right)}{1848}\right)}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(225 a^4 A+275 a^4 C+1100 a^3 b B+1650 a^2 A b^2+2310 a^2 b^2 C+1540 a b^3 B+385 A b^4+1155 b^4 C\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(539 a^4 B+2156 a^3 A b+2772 a^3 b C+4158 a^2 b^2 B+2772 a A b^3+4620 a b^3 C+1155 b^4 B\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{1155 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a+b \sec (c+d x))^2}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right)}{3465 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right)}{693 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(2*Cos[c + d*x]^6*((2*(2156*a^3*A*b + 2772*a*A*b^3 + 539*a^4*B + 4158*a^2*b^2*B + 1155*b^4*B + 2772*a^3*b*C + 4620*a*b^3*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(225*a^4*A + 1650*a^2*A*b^2 + 385*A*b^4 + 1100*a^3*b*B + 1540*a*b^3*B + 275*a^4*C + 2310*a^2*b^2*C + 1155*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(1155*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((a*(76*a^2*A*b + 72*A*b^3 + 19*a^3*B + 108*a*b^2*B + 72*a^2*b*C)*Sin[c + d*x])/90 + ((1041*a^4*A + 6864*a^2*A*b^2 + 1232*A*b^4 + 4576*a^3*b*B + 4928*a*b^3*B + 1144*a^4*C + 7392*a^2*b^2*C)*Sin[2*(c + d*x)])/1848 + (a*(172*a^2*A*b + 144*A*b^3 + 43*a^3*B + 216*a*b^2*B + 144*a^2*b*C)*Sin[3*(c + d*x)])/180 + (a^2*(16*a^2*A + 66*A*b^2 + 44*a*b*B + 11*a^2*C)*Sin[4*(c + d*x)])/154 + (a^3*(4*A*b + a*B)*Sin[5*(c + d*x)])/36 + (a^4*A*Sin[6*(c + d*x)])/88))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))","A",1
1011,1,658,516,7.2041397,"\int \frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{13}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]","\frac{(a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{1}{208} a^4 A \sin (7 (c+d x))+\frac{1}{88} a^3 (a B+4 A b) \sin (6 (c+d x))+\frac{a^2 \sin (5 (c+d x)) \left(89 a^2 A+52 a^2 C+208 a b B+312 A b^2\right)}{1872}+\frac{1}{77} a \sin (4 (c+d x)) \left(8 a^3 B+32 a^2 A b+22 a^2 b C+33 a b^2 B+22 A b^3\right)+\frac{\sin (c+d x) \left(1897 a^4 A+1976 a^4 C+7904 a^3 b B+11856 a^2 A b^2+11232 a^2 b^2 C+7488 a b^3 B+1872 A b^4\right)}{9360}+\frac{\sin (2 (c+d x)) \left(1041 a^4 B+4164 a^3 A b+4576 a^3 b C+6864 a^2 b^2 B+4576 a A b^3+4928 a b^3 C+1232 b^4 B\right)}{1848}+\frac{\sin (3 (c+d x)) \left(2297 a^4 A+2236 a^4 C+8944 a^3 b B+13416 a^2 A b^2+11232 a^2 b^2 C+7488 a b^3 B+1872 A b^4\right)}{9360}\right)}{d \sec ^{\frac{11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2925 a^4 B+11700 a^3 A b+14300 a^3 b C+21450 a^2 b^2 B+14300 a A b^3+20020 a b^3 C+5005 b^4 B\right)+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5929 a^4 A+7007 a^4 C+28028 a^3 b B+42042 a^2 A b^2+54054 a^2 b^2 C+36036 a b^3 B+9009 A b^4+15015 b^4 C\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{15015 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \left(11 a^2 (11 A+13 C)+221 a b B+48 A b^2\right) (a+b \sec (c+d x))^2}{1287 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(1053 a^3 B+a^2 (2518 A b+3146 b C)+2171 a b^2 B+192 A b^3\right)}{9009 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(77 a^4 (11 A+13 C)+4004 a^3 b B+11 a^2 b^2 (491 A+637 C)+3458 a b^3 B+192 A b^4\right)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(45 a^4 B+20 a^3 b (9 A+11 C)+330 a^2 b^2 B+44 a b^3 (5 A+7 C)+77 b^4 B\right)}{231 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (77 A+91 C)+364 a^3 b B+78 a^2 b^2 (7 A+9 C)+468 a b^3 B+39 b^4 (3 A+5 C)\right)}{195 d}+\frac{2 (13 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{143 d \sec ^{\frac{9}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^4}{13 d \sec ^{\frac{11}{2}}(c+d x)}",1,"(2*Cos[c + d*x]^6*((2*(5929*a^4*A + 42042*a^2*A*b^2 + 9009*A*b^4 + 28028*a^3*b*B + 36036*a*b^3*B + 7007*a^4*C + 54054*a^2*b^2*C + 15015*b^4*C)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(11700*a^3*A*b + 14300*a*A*b^3 + 2925*a^4*B + 21450*a^2*b^2*B + 5005*b^4*B + 14300*a^3*b*C + 20020*a*b^3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(15015*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((1897*a^4*A + 11856*a^2*A*b^2 + 1872*A*b^4 + 7904*a^3*b*B + 7488*a*b^3*B + 1976*a^4*C + 11232*a^2*b^2*C)*Sin[c + d*x])/9360 + ((4164*a^3*A*b + 4576*a*A*b^3 + 1041*a^4*B + 6864*a^2*b^2*B + 1232*b^4*B + 4576*a^3*b*C + 4928*a*b^3*C)*Sin[2*(c + d*x)])/1848 + ((2297*a^4*A + 13416*a^2*A*b^2 + 1872*A*b^4 + 8944*a^3*b*B + 7488*a*b^3*B + 2236*a^4*C + 11232*a^2*b^2*C)*Sin[3*(c + d*x)])/9360 + (a*(32*a^2*A*b + 22*A*b^3 + 8*a^3*B + 33*a*b^2*B + 22*a^2*b*C)*Sin[4*(c + d*x)])/77 + (a^2*(89*a^2*A + 312*A*b^2 + 208*a*b*B + 52*a^2*C)*Sin[5*(c + d*x)])/1872 + (a^3*(4*A*b + a*B)*Sin[6*(c + d*x)])/88 + (a^4*A*Sin[7*(c + d*x)])/208))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(11/2))","A",1
1012,0,0,296,84.3433896,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}-\frac{2 a \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d}+\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 b d}",1,"Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]), x]","F",-1
1013,0,0,218,60.5840742,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (b B-a C) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d}-\frac{2 (b B-a C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}",1,"Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]), x]","F",-1
1014,0,0,178,46.4738054,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{2 C \sin (c+d x) \sqrt{\sec (c+d x)}}{b d}-\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}",1,"Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]), x]","F",-1
1015,0,0,157,13.6347917,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),x]","\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])), x]","F",-1
1016,0,0,207,61.7696524,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","-\frac{2 b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)-3 a b B+3 A b^2\right)}{3 a^3 d}+\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}",1,"Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])), x]","F",-1
1017,0,0,269,55.6356729,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),x]","\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","\frac{2 b^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (-B)+a^2 b (A+3 C)-3 a b^2 B+3 A b^3\right)}{3 a^4 d}+\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])), x]","F",-1
1018,0,0,342,69.901876,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + b*Sec[c + d*x])),x]","\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","-\frac{2 b^3 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^5 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x)}{5 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(a^2 (5 A+7 C)-7 a b B+7 A b^2\right)}{21 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^3 B+a^2 b (3 A+5 C)-5 a b^2 B+5 A b^3\right)}{5 a^4 d}+\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)-7 a^3 b B+7 a^2 b^2 (A+3 C)-21 a b^3 B+21 A b^4\right)}{21 a^5 d}+\frac{2 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}",1,"Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*(a + b*Sec[c + d*x])), x]","F",-1
1019,1,926,447,7.4555717,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(45 C a^4-27 b B a^3+9 A b^2 a^2-44 b^2 C a^2+30 b^3 B a-12 A b^4-4 b^4 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(12 B b^4+12 a A b^3-28 a C b^3-24 a^2 B b^2+40 a^3 C b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(15 C a^4-9 b B a^3+3 A b^2 a^2-12 b^2 C a^2+6 b^3 B a\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^2}{6 (a-b) b^3 (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(5 C a^3-3 b B a^2+A b^2 a-4 b^2 C a+2 b^3 B\right) \sin (c+d x)}{b^3 \left(b^2-a^2\right)}-\frac{2 \left(C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a\right)}{b^2 \left(b^2-a^2\right) (b+a \cos (c+d x))}+\frac{4 C \tan (c+d x)}{3 b^2}\right) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}","-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-5 a^4 C+3 a^3 b B-a^2 b^2 (A-7 C)-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(9*a^2*A*b^2 - 12*A*b^4 - 27*a^3*b*B + 30*a*b^3*B + 45*a^4*C - 44*a^2*b^2*C - 4*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(12*a*A*b^3 - 24*a^2*b^2*B + 12*b^4*B + 40*a^3*b*C - 28*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((3*a^2*A*b^2 - 9*a^3*b*B + 6*a*b^3*B + 15*a^4*C - 12*a^2*b^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(6*(a - b)*b^3*(a + b)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(a*A*b^2 - 3*a^2*b*B + 2*b^3*B + 5*a^3*C - 4*a*b^2*C)*Sin[c + d*x])/(b^3*(-a^2 + b^2)) - (2*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (4*C*Tan[c + d*x])/(3*b^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2)","B",0
1020,1,860,363,7.2038972,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(9 C a^3-3 b B a^2-A b^2 a-10 b^2 C a+4 b^3 B\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(4 A b^3-4 C b^3-4 a B b^2+8 a^2 C b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(3 C a^3-b B a^2+A b^2 a-2 b^2 C a\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^2}{2 b^2 (b-a) (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)\right)}{b \left(b^2-a^2\right) (b+a \cos (c+d x))}-\frac{2 \left(3 C a^2-b B a+A b^2-2 b^2 C\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}\right) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^4 C+a^3 b B+a^2 b^2 (A+5 C)-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a-b) (a+b)^2}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-(a*A*b^2) - 3*a^2*b*B + 4*b^3*B + 9*a^3*C - 10*a*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*A*b^3 - 4*a*b^2*B + 8*a^2*b*C - 4*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((a*A*b^2 - a^2*b*B + 3*a^3*C - 2*a*b^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(2*b^2*(-a + b)*(a + b)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) + (2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2)","B",0
1021,1,824,299,7.0369093,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(3 C a^2+b B a-A b^2-4 b^2 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-4 B b^2+4 a A b+4 a C b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(C a^2-b B a+A b^2\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^2}{2 (a-b) b (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(C a^2-b B a+A b^2\right) \sin (c+d x)}{a b \left(b^2-a^2\right)}+\frac{2 \left(C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A+C)\right)+a b B+A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^4 C+a^3 b B-3 a^2 b^2 (A+C)+a b^3 B+A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-(A*b^2) + a*b*B + 3*a^2*C - 4*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A*b - 4*b^2*B + 4*a*b*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((A*b^2 - a*b*B + a^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(2*(a - b)*b*(a + b)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a*b*(-a^2 + b^2)) + (2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2)","B",0
1022,1,830,317,7.1013605,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(-2 A a^2-C a^2+b B a+A b^2\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-4 B a^2+4 A b a+4 b C a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(-2 A a^2+C a^2-b B a+3 A b^2\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^2}{2 a (b-a) (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(C a^2-b B a+A b^2\right) \sin (c+d x)}{a^2 \left(a^2-b^2\right)}-\frac{2 \left(A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b\right)}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A-C)\right)-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2 a^3 B-a^2 b (4 A+C)-a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-2*a^2*A + A*b^2 + a*b*B - a^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A*b - 4*a^2*B + 4*a*b*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-2*a^2*A + 3*A*b^2 - a*b*B + a^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(2*a*(-a + b)*(a + b)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)) - (2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2)","B",0
1023,1,882,406,7.2255273,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(6 B a^3-8 A b a^2-3 b C a^2-3 b^2 B a+5 A b^3\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(4 A a^3+12 C a^3-12 b B a^2+8 A b^2 a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(6 B a^3-12 A b a^2+3 b C a^2-9 b^2 B a+15 A b^3\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^2}{6 a^2 (a-b) (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 b \left(C a^2-b B a+A b^2\right) \sin (c+d x)}{a^3 \left(b^2-a^2\right)}+\frac{2 \left(A \sin (c+d x) b^4-a B \sin (c+d x) b^3+a^2 C \sin (c+d x) b^2\right)}{a^3 \left(a^2-b^2\right) (b+a \cos (c+d x))}+\frac{2 A \sin (2 (c+d x))}{3 a^2}\right) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}","-\frac{\sin (c+d x) \left(-\left(a^2 (2 A-3 C)\right)-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2 a^3 B-a^2 b (4 A-C)-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-2 a^4 (A+3 C)+12 a^3 b B-a^2 b^2 (16 A-3 C)-9 a b^3 B+15 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}+\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^4 C+5 a^3 b B-a^2 b^2 (7 A-C)-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-8*a^2*A*b + 5*A*b^3 + 6*a^3*B - 3*a*b^2*B - 3*a^2*b*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a^3*A + 8*a*A*b^2 - 12*a^2*b*B + 12*a^3*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-12*a^2*A*b + 15*A*b^3 + 6*a^3*B - 9*a*b^2*B + 3*a^2*b*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(6*a^2*(a - b)*(a + b)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*b*(A*b^2 - a*b*B + a^2*C)*Sin[c + d*x])/(a^3*(-a^2 + b^2)) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (2*A*Sin[2*(c + d*x)])/(3*a^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2)","B",0
1024,1,971,507,7.4716333,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(-18 A a^4-30 C a^4+40 b B a^3-32 A b^2 a^2+15 b^2 C a^2-25 b^3 B a+35 A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-20 B a^4+4 A b a^3+60 b C a^3-40 b^2 B a^2+56 A b^3 a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(-18 A a^4-30 C a^4+60 b B a^3-72 A b^2 a^2+45 b^2 C a^2-75 b^3 B a+105 A b^4\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^2}{30 a^3 (b-a) (a+b) d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}+\frac{\sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\left(A a^4-A b^2 a^2+10 b^2 C a^2-10 b^3 B a+10 A b^4\right) \sin (c+d x)}{5 a^4 \left(a^2-b^2\right)}-\frac{2 \left(A \sin (c+d x) b^5-a B \sin (c+d x) b^4+a^2 C \sin (c+d x) b^3\right)}{a^4 \left(a^2-b^2\right) (b+a \cos (c+d x))}+\frac{2 (a B-2 A b) \sin (2 (c+d x))}{3 a^3}+\frac{A \sin (3 (c+d x))}{5 a^2}\right) (b+a \cos (c+d x))^2}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^2}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}-\frac{\sin (c+d x) \left(-\left(a^2 (2 A-5 C)\right)-5 a b B+7 A b^2\right)}{5 a^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(2 a^3 B-a^2 (4 A b-3 b C)-5 a b^2 B+7 A b^3\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-2 a^4 (3 A+5 C)+20 a^3 b B-3 a^2 b^2 (8 A-5 C)-25 a b^3 B+35 A b^4\right)}{5 a^4 d \left(a^2-b^2\right)}-\frac{b^2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-5 a^4 C+7 a^3 b B-3 a^2 b^2 (3 A-C)-5 a b^3 B+7 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^5 d (a-b) (a+b)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2 a^5 B-4 a^4 b (A+3 C)+16 a^3 b^2 B-a^2 b^3 (20 A-9 C)-15 a b^4 B+21 A b^5\right)}{3 a^5 d \left(a^2-b^2\right)}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-18*a^4*A - 32*a^2*A*b^2 + 35*A*b^4 + 40*a^3*b*B - 25*a*b^3*B - 30*a^4*C + 15*a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a^3*A*b + 56*a*A*b^3 - 20*a^4*B - 40*a^2*b^2*B + 60*a^3*b*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-18*a^4*A - 72*a^2*A*b^2 + 105*A*b^4 + 60*a^3*b*B - 75*a*b^3*B - 30*a^4*C + 45*a^2*b^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(30*a^3*(-a + b)*(a + b)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((a^4*A - a^2*A*b^2 + 10*A*b^4 - 10*a*b^3*B + 10*a^2*b^2*C)*Sin[c + d*x])/(5*a^4*(a^2 - b^2)) - (2*(A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (2*(-2*A*b + a*B)*Sin[2*(c + d*x)])/(3*a^3) + (A*Sin[3*(c + d*x)])/(5*a^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^2)","A",0
1025,1,1156,667,7.7647752,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(315 C a^6-135 b B a^5+27 A b^2 a^4-641 b^2 C a^4+285 b^3 B a^3-57 A b^4 a^2+328 b^4 C a^2-168 b^5 B a+48 A b^6+16 b^6 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-48 B b^6-96 a A b^5+160 a C b^5+240 a^2 B b^4+24 a^3 A b^3-512 a^3 C b^3-120 a^4 B b^2+280 a^5 C b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(105 C a^6-45 b B a^5+9 A b^2 a^4-195 b^2 C a^4+87 b^3 B a^3-27 A b^4 a^2+72 b^4 C a^2-24 b^5 B a\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^3}{24 (a-b)^2 b^4 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{\left(35 C a^5-15 b B a^4+3 A b^2 a^3-65 b^2 C a^3+29 b^3 B a^2-9 A b^4 a+24 b^4 C a-8 b^5 B\right) \sin (c+d x)}{2 b^4 \left(a^2-b^2\right)^2}+\frac{-C \sin (c+d x) a^3+b B \sin (c+d x) a^2-A b^2 \sin (c+d x) a}{b^2 \left(b^2-a^2\right) (b+a \cos (c+d x))^2}+\frac{9 C \sin (c+d x) a^5-5 b B \sin (c+d x) a^4+A b^2 \sin (c+d x) a^3-15 b^2 C \sin (c+d x) a^3+11 b^3 B \sin (c+d x) a^2-7 A b^4 \sin (c+d x) a}{2 b^3 \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}+\frac{4 C \tan (c+d x)}{3 b^3}\right) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","-\frac{\sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(-7 a^4 C+3 a^3 b B+a^2 b^2 (A+13 C)-9 a b^3 B+5 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-35 a^4 C+15 a^3 b B-a^2 b^2 (3 A-61 C)-33 a b^3 B+b^4 (21 A-8 C)\right)}{12 b^3 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-35 a^5 C+15 a^4 b B-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+3 a b^4 (3 A-8 C)+8 b^5 B\right)}{4 b^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(35 a^6 C-15 a^5 b B+a^4 b^2 (3 A-86 C)+38 a^3 b^3 B-3 a^2 b^4 (2 A-21 C)-35 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(27*a^4*A*b^2 - 57*a^2*A*b^4 + 48*A*b^6 - 135*a^5*b*B + 285*a^3*b^3*B - 168*a*b^5*B + 315*a^6*C - 641*a^4*b^2*C + 328*a^2*b^4*C + 16*b^6*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(24*a^3*A*b^3 - 96*a*A*b^5 - 120*a^4*b^2*B + 240*a^2*b^4*B - 48*b^6*B + 280*a^5*b*C - 512*a^3*b^3*C + 160*a*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^4*A*b^2 - 27*a^2*A*b^4 - 45*a^5*b*B + 87*a^3*b^3*B - 24*a*b^5*B + 105*a^6*C - 195*a^4*b^2*C + 72*a^2*b^4*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(24*(a - b)^2*b^4*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/2*((3*a^3*A*b^2 - 9*a*A*b^4 - 15*a^4*b*B + 29*a^2*b^3*B - 8*b^5*B + 35*a^5*C - 65*a^3*b^2*C + 24*a*b^4*C)*Sin[c + d*x])/(b^4*(a^2 - b^2)^2) + (-(a*A*b^2*Sin[c + d*x]) + a^2*b*B*Sin[c + d*x] - a^3*C*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (a^3*A*b^2*Sin[c + d*x] - 7*a*A*b^4*Sin[c + d*x] - 5*a^4*b*B*Sin[c + d*x] + 11*a^2*b^3*B*Sin[c + d*x] + 9*a^5*C*Sin[c + d*x] - 15*a^3*b^2*C*Sin[c + d*x])/(2*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (4*C*Tan[c + d*x])/(3*b^3)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","A",0
1026,1,1087,556,7.5488233,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{(b+a \cos (c+d x))^3 \sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\left(15 C a^4-3 b B a^3-A b^2 a^2-29 b^2 C a^2+9 b^3 B a-5 A b^4+8 b^4 C\right) \sin (c+d x)}{2 b^3 \left(b^2-a^2\right)^2}+\frac{C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)}{b \left(b^2-a^2\right) (b+a \cos (c+d x))^2}+\frac{-5 C \sin (c+d x) a^4+b B \sin (c+d x) a^3+3 A b^2 \sin (c+d x) a^2+11 b^2 C \sin (c+d x) a^2-7 b^3 B \sin (c+d x) a+3 A b^4 \sin (c+d x)}{2 b^2 \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}-\frac{(b+a \cos (c+d x))^3 \sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(45 C a^5-9 b B a^4-3 A b^2 a^3-95 b^2 C a^3+19 b^3 B a^2+9 A b^4 a+56 b^4 C a-16 b^5 B\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-16 A b^5+16 C b^5+32 a B b^4-8 a^2 A b^3-80 a^2 C b^3-8 a^3 B b^2+40 a^4 C b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(15 C a^5-3 b B a^4-A b^2 a^3-29 b^2 C a^3+9 b^3 B a^2-5 A b^4 a+8 b^4 C a\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right)}{8 (a-b)^2 b^3 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","-\frac{\sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"-1/8*((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-3*a^3*A*b^2 + 9*a*A*b^4 - 9*a^4*b*B + 19*a^2*b^3*B - 16*b^5*B + 45*a^5*C - 95*a^3*b^2*C + 56*a*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-8*a^2*A*b^3 - 16*A*b^5 - 8*a^3*b^2*B + 32*a*b^4*B + 40*a^4*b*C - 80*a^2*b^3*C + 16*b^5*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-(a^3*A*b^2) - 5*a*A*b^4 - 3*a^4*b*B + 9*a^2*b^3*B + 15*a^5*C - 29*a^3*b^2*C + 8*a*b^4*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/((a - b)^2*b^3*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-(a^2*A*b^2) - 5*A*b^4 - 3*a^3*b*B + 9*a*b^3*B + 15*a^4*C - 29*a^2*b^2*C + 8*b^4*C)*Sin[c + d*x])/(2*b^3*(-a^2 + b^2)^2) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (3*a^2*A*b^2*Sin[c + d*x] + 3*A*b^4*Sin[c + d*x] + a^3*b*B*Sin[c + d*x] - 7*a*b^3*B*Sin[c + d*x] - 5*a^4*C*Sin[c + d*x] + 11*a^2*b^2*C*Sin[c + d*x])/(2*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","A",0
1027,1,1046,469,7.2878438,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(9 C a^4+3 b B a^3+A b^2 a^2-19 b^2 C a^2-9 b^3 B a+5 A b^4+16 b^4 C\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 B b^4-24 a A b^3-32 a C b^3+8 a^2 B b^2+8 a^3 C b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(3 C a^4+b B a^3-5 A b^2 a^2-9 b^2 C a^2+5 b^3 B a-A b^4\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^3}{8 (a-b)^2 b^2 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\left(-3 C a^4-b B a^3+5 A b^2 a^2+9 b^2 C a^2-5 b^3 B a+A b^4\right) \sin (c+d x)}{2 a b^2 \left(b^2-a^2\right)^2}+\frac{C \sin (c+d x) a^2-b B \sin (c+d x) a+A b^2 \sin (c+d x)}{a \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{C \sin (c+d x) a^4+3 b B \sin (c+d x) a^3-7 A b^2 \sin (c+d x) a^2-7 b^2 C \sin (c+d x) a^2+3 b^3 B \sin (c+d x) a+A b^4 \sin (c+d x)}{2 a b \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","-\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(a^2*A*b^2 + 5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B + 9*a^4*C - 19*a^2*b^2*C + 16*b^4*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a*A*b^3 + 8*a^2*b^2*B + 16*b^4*B + 8*a^3*b*C - 32*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-5*a^2*A*b^2 - A*b^4 + a^3*b*B + 5*a*b^3*B + 3*a^4*C - 9*a^2*b^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(8*(a - b)^2*b^2*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((5*a^2*A*b^2 + A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + 9*a^2*b^2*C)*Sin[c + d*x])/(2*a*b^2*(-a^2 + b^2)^2) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x])/(a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (-7*a^2*A*b^2*Sin[c + d*x] + A*b^4*Sin[c + d*x] + 3*a^3*b*B*Sin[c + d*x] + 3*a*b^3*B*Sin[c + d*x] + a^4*C*Sin[c + d*x] - 7*a^2*b^2*C*Sin[c + d*x])/(2*a*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","B",0
1028,1,1046,478,7.2887534,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(3 C a^4+b B a^3-5 A b^2 a^2-9 b^2 C a^2+5 b^3 B a-A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 A b a^3+8 b C a^3-24 b^2 B a^2+8 A b^3 a+16 b^3 C a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(C a^4-5 b B a^3+9 A b^2 a^2+5 b^2 C a^2-b^3 B a-3 A b^4\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^3}{8 a (a-b)^2 b (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\left(-C a^4+5 b B a^3-9 A b^2 a^2-5 b^2 C a^2+b^3 B a+3 A b^4\right) \sin (c+d x)}{2 a^2 b \left(b^2-a^2\right)^2}-\frac{A \sin (c+d x) b^3-a B \sin (c+d x) b^2+a^2 C \sin (c+d x) b}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{3 C \sin (c+d x) a^4-7 b B \sin (c+d x) a^3+11 A b^2 \sin (c+d x) a^2+3 b^2 C \sin (c+d x) a^2+b^3 B \sin (c+d x) a-5 A b^4 \sin (c+d x)}{2 a^2 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-5*a^2*A*b^2 - A*b^4 + a^3*b*B + 5*a*b^3*B + 3*a^4*C - 9*a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^3*A*b + 8*a*A*b^3 - 24*a^2*b^2*B + 8*a^3*b*C + 16*a*b^3*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^2*A*b^2 - 3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(8*a*(a - b)^2*b*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((-9*a^2*A*b^2 + 3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - 5*a^2*b^2*C)*Sin[c + d*x])/(2*a^2*b*(-a^2 + b^2)^2) - (A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x])/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (11*a^2*A*b^2*Sin[c + d*x] - 5*A*b^4*Sin[c + d*x] - 7*a^3*b*B*Sin[c + d*x] + a*b^3*B*Sin[c + d*x] + 3*a^4*C*Sin[c + d*x] + 3*a^2*b^2*C*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","B",0
1029,1,1059,486,7.3901647,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(8 A a^4+C a^4-5 b B a^3-7 A b^2 a^2+5 b^2 C a^2-b^3 B a+5 A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 B a^4-32 A b a^3-24 b C a^3+8 b^2 B a^2+8 A b^3 a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(8 A a^4-5 C a^4+9 b B a^3-29 A b^2 a^2-b^2 C a^2-3 b^3 B a+15 A b^4\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^3}{8 a^2 (a-b)^2 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{\left(-5 C a^4+9 b B a^3-13 A b^2 a^2-b^2 C a^2-3 b^3 B a+7 A b^4\right) \sin (c+d x)}{2 a^3 \left(b^2-a^2\right)^2}-\frac{-A \sin (c+d x) b^4+a B \sin (c+d x) b^3-a^2 C \sin (c+d x) b^2}{a^3 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{9 A \sin (c+d x) b^5-5 a B \sin (c+d x) b^4-15 a^2 A \sin (c+d x) b^3+a^2 C \sin (c+d x) b^3+11 a^3 B \sin (c+d x) b^2-7 a^4 C \sin (c+d x) b}{2 a^3 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-3 a^4 C+7 a^3 b B-a^2 b^2 (11 A+3 C)-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-8 a^5 B+a^4 b (24 A+7 C)+5 a^3 b^2 B-a^2 b^3 (33 A+C)-3 a b^4 B+15 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(3 a^6 C-15 a^5 b B+5 a^4 b^2 (7 A+2 C)+6 a^3 b^3 B-a^2 b^4 (38 A+C)-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(8*a^4*A - 7*a^2*A*b^2 + 5*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + 5*a^2*b^2*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-32*a^3*A*b + 8*a*A*b^3 + 16*a^4*B + 8*a^2*b^2*B - 24*a^3*b*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(8*a^2*(a - b)^2*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/2*((-13*a^2*A*b^2 + 7*A*b^4 + 9*a^3*b*B - 3*a*b^3*B - 5*a^4*C - a^2*b^2*C)*Sin[c + d*x])/(a^3*(-a^2 + b^2)^2) - (-(A*b^4*Sin[c + d*x]) + a*b^3*B*Sin[c + d*x] - a^2*b^2*C*Sin[c + d*x])/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (-15*a^2*A*b^3*Sin[c + d*x] + 9*A*b^5*Sin[c + d*x] + 11*a^3*b^2*B*Sin[c + d*x] - 5*a*b^4*B*Sin[c + d*x] - 7*a^4*b*C*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","B",0
1030,1,1116,598,7.6290205,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\sec (c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(24 B a^5-56 A b a^4-15 b C a^4-21 b^2 B a^3+73 A b^3 a^2-3 b^3 C a^2+15 b^4 B a-35 A b^5\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 A a^5+48 C a^5-96 b B a^4+112 A b^2 a^3+24 b^2 C a^3+24 b^3 B a^2-56 A b^4 a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(24 B a^5-72 A b a^4+27 b C a^4-87 b^2 B a^3+195 A b^3 a^2-9 b^3 C a^2+45 b^4 B a-105 A b^5\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^3}{24 a^3 (a-b)^2 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{b \left(-9 C a^4+13 b B a^3-17 A b^2 a^2+3 b^2 C a^2-7 b^3 B a+11 A b^4\right) \sin (c+d x)}{2 a^4 \left(b^2-a^2\right)^2}-\frac{A \sin (c+d x) b^5-a B \sin (c+d x) b^4+a^2 C \sin (c+d x) b^3}{a^4 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{-13 A \sin (c+d x) b^6+9 a B \sin (c+d x) b^5+19 a^2 A \sin (c+d x) b^4-5 a^2 C \sin (c+d x) b^4-15 a^3 B \sin (c+d x) b^3+11 a^4 C \sin (c+d x) b^2}{2 a^4 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}+\frac{2 A \sin (2 (c+d x))}{3 a^3}\right) (b+a \cos (c+d x))^3}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^3}","\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}+\frac{\sin (c+d x) \left(a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{\sin (c+d x) \left(-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}-\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-8 a^5 B+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-56*a^4*A*b + 73*a^2*A*b^3 - 35*A*b^5 + 24*a^5*B - 21*a^3*b^2*B + 15*a*b^4*B - 15*a^4*b*C - 3*a^2*b^3*C)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^5*A + 112*a^3*A*b^2 - 56*a*A*b^4 - 96*a^4*b*B + 24*a^2*b^3*B + 48*a^5*C + 24*a^3*b^2*C)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-72*a^4*A*b + 195*a^2*A*b^3 - 105*A*b^5 + 24*a^5*B - 87*a^3*b^2*B + 45*a*b^4*B + 27*a^4*b*C - 9*a^2*b^3*C)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(24*a^3*(a - b)^2*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((b*(-17*a^2*A*b^2 + 11*A*b^4 + 13*a^3*b*B - 7*a*b^3*B - 9*a^4*C + 3*a^2*b^2*C)*Sin[c + d*x])/(2*a^4*(-a^2 + b^2)^2) - (A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x])/(a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (19*a^2*A*b^4*Sin[c + d*x] - 13*A*b^6*Sin[c + d*x] - 15*a^3*b^3*B*Sin[c + d*x] + 9*a*b^5*B*Sin[c + d*x] + 11*a^4*b^2*C*Sin[c + d*x] - 5*a^2*b^4*C*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + (2*A*Sin[2*(c + d*x)])/(3*a^3)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^3)","A",0
1031,1,782,447,6.9386217,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{\sec (c+d x) \left(-3 a^2 C \sin (c+d x)+6 a b B \sin (c+d x)+24 A b^2 \sin (c+d x)+16 b^2 C \sin (c+d x)\right)}{12 b^2}+\frac{\sec ^2(c+d x) (a C \sin (c+d x)+6 b B \sin (c+d x))}{6 b}+\frac{2}{3} C \tan (c+d x) \sec ^2(c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(4 a^2 b C+24 a b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(3 a^3 C-6 a^2 b B-24 a A b^2-16 a b^2 C\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(9 a^3 C-18 a^2 b B+24 a A b^2+8 a b^2 C+48 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{48 b^2 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \cos (c+d x)+b} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{24 b^2 d}+\frac{\sqrt{\sec (c+d x)} \left(a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 b d \sqrt{a+b \sec (c+d x)}}-\frac{\left(-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 b^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{\sqrt{\sec (c+d x)} \left(a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{(a C+6 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{12 b d}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(24*a*b^2*B + 4*a^2*b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(24*a*A*b^2 - 18*a^2*b*B + 48*b^3*B + 9*a^3*C + 8*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(-24*a*A*b^2 - 6*a^2*b*B + 3*a^3*C - 16*a*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(48*b^2*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) + (Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]^2*(6*b*B*Sin[c + d*x] + a*C*Sin[c + d*x]))/(6*b) + (Sec[c + d*x]*(24*A*b^2*Sin[c + d*x] + 6*a*b*B*Sin[c + d*x] - 3*a^2*C*Sin[c + d*x] + 16*b^2*C*Sin[c + d*x]))/(12*b^2) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/3))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))","C",0
1032,1,478,346,6.3922591,"\int \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(-3 a^2 C+4 a b B+16 A b^2+8 b^2 C\right) \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b (a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{8 a (4 A+C) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 i (a C+4 b B) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b^2 \sqrt{\frac{1}{a-b}} \sqrt{a \cos (c+d x)+b}}+\frac{4 (a C+4 b B) \tan (c+d x)}{b}+8 C \tan (c+d x) \sec (c+d x)\right)}{8 d \sec ^{\frac{5}{2}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\sqrt{\sec (c+d x)} \left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} (8 a A+3 a C+4 b B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 b d}-\frac{(a C+4 b B) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*a*(4*A + C)*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(16*A*b^2 + 4*a*b*B - 3*a^2*C + 8*b^2*C)*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - ((2*I)*(4*b*B + a*C)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b^2*Sqrt[b + a*Cos[c + d*x]]) + (4*(4*b*B + a*C)*Tan[c + d*x])/b + 8*C*Sec[c + d*x]*Tan[c + d*x]))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2))","C",0
1033,1,438,258,3.6798458,"\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 (a (2 A+C)+4 b B) \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{8 (a B+A b) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 i (2 A-C) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}} \sqrt{a \cos (c+d x)+b}}+4 C \tan (c+d x)\right)}{2 d \sec ^{\frac{5}{2}}(c+d x) (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(2 A-C) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 a B+b C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}",1,"(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(A*b + a*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(4*b*B + a*(2*A + C))*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((2*I)*(2*A - C)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b*Sqrt[b + a*Cos[c + d*x]]) + 4*C*Tan[c + d*x]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(5/2))","C",1
1034,0,0,277,33.3943188,"\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","-\frac{2 \left(A b^2-a^2 (A+3 C)\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 (3 a B+A b) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2), x]","F",-1
1035,1,3426,273,6.6408003,"\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\text{Result too large to show}","-\frac{2 \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \left(a^2-b^2\right) (2 A b-5 a B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 (5 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(9*a^2*A - 2*A*b^2 + 5*a*b*B + 15*a^2*C)*Cot[c])/(15*a^2*d) + (4*(A*b + 5*a*B)*Cos[d*x]*Sin[c])/(15*a*d) + (2*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*(A*b + 5*a*B)*Cos[c]*Sin[d*x])/(15*a*d) + (2*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (28*A*b*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(15*a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (4*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (4*b*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (6*a*A*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) + (4*A*b^2*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (2*b*B*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (2*a*C*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))","C",0
1036,1,4441,360,6.8227106,"\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \left(25 a^2 A+35 a^2 C-14 a b B+8 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(63 a^3 B+a^2 b (19 A+35 C)-14 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B + 35*a^2*b*C)*Cot[c])/(105*a^3*d) + ((115*a^2*A - 16*A*b^2 + 28*a*b*B + 140*a^2*C)*Cos[d*x]*Sin[c])/(105*a^2*d) + (2*(A*b + 7*a*B)*Cos[2*d*x]*Sin[2*c])/(35*a*d) + (A*Cos[3*d*x]*Sin[3*c])/(7*d) + ((115*a^2*A - 16*A*b^2 + 28*a*b*B + 140*a^2*C)*Cos[c]*Sin[d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Cos[2*c]*Sin[2*d*x])/(35*a*d) + (A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (20*A*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (8*A*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(105*a^2*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (28*b*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(15*a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (4*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(5/2)) - (38*A*b*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(105*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (16*A*b^3*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(105*a^2*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (6*a*B*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) + (4*b^2*B*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*a*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2)) - (2*b*C*Csc[c]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*d*Sqrt[b + a*Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(5/2))","C",0
1037,1,5993,457,7.0396277,"\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right) \sqrt{a+b \sec (c+d x)}}{315 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)}}{315 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \left(-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^4 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (9 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{63 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"Result too large to show","C",0
1038,1,916,551,7.1751063,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(12 b C a^3+224 b^2 B a^2+192 A b^3 a+144 b^3 C a\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 \left(27 C a^4-72 b B a^3+48 A b^2 a^2-12 b^2 C a^2+448 b^3 B a+384 A b^4+288 b^4 C\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 i \left(9 C a^4-24 b B a^3-240 A b^2 a^2-156 b^2 C a^2-128 b^3 B a\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{\cos (c+d x) a+a}{a-b}} \cos (2 (c+d x)) \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right) \sin (c+d x)}{\sqrt{\frac{1}{a-b}} b \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 b^2+2 (b+a \cos (c+d x))^2-4 b (b+a \cos (c+d x))\right)}\right) (a+b \sec (c+d x))^{3/2}}{384 b^2 d (b+a \cos (c+d x))^{3/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{1}{12} (8 b B \sin (c+d x)+9 a C \sin (c+d x)) \sec ^3(c+d x)+\frac{1}{2} b C \tan (c+d x) \sec ^3(c+d x)+\frac{\left(3 C \sin (c+d x) a^2+56 b B \sin (c+d x) a+48 A b^2 \sin (c+d x)+36 b^2 C \sin (c+d x)\right) \sec ^2(c+d x)}{48 b}+\frac{\left(-9 C \sin (c+d x) a^3+24 b B \sin (c+d x) a^2+240 A b^2 \sin (c+d x) a+156 b^2 C \sin (c+d x) a+128 b^3 B \sin (c+d x)\right) \sec (c+d x)}{96 b^2}\right) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(3 a^2 C+56 a b B+48 A b^2+36 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{96 b d}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{192 b^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-3 a^3 C+136 a^2 b B+12 a b^2 (28 A+19 C)+128 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 b d \sqrt{a+b \sec (c+d x)}}-\frac{\left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 b^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{\sqrt{\sec (c+d x)} \left(-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{64 b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{(3 a C+8 b B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{C \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}",1,"((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(192*a*A*b^3 + 224*a^2*b^2*B + 12*a^3*b*C + 144*a*b^3*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(48*a^2*A*b^2 + 384*A*b^4 - 72*a^3*b*B + 448*a*b^3*B + 27*a^4*C - 12*a^2*b^2*C + 288*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(-240*a^2*A*b^2 - 24*a^3*b*B - 128*a*b^3*B + 9*a^4*C - 156*a^2*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(384*b^2*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + ((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]^3*(8*b*B*Sin[c + d*x] + 9*a*C*Sin[c + d*x]))/12 + (Sec[c + d*x]^2*(48*A*b^2*Sin[c + d*x] + 56*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 36*b^2*C*Sin[c + d*x]))/(48*b) + (Sec[c + d*x]*(240*a*A*b^2*Sin[c + d*x] + 24*a^2*b*B*Sin[c + d*x] + 128*b^3*B*Sin[c + d*x] - 9*a^3*C*Sin[c + d*x] + 156*a*b^2*C*Sin[c + d*x]))/(96*b^2) + (b*C*Sec[c + d*x]^3*Tan[c + d*x])/2))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",0
1039,1,800,446,6.9795248,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{1}{6} (6 b B \sin (c+d x)+7 a C \sin (c+d x)) \sec ^2(c+d x)+\frac{2}{3} b C \tan (c+d x) \sec ^2(c+d x)+\frac{\left(3 C \sin (c+d x) a^2+30 b B \sin (c+d x) a+24 A b^2 \sin (c+d x)+16 b^2 C \sin (c+d x)\right) \sec (c+d x)}{12 b}\right)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}-\frac{(a+b \sec (c+d x))^{3/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(-96 A b a^2-28 b C a^2-24 b^2 B a\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 \left(9 C a^3-6 b B a^2-120 A b^2 a-56 b^2 C a-48 b^3 B\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 i \left(3 C a^3+30 b B a^2+24 A b^2 a+16 b^2 C a\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{\cos (c+d x) a+a}{a-b}} \cos (2 (c+d x)) \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right) \sin (c+d x)}{\sqrt{\frac{1}{a-b}} b \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 b^2+2 (b+a \cos (c+d x))^2-4 b (b+a \cos (c+d x))\right)}\right)}{48 b d (b+a \cos (c+d x))^{3/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{7}{2}}(c+d x)}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{24 b d}+\frac{\sqrt{\sec (c+d x)} \left(a^2 (48 A+17 C)+42 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{a+b \sec (c+d x)}}-\frac{\left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sqrt{\sec (c+d x)} \left(a^3 (-C)+6 a^2 b B+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 b d \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}",1,"-1/48*((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-96*a^2*A*b - 24*a*b^2*B - 28*a^2*b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(-120*a*A*b^2 - 6*a^2*b*B - 48*b^3*B + 9*a^3*C - 56*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(24*a*A*b^2 + 30*a^2*b*B + 3*a^3*C + 16*a*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(b*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + ((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]^2*(6*b*B*Sin[c + d*x] + 7*a*C*Sin[c + d*x]))/6 + (Sec[c + d*x]*(24*A*b^2*Sin[c + d*x] + 30*a*b*B*Sin[c + d*x] + 3*a^2*C*Sin[c + d*x] + 16*b^2*C*Sin[c + d*x]))/(12*b) + (2*b*C*Sec[c + d*x]^2*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",0
1040,1,709,353,7.0130902,"\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{1}{2} \sec (c+d x) (5 a C \sin (c+d x)+4 b B \sin (c+d x))+b C \tan (c+d x) \sec (c+d x)\right)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(8 a^2 A+a^2 C+20 a b B+16 A b^2+8 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(8 a^2 A-5 a^2 C-4 a b B\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(16 a^2 B+32 a A b+4 a b C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{8 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{\sqrt{\sec (c+d x)} \left(8 a^2 B+a b (8 A+7 C)+4 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{(8 a A-5 a C-4 b B) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(3 a C+4 b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}",1,"((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(32*a*A*b + 16*a^2*B + 4*a*b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(8*a^2*A + 16*A*b^2 + 20*a*b*B + a^2*C + 8*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(8*a^2*A - 4*a*b*B - 5*a^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(8*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + ((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]*(4*b*B*Sin[c + d*x] + 5*a*C*Sin[c + d*x]))/2 + b*C*Sec[c + d*x]*Tan[c + d*x]))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",0
1041,1,685,340,6.9072774,"\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4}{3} a A \sin (c+d x)+2 b C \tan (c+d x)\right)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b) (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(4 a^2 A+12 a^2 C+24 a b B+12 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 \left(6 a^2 B+8 a A b+15 a b C+12 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(6 a^2 B+8 a A b-3 a b C\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}\right)}{6 d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{\sqrt{\sec (c+d x)} \left(2 a^2 (A+3 C)+6 a b B-b^2 (2 A-3 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{a+b \sec (c+d x)}}+\frac{(6 a B+8 A b-3 b C) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{b (2 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{b (3 a C+2 b B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(4*a^2*A + 12*A*b^2 + 24*a*b*B + 12*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(8*a*A*b + 6*a^2*B + 12*b^2*B + 15*a*b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(8*a*A*b + 6*a^2*B - 3*a*b*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(6*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + ((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*a*A*Sin[c + d*x])/3 + 2*b*C*Tan[c + d*x]))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",0
1042,0,0,356,40.690396,"\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","\frac{2 \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \sqrt{\sec (c+d x)} \left(-5 a^3 B-3 a^2 b (A+5 C)+5 a b^2 B+3 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2), x]","F",-1
1043,1,4862,359,6.8584044,"\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 a d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \left(25 a^2 A+35 a^2 C+21 a b B-6 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-63 a^3 B-2 a^2 b (41 A+70 C)-21 a b^2 B+6 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+3 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 140*a^2*b*C)*Cot[c])/(105*a^2*d) + ((115*a^2*A + 12*A*b^2 + 168*a*b*B + 140*a^2*C)*Cos[d*x]*Sin[c])/(105*a*d) + (2*(8*A*b + 7*a*B)*Cos[2*d*x]*Sin[2*c])/(35*d) + (a*A*Cos[3*d*x]*Sin[3*c])/(7*d) + ((115*a^2*A + 12*A*b^2 + 168*a*b*B + 140*a^2*C)*Cos[c]*Sin[d*x])/(105*a*d) + (2*(8*A*b + 7*a*B)*Cos[2*c]*Sin[2*d*x])/(35*d) + (a*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) - (20*a*A*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(7/2)) - (68*A*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(35*a*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(7/2)) - (16*b*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(5*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(7/2)) - (4*a*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(7/2)) - (4*b^2*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(a*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(7/2)) - (164*a*A*b*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(105*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) + (4*A*b^3*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(35*a*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) - (6*a^2*B*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) - (2*b^2*B*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)) - (8*a*b*C*Csc[c]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*d*(b + a*Cos[c + d*x])^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2))","C",0
1044,1,5997,455,7.0660084,"\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)}}{315 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right) \sqrt{a+b \sec (c+d x)}}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \left(75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (3 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{21 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"Result too large to show","C",0
1045,1,925,550,7.2361655,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{1}{12} \left(8 B \sin (c+d x) b^2+17 a C \sin (c+d x) b\right) \sec ^3(c+d x)+\frac{1}{2} b^2 C \tan (c+d x) \sec ^3(c+d x)+\frac{1}{48} \left(59 C \sin (c+d x) a^2+104 b B \sin (c+d x) a+48 A b^2 \sin (c+d x)+36 b^2 C \sin (c+d x)\right) \sec ^2(c+d x)+\frac{\left(15 C \sin (c+d x) a^3+264 b B \sin (c+d x) a^2+432 A b^2 \sin (c+d x) a+284 b^2 C \sin (c+d x) a+128 b^3 B \sin (c+d x)\right) \sec (c+d x)}{96 b}\right)}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}-\frac{(a+b \sec (c+d x))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(-768 A b a^3-236 b C a^3-416 b^2 B a^2-192 A b^3 a-144 b^3 C a\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 \left(45 C a^4+24 b B a^3-1008 A b^2 a^2-436 b^2 C a^2-832 b^3 B a-384 A b^4-288 b^4 C\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 i \left(15 C a^4+264 b B a^3+432 A b^2 a^2+284 b^2 C a^2+128 b^3 B a\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{\cos (c+d x) a+a}{a-b}} \cos (2 (c+d x)) \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right) \sin (c+d x)}{\sqrt{\frac{1}{a-b}} b \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 b^2+2 (b+a \cos (c+d x))^2-4 b (b+a \cos (c+d x))\right)}\right)}{384 b d (b+a \cos (c+d x))^{5/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{32 d}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{192 b d}+\frac{\sqrt{\sec (c+d x)} \left(a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 d \sqrt{a+b \sec (c+d x)}}-\frac{\left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sqrt{\sec (c+d x)} \left(-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{64 b d \sqrt{a+b \sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{24 d}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{4 d}",1,"-1/384*((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-768*a^3*A*b - 192*a*A*b^3 - 416*a^2*b^2*B - 236*a^3*b*C - 144*a*b^3*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(-1008*a^2*A*b^2 - 384*A*b^4 + 24*a^3*b*B - 832*a*b^3*B + 45*a^4*C - 436*a^2*b^2*C - 288*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(432*a^2*A*b^2 + 264*a^3*b*B + 128*a*b^3*B + 15*a^4*C + 284*a^2*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(b*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + ((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]^3*(8*b^2*B*Sin[c + d*x] + 17*a*b*C*Sin[c + d*x]))/12 + (Sec[c + d*x]^2*(48*A*b^2*Sin[c + d*x] + 104*a*b*B*Sin[c + d*x] + 59*a^2*C*Sin[c + d*x] + 36*b^2*C*Sin[c + d*x]))/48 + (Sec[c + d*x]*(432*a*A*b^2*Sin[c + d*x] + 264*a^2*b*B*Sin[c + d*x] + 128*b^3*B*Sin[c + d*x] + 15*a^3*C*Sin[c + d*x] + 284*a*b^2*C*Sin[c + d*x]))/(96*b) + (b^2*C*Sec[c + d*x]^3*Tan[c + d*x])/2))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",0
1046,1,817,453,7.2994954,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(96 B a^3+288 A b a^2+52 b C a^2+24 b^2 B a\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 \left(48 A a^3-3 C a^3+126 b B a^2+216 A b^2 a+104 b^2 C a+48 b^3 B\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 i \left(48 A a^3-33 C a^3-54 b B a^2-24 A b^2 a-16 b^2 C a\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{\cos (c+d x) a+a}{a-b}} \cos (2 (c+d x)) \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right) \sin (c+d x)}{\sqrt{\frac{1}{a-b}} b \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 b^2+2 (b+a \cos (c+d x))^2-4 b (b+a \cos (c+d x))\right)}\right) (a+b \sec (c+d x))^{5/2}}{48 d (b+a \cos (c+d x))^{5/2} (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{1}{6} \left(6 B \sin (c+d x) b^2+13 a C \sin (c+d x) b\right) \sec ^2(c+d x)+\frac{2}{3} b^2 C \tan (c+d x) \sec ^2(c+d x)+\frac{1}{12} \left(33 C \sin (c+d x) a^2+54 b B \sin (c+d x) a+24 A b^2 \sin (c+d x)+16 b^2 C \sin (c+d x)\right) \sec (c+d x)\right) (a+b \sec (c+d x))^{5/2}}{d (b+a \cos (c+d x))^2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{24 d}-\frac{\left(-\left(a^2 (48 A-33 C)\right)+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sqrt{\sec (c+d x)} \left(48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \left(5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{a+b \sec (c+d x)}}+\frac{(5 a C+6 b B) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{12 d}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}",1,"((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(288*a^2*A*b + 96*a^3*B + 24*a*b^2*B + 52*a^2*b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(48*a^3*A + 216*a*A*b^2 + 126*a^2*b*B + 48*b^3*B - 3*a^3*C + 104*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(48*a^3*A - 24*a*A*b^2 - 54*a^2*b*B - 33*a^3*C - 16*a*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(48*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + ((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Sec[c + d*x]^2*(6*b^2*B*Sin[c + d*x] + 13*a*b*C*Sin[c + d*x]))/6 + (Sec[c + d*x]*(24*A*b^2*Sin[c + d*x] + 54*a*b*B*Sin[c + d*x] + 33*a^2*C*Sin[c + d*x] + 16*b^2*C*Sin[c + d*x]))/12 + (2*b^2*C*Sec[c + d*x]^2*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",0
1047,1,766,427,7.1243107,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4}{3} a^2 A \sin (c+d x)+\frac{1}{2} \sec (c+d x) \left(9 a b C \sin (c+d x)+4 b^2 B \sin (c+d x)\right)+b^2 C \tan (c+d x) \sec (c+d x)\right)}{d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(16 a^3 A+48 a^3 C+144 a^2 b B+144 a A b^2+12 a b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(24 a^3 B+56 a^2 A b-27 a^2 b C-12 a b^2 B\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(24 a^3 B+56 a^2 A b+63 a^2 b C+108 a b^2 B+48 A b^3+24 b^3 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{24 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{\left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{12 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b \sqrt{\sec (c+d x)} \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\sec (c+d x)} \left(8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{12 d \sqrt{a+b \sec (c+d x)}}-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} (8 a A-21 a C-12 b B) \sqrt{a+b \sec (c+d x)}}{12 d}-\frac{b (4 A-3 C) \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{6 d}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt{\sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(16*a^3*A + 144*a*A*b^2 + 144*a^2*b*B + 48*a^3*C + 12*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(56*a^2*A*b + 48*A*b^3 + 24*a^3*B + 108*a*b^2*B + 63*a^2*b*C + 24*b^3*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(56*a^2*A*b + 24*a^3*B - 12*a*b^2*B - 27*a^2*b*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(24*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + ((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*a^2*A*Sin[c + d*x])/3 + (Sec[c + d*x]*(4*b^2*B*Sin[c + d*x] + 9*a*b*C*Sin[c + d*x]))/2 + b^2*C*Sec[c + d*x]*Tan[c + d*x]))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",0
1048,1,755,419,7.1536423,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2}{5} a^2 A \sin (2 (c+d x))+\frac{4}{15} a (5 a B+11 A b) \sin (c+d x)+2 b^2 C \tan (c+d x)\right)}{d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(18 a^3 A+30 a^3 C+70 a^2 b B+46 a A b^2-15 a b^2 C\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(20 a^3 B+68 a^2 A b+180 a^2 b C+180 a b^2 B+60 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 \left(18 a^3 A+30 a^3 C+70 a^2 b B+46 a A b^2+135 a b^2 C+60 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{30 d \sec ^{\frac{9}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{\left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\sqrt{\sec (c+d x)} \left(10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{a+b \sec (c+d x)}}-\frac{b \sin (c+d x) \sqrt{\sec (c+d x)} (10 a B+16 A b-15 b C) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 (a B+A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{b^2 (5 a C+2 b B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(68*a^2*A*b + 60*A*b^3 + 20*a^3*B + 180*a*b^2*B + 180*a^2*b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(18*a^3*A + 46*a*A*b^2 + 70*a^2*b*B + 60*b^3*B + 30*a^3*C + 135*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(18*a^3*A + 46*a*A*b^2 + 70*a^2*b*B + 30*a^3*C - 15*a*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(30*d*(b + a*Cos[c + d*x])^(5/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)) + ((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*a*(11*A*b + 5*a*B)*Sin[c + d*x])/15 + (2*a^2*A*Sin[2*(c + d*x)])/5 + 2*b^2*C*Tan[c + d*x]))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","C",0
1049,0,0,441,54.931395,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \left(63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \sqrt{\sec (c+d x)} \left(-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 (7 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 b^3 C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2), x]","F",-1
1050,1,6410,452,7.1040901,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \sec (c+d x)}}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right) \sqrt{a+b \sec (c+d x)}}{315 a d \sqrt{\sec (c+d x)}}-\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \left(-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (9 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"Result too large to show","C",0
1051,1,7479,565,7.3559769,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right) \sqrt{a+b \sec (c+d x)}}{231 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right) \sqrt{a+b \sec (c+d x)}}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \left(75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (11 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"Result too large to show","C",0
1052,1,503,350,5.3408383,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(9 a^2 C-12 a b B+16 A b^2+8 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2}+\frac{2 i (3 a C-4 b B) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \sqrt{a \cos (c+d x)+b} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b^3 \sqrt{\frac{1}{a-b}}}-\frac{4 a (3 a C-4 b B) \sin (c+d x)}{b^2}+\frac{4 (4 b B-3 a C) \tan (c+d x)}{b}+\frac{8 a C \tan (c+d x)}{b}+\frac{8 a C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b}+8 C \tan (c+d x) \sec (c+d x)\right)}{8 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{\sqrt{\sec (c+d x)} \left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 b^2 d}-\frac{(4 b B-3 a C) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(4 b B-a C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{a+b \sec (c+d x)}}+\frac{C \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d}",1,"((A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*a*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/b + (2*(16*A*b^2 - 12*a*b*B + 9*a^2*C + 8*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/b^2 + ((2*I)*(-4*b*B + 3*a*C)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Sqrt[b + a*Cos[c + d*x]]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b^3) - (4*a*(-4*b*B + 3*a*C)*Sin[c + d*x])/b^2 + (8*a*C*Tan[c + d*x])/b + (4*(4*b*B - 3*a*C)*Tan[c + d*x])/b + 8*C*Sec[c + d*x]*Tan[c + d*x]))/(8*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]])","C",1
1053,1,623,260,6.7189551,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 C \sin (c+d x) (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{b d \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{\sqrt{a \cos (c+d x)+b} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{2 i a C \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{8 A b \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 (4 b B-3 a C) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{2 b d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{(2 A+C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{(2 b B-a C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b d}-\frac{C \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(2*C*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sin[c + d*x])/(b*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (Sqrt[b + a*Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*A*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(4*b*B - 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] - ((2*I)*a*C*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(2*b*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]])","C",0
1054,0,0,219,17.9252108,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","-\frac{2 (A b-a B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]), x]","F",-1
1055,1,1959,216,6.5860891,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","-\frac{4 A F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{\csc (c) \left(b-a \sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)\right)}{a \sqrt{\cot ^2(c)+1} \left(\frac{b \csc (c)}{a \sqrt{\cot ^2(c)+1}}+1\right)},\frac{\csc (c) \left(b-a \sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)\right)}{a \sqrt{\cot ^2(c)+1} \left(\frac{b \csc (c)}{a \sqrt{\cot ^2(c)+1}}-1\right)}\right) \sqrt{b+a \cos (c+d x)} \csc (c) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{a \sqrt{\cot ^2(c)+1}-a \sqrt{\cot ^2(c)+1} \sin \left(d x-\tan ^{-1}(\cot (c))\right)}{a \sqrt{\cot ^2(c)+1}-b \csc (c)}} \sqrt{\frac{\sqrt{\cot ^2(c)+1} \sin \left(d x-\tan ^{-1}(\cot (c))\right) a+\sqrt{\cot ^2(c)+1} a}{\sqrt{\cot ^2(c)+1} a+b \csc (c)}} \sqrt{b-a \sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right)}{3 a d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}-\frac{4 C F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{2};\frac{3}{2};\frac{\csc (c) \left(b-a \sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)\right)}{a \sqrt{\cot ^2(c)+1} \left(\frac{b \csc (c)}{a \sqrt{\cot ^2(c)+1}}+1\right)},\frac{\csc (c) \left(b-a \sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)\right)}{a \sqrt{\cot ^2(c)+1} \left(\frac{b \csc (c)}{a \sqrt{\cot ^2(c)+1}}-1\right)}\right) \sqrt{b+a \cos (c+d x)} \csc (c) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{a \sqrt{\cot ^2(c)+1}-a \sqrt{\cot ^2(c)+1} \sin \left(d x-\tan ^{-1}(\cot (c))\right)}{a \sqrt{\cot ^2(c)+1}-b \csc (c)}} \sqrt{\frac{\sqrt{\cot ^2(c)+1} \sin \left(d x-\tan ^{-1}(\cot (c))\right) a+\sqrt{\cot ^2(c)+1} a}{\sqrt{\cot ^2(c)+1} a+b \csc (c)}} \sqrt{b-a \sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right)}{a d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{(b+a \cos (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 (3 a B-2 A b) \cot (c)}{3 a^2 d}+\frac{4 A \cos (d x) \sin (c)}{3 a d}+\frac{4 A \cos (c) \sin (d x)}{3 a d}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{4 A b \sqrt{b+a \cos (c+d x)} \csc (c) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\sec (c) \left(b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}\right)}{a \sqrt{\tan ^2(c)+1} \left(1-\frac{b \sec (c)}{a \sqrt{\tan ^2(c)+1}}\right)},-\frac{\sec (c) \left(b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}\right)}{a \sqrt{\tan ^2(c)+1} \left(-\frac{b \sec (c)}{a \sqrt{\tan ^2(c)+1}}-1\right)}\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1} \sqrt{\frac{a \sqrt{\tan ^2(c)+1}-a \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}{\sqrt{\tan ^2(c)+1} a+b \sec (c)}} \sqrt{\frac{\cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} a+\sqrt{\tan ^2(c)+1} a}{a \sqrt{\tan ^2(c)+1}-b \sec (c)}} \sqrt{b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}-\frac{\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}+\frac{2 a \cos (c) \left(b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}\right)}{a^2 \cos ^2(c)+a^2 \sin ^2(c)}}{\sqrt{b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right)}{3 a d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}-\frac{2 B \sqrt{b+a \cos (c+d x)} \csc (c) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{F_1\left(-\frac{1}{2};-\frac{1}{2},-\frac{1}{2};\frac{1}{2};-\frac{\sec (c) \left(b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}\right)}{a \sqrt{\tan ^2(c)+1} \left(1-\frac{b \sec (c)}{a \sqrt{\tan ^2(c)+1}}\right)},-\frac{\sec (c) \left(b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}\right)}{a \sqrt{\tan ^2(c)+1} \left(-\frac{b \sec (c)}{a \sqrt{\tan ^2(c)+1}}-1\right)}\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1} \sqrt{\frac{a \sqrt{\tan ^2(c)+1}-a \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}{\sqrt{\tan ^2(c)+1} a+b \sec (c)}} \sqrt{\frac{\cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} a+\sqrt{\tan ^2(c)+1} a}{a \sqrt{\tan ^2(c)+1}-b \sec (c)}} \sqrt{b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}-\frac{\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}+\frac{2 a \cos (c) \left(b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}\right)}{a^2 \cos ^2(c)+a^2 \sin ^2(c)}}{\sqrt{b+a \cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}","\frac{2 \sqrt{\sec (c+d x)} \left(a^2 (A+3 C)-3 a b B+2 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{2 (2 A b-3 a B) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d \sqrt{\sec (c+d x)}}",1,"((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(-2*A*b + 3*a*B)*Cot[c])/(3*a^2*d) + (4*A*Cos[d*x]*Sin[c])/(3*a*d) + (4*A*Cos[c]*Sin[d*x])/(3*a*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (4*A*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (4*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (4*A*b*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (2*B*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]])","C",0
1056,1,3039,291,6.729986,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","-\frac{2 (4 A b-5 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \sqrt{\sec (c+d x)} \left(-5 a^3 B+a^2 b (7 A+15 C)-10 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(9*a^2*A + 8*A*b^2 - 10*a*b*B + 15*a^2*C)*Cot[c])/(15*a^3*d) + (4*(-4*A*b + 5*a*B)*Cos[d*x]*Sin[c])/(15*a^2*d) + (2*A*Cos[2*d*x]*Sin[2*c])/(5*a*d) + (4*(-4*A*b + 5*a*B)*Cos[c]*Sin[d*x])/(15*a^2*d) + (2*A*Cos[2*c]*Sin[2*d*x])/(5*a*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (8*A*b*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(15*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (4*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (6*A*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (16*A*b^2*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (4*b*B*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (2*C*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]])","C",0
1057,1,4470,380,6.951486,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","-\frac{2 (6 A b-7 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{35 a^2 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 a^3 d \sqrt{\sec (c+d x)}}-\frac{2 \left(-63 a^3 B+a^2 (44 A b+70 b C)-56 a b^2 B+48 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sqrt{\sec (c+d x)} \left(5 a^4 (5 A+7 C)-49 a^3 b B+2 a^2 b^2 (16 A+35 C)-56 a b^3 B+48 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^4 d \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 a d \sec ^{\frac{5}{2}}(c+d x)}",1,"((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(-44*a^2*A*b - 48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 70*a^2*b*C)*Cot[c])/(105*a^4*d) + ((115*a^2*A + 96*A*b^2 - 112*a*b*B + 140*a^2*C)*Cos[d*x]*Sin[c])/(105*a^3*d) + (2*(-6*A*b + 7*a*B)*Cos[2*d*x]*Sin[2*c])/(35*a^2*d) + (A*Cos[3*d*x]*Sin[3*c])/(7*a*d) + ((115*a^2*A + 96*A*b^2 - 112*a*b*B + 140*a^2*C)*Cos[c]*Sin[d*x])/(105*a^3*d) + (2*(-6*A*b + 7*a*B)*Cos[2*c]*Sin[2*d*x])/(35*a^2*d) + (A*Cos[3*c]*Sin[3*d*x])/(7*a*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (20*A*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(21*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (16*A*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(35*a^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (8*b*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(15*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (4*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (88*A*b*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(105*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (32*A*b^3*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(35*a^3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (6*B*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (16*b^2*B*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (4*b*C*Sqrt[b + a*Cos[c + d*x]]*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*a*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]])","C",0
1058,1,377,253,4.5389035,"\int \frac{\sqrt{\sec (c+d x)} \left(a A+(A b+a B) \sec (c+d x)+b B \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 (a B+4 A b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+8 a A \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 B \tan (c+d x) (a \cos (c+d x)+b)-\frac{2 i B \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \sqrt{a \cos (c+d x)+b} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}}}\right)}{4 d \sqrt{a+b \sec (c+d x)}}","\frac{(2 a A+b B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{(a B+2 A b) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}-\frac{B \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(Sqrt[Sec[c + d*x]]*(8*a*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 2*(4*A*b + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)] - ((2*I)*B*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Sqrt[b + a*Cos[c + d*x]]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b) + 4*B*(b + a*Cos[c + d*x])*Tan[c + d*x]))/(4*d*Sqrt[a + b*Sec[c + d*x]])","C",1
1059,1,774,393,7.0761902,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a \cos (c+d x)+b)^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 C \tan (c+d x)}{b^2}-\frac{4 \left(a^3 C \sin (c+d x)-a^2 b B \sin (c+d x)+a A b^2 \sin (c+d x)\right)}{b^2 \left(b^2-a^2\right) (a \cos (c+d x)+b)}\right)}{d \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{(a \cos (c+d x)+b)^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{2 \left(4 a^2 b C-4 a b^2 B+4 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \sin (c+d x) \cos (2 (c+d x)) \left(3 a^3 C-2 a^2 b B+2 a A b^2-a b^2 C\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(9 a^3 C-6 a^2 b B+2 a A b^2-7 a b^2 C+4 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{2 b^2 d (b-a) (a+b) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}-\frac{\left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 b B-3 a C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(4*A*b^3 - 4*a*b^2*B + 4*a^2*b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(2*a*A*b^2 - 6*a^2*b*B + 4*b^3*B + 9*a^3*C - 7*a*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(2*a*A*b^2 - 2*a^2*b*B + 3*a^3*C - a*b^2*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(2*b^2*(-a + b)*(a + b)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*C*Tan[c + d*x])/b^2))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2))","C",0
1060,0,0,311,34.6457371,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 A \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{a+b \sec (c+d x)}}+\frac{2 C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{a+b \sec (c+d x)}}",1,"Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2), x]","F",-1
1061,1,3541,249,7.1301462,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-\left(a^2 (A-C)\right)-a b B+2 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 (2 A b-a B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(a^2*A - 3*A*b^2 + 2*a*b*B - 2*a^2*C + a^2*A*Cos[2*c] - A*b^2*Cos[2*c])*Csc[c]*Sec[c])/(a^2*(a^2 - b^2)*d) - (4*Sec[c]*(A*b^3*Sin[c] - a*b^2*B*Sin[c] + a^2*b*C*Sin[c] - a*A*b^2*Sin[d*x] + a^2*b*B*Sin[d*x] - a^3*C*Sin[d*x]))/(a^2*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*A*b*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(a*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (4*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*b*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(a*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*a*A*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*A*b^2*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(a*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*b*B*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*a*C*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2))","C",0
1062,1,4557,350,7.5976661,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(-\left(a^2 (A-3 C)\right)-3 a b B+4 A b^2\right) \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \left(a^2 (A+3 C)-6 a b B+8 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^3 B-a^2 (5 A b-3 b C)-6 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(-5*a^2*A*b + 11*A*b^3 + 3*a^3*B - 9*a*b^2*B + 6*a^2*b*C - 5*a^2*A*b*Cos[2*c] + 5*A*b^3*Cos[2*c] + 3*a^3*B*Cos[2*c] - 3*a*b^2*B*Cos[2*c])*Csc[c]*Sec[c])/(3*a^3*(a^2 - b^2)*d) + (4*A*Cos[d*x]*Sin[c])/(3*a^2*d) + (4*A*Cos[c]*Sin[d*x])/(3*a^2*d) + (4*Sec[c]*(A*b^4*Sin[c] - a*b^3*B*Sin[c] + a^2*b^2*C*Sin[c] - a*A*b^3*Sin[d*x] + a^2*b^2*B*Sin[d*x] - a^3*b*C*Sin[d*x]))/(a^3*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (4*A*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (8*A*b^2*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(3*a^2*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*b*B*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/(a*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (4*C*AppellF1[1/2, 1/2, 1/2, 3/2, (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2]))), (Csc[c]*(b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]))/(a*Sqrt[1 + Cot[c]^2]*(-1 + (b*Csc[c])/(a*Sqrt[1 + Cot[c]^2])))]*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[(a*Sqrt[1 + Cot[c]^2] - a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] - b*Csc[c])]*Sqrt[(a*Sqrt[1 + Cot[c]^2] + a*Sqrt[1 + Cot[c]^2]*Sin[d*x - ArcTan[Cot[c]]])/(a*Sqrt[1 + Cot[c]^2] + b*Csc[c])]*Sqrt[b - a*Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]]])/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (10*A*b*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (16*A*b^3*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(3*a^2*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*a*B*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (4*b^2*B*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(a*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*b*C*(b + a*Cos[c + d*x])^(3/2)*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((AppellF1[-1/2, -1/2, -1/2, 1/2, -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2])))), -((Sec[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a*Sqrt[1 + Tan[c]^2]*(-1 - (b*Sec[c])/(a*Sqrt[1 + Tan[c]^2]))))]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 + Tan[c]^2]*Sqrt[(a*Sqrt[1 + Tan[c]^2] - a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(b*Sec[c] + a*Sqrt[1 + Tan[c]^2])]*Sqrt[(a*Sqrt[1 + Tan[c]^2] + a*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(-(b*Sec[c]) + a*Sqrt[1 + Tan[c]^2])]*Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*a*Cos[c]*(b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]))/(a^2*Cos[c]^2 + a^2*Sin[c]^2))/Sqrt[b + a*Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/((a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2))","C",0
1063,1,6134,461,8.1997701,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(-\left(a^2 (A-5 C)\right)-5 a b B+6 A b^2\right) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left(5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \sqrt{\sec (c+d x)} \left(-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
1064,1,938,563,7.4339342,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(b+a \cos (c+d x))^3 \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \left(C \sin (c+d x) a^3-b B \sin (c+d x) a^2+A b^2 \sin (c+d x) a\right)}{3 b^2 \left(b^2-a^2\right) (b+a \cos (c+d x))^2}-\frac{4 \left(-6 C \sin (c+d x) a^5+3 b B \sin (c+d x) a^4+10 b^2 C \sin (c+d x) a^3-7 b^3 B \sin (c+d x) a^2+4 A b^4 \sin (c+d x) a\right)}{3 b^3 \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}+\frac{2 C \tan (c+d x)}{b^3}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}}-\frac{(b+a \cos (c+d x))^{5/2} \sqrt{\sec (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 \left(-12 A b^5+24 a B b^4-4 a^2 A b^3-36 a^2 C b^3-8 a^3 B b^2+20 a^4 C b\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 \left(45 C a^5-18 b B a^4-86 b^2 C a^3+38 b^3 B a^2-8 A b^4 a+33 b^4 C a-12 b^5 B\right) \sqrt{\frac{b+a \cos (c+d x)}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{b+a \cos (c+d x)}}+\frac{2 i \left(15 C a^5-6 b B a^4-26 b^2 C a^3+14 b^3 B a^2-8 A b^4 a+3 b^4 C a\right) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{\cos (c+d x) a+a}{a-b}} \cos (2 (c+d x)) \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right) \sin (c+d x)}{\sqrt{\frac{1}{a-b}} b \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 b^2+2 (b+a \cos (c+d x))^2-4 b (b+a \cos (c+d x))\right)}\right)}{6 (a-b)^2 b^3 (a+b)^2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (a+b \sec (c+d x))^{5/2}}","-\frac{2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{\sqrt{\sec (c+d x)} \left(5 a^2 C-2 a b B+2 A b^2-3 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 b B-5 a C) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^3 d \sqrt{a+b \sec (c+d x)}}",1,"-1/6*((b + a*Cos[c + d*x])^(5/2)*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(-4*a^2*A*b^3 - 12*A*b^5 - 8*a^3*b^2*B + 24*a*b^4*B + 20*a^4*b*C - 36*a^2*b^3*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(-8*a*A*b^4 - 18*a^4*b*B + 38*a^2*b^3*B - 12*b^5*B + 45*a^5*C - 86*a^3*b^2*C + 33*a*b^4*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(-8*a*A*b^4 - 6*a^4*b*B + 14*a^2*b^3*B + 15*a^5*C - 26*a^3*b^2*C + 3*a*b^4*C)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/((a - b)^2*b^3*(a + b)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) + ((b + a*Cos[c + d*x])^3*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(a*A*b^2*Sin[c + d*x] - a^2*b*B*Sin[c + d*x] + a^3*C*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (4*(4*a*A*b^4*Sin[c + d*x] + 3*a^4*b*B*Sin[c + d*x] - 7*a^2*b^3*B*Sin[c + d*x] - 6*a^5*C*Sin[c + d*x] + 10*a^3*b^2*C*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (2*C*Tan[c + d*x])/b^3))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2))","C",0
1065,0,0,447,53.9282856,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\int \frac{\sec ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","-\frac{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-3 a^4 C+a^2 b^2 (3 A+7 C)-4 a b^3 B+A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^4 C+a^2 b^2 (3 A+7 C)-4 a b^3 B+A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a b^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 C \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{a+b \sec (c+d x)}}",1,"Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2), x]","F",-1
1066,1,5040,378,7.7130765,"\int \frac{\sqrt{\sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\frac{2 \sqrt{\sec (c+d x)} \left(-\left(a^2 (3 A+C)\right)+a b B+2 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right)}{3 a b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1067,1,6142,401,8.4250166,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sqrt{\sec (c+d x)} \left(3 a^3 B-a^2 b (9 A+C)-2 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^4 C+5 a^3 b B-2 a^2 b^2 (4 A+C)-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4 (A-C)+6 a^3 b B-a^2 b^2 (15 A+C)-2 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
1068,1,7608,521,9.4890014,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \left(a^4 (A-5 C)+8 a^3 b B-a^2 b^2 (13 A-C)-4 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{2 \sin (c+d x) \left(4 a^4 C-7 a^3 b B+10 a^2 A b^2+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\sec (c+d x)} \left(-\left(a^4 (A+3 C)\right)+9 a^3 b B-2 a^2 b^2 (8 A-C)-8 a b^3 B+16 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^5 B+a^4 (8 A b-6 b C)+15 a^3 b^2 B-2 a^2 b^3 (14 A-C)-8 a b^4 B+16 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
1069,1,9192,663,10.4571547,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \left(a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}-\frac{2 \sin (c+d x) \left(-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \left(-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \left(5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
1070,0,0,248,52.4759164,"\int (a+b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\int (a+b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","A \text{Int}\left((a+b \sec (c+d x))^{2/3},x\right)+\frac{\sqrt{2} (b B-a C) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}+\frac{\sqrt{2} C (a+b) \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{5}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",0,"Integrate[(a + b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x]","A",-1
1071,0,0,248,27.4779843,"\int \sqrt[3]{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\int \sqrt[3]{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","A \text{Int}\left(\sqrt[3]{a+b \sec (c+d x)},x\right)+\frac{\sqrt{2} (b B-a C) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}+\frac{\sqrt{2} C (a+b) \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{4}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",0,"Integrate[(a + b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2), x]","A",-1
1072,0,0,245,49.1833734,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3),x]","\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","A \text{Int}\left(\frac{1}{\sqrt[3]{a+b \sec (c+d x)}},x\right)+\frac{\sqrt{2} (b B-a C) \tan (c+d x) \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}} F_1\left(\frac{1}{2};\frac{1}{2},\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}+\frac{\sqrt{2} C \tan (c+d x) (a+b \sec (c+d x))^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",0,"Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3), x]","A",-1
1073,0,0,245,26.9379635,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3),x]","\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","A \text{Int}\left(\frac{1}{(a+b \sec (c+d x))^{2/3}},x\right)+\frac{\sqrt{2} (b B-a C) \tan (c+d x) \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3} F_1\left(\frac{1}{2};\frac{1}{2},\frac{2}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}+\frac{\sqrt{2} C \tan (c+d x) \sqrt[3]{a+b \sec (c+d x)} F_1\left(\frac{1}{2};\frac{1}{2},-\frac{1}{3};\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{b d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",0,"Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3), x]","A",-1
1074,0,0,137,9.2656002,"\int (a+b \sec (c+d x))^m \left(a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right) \, dx","Integrate[(a + b*Sec[c + d*x])^m*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2),x]","\int (a+b \sec (c+d x))^m \left(a b B-a^2 C+b^2 B \sec (c+d x)+b^2 C \sec ^2(c+d x)\right) \, dx","(b B-a C) \text{Int}\left((a+b \sec (c+d x))^{m+1},x\right)+\frac{\sqrt{2} b C (a+b) \tan (c+d x) (a+b \sec (c+d x))^m \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{-m} F_1\left(\frac{1}{2};\frac{1}{2},-m-1;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right)}{d \sqrt{\sec (c+d x)+1}}",0,"Integrate[(a + b*Sec[c + d*x])^m*(a*b*B - a^2*C + b^2*B*Sec[c + d*x] + b^2*C*Sec[c + d*x]^2), x]","A",-1
1075,1,65,80,0.335266,"\int \cos ^{\frac{9}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(A + C*Sec[c + d*x]^2),x]","\frac{12 (7 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (2 (c+d x)) \sqrt{\cos (c+d x)} (5 A \cos (2 (c+d x))+19 A+18 C)}{90 d}","\frac{2 (7 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 (7 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(12*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(19*A + 18*C + 5*A*Cos[2*(c + d*x)])*Sin[2*(c + d*x)])/(90*d)","A",1
1076,1,63,80,0.2895754,"\int \cos ^{\frac{7}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2),x]","\frac{2 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (3 A \cos (2 (c+d x))+13 A+14 C)}{21 d}","\frac{2 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (5 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(2*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(13*A + 14*C + 3*A*Cos[2*(c + d*x)])*Sin[c + d*x])/(21*d)","A",1
1077,1,48,50,0.1118993,"\int \cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \sin (2 (c+d x)) \sqrt{\cos (c+d x)}}{5 d}","\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(2*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2] + A*Sqrt[Cos[c + d*x]]*Sin[2*(c + d*x)])/(5*d)","A",1
1078,1,124,48,0.9092233,"\int \cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2),x]","-\frac{4 \sin (c) \sqrt{\cos (c+d x)} \left(A \cos ^2(c+d x)+C\right) \left((A+3 C) \sqrt{\csc ^2(c)} \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)-A \csc (c) \sin (c+d x)\right)}{3 d (A \cos (2 (c+d x))+A+2 C)}","\frac{2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(-4*Sqrt[Cos[c + d*x]]*(C + A*Cos[c + d*x]^2)*Sin[c]*((A + 3*C)*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]] - A*Csc[c]*Sin[c + d*x]))/(3*d*(A + 2*C + A*Cos[2*(c + d*x)]))","C",0
1079,1,289,44,1.65382,"\int \sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right) \left(-\frac{4 \csc (c) (A \cos (2 c+d x)+(A-2 C) \cos (d x))}{d \sqrt{\cos (c+d x)}}+\frac{2 (A-C) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sqrt{e^{-i d x} \left(2 i \sin (c) \left(-1+e^{2 i d x}\right)+2 \cos (c) \left(1+e^{2 i d x}\right)\right)} \sqrt{i \sin (2 c) e^{2 i d x}+\cos (2 c) e^{2 i d x}+1} \left(3 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)+e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right)\right)}{3 d \left(i \sin (c) \left(-1+e^{2 i d x}\right)+\cos (c) \left(1+e^{2 i d x}\right)\right)}\right)}{2 (A \cos (2 (c+d x))+A+2 C)}","\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2)*((-4*((A - 2*C)*Cos[d*x] + A*Cos[2*c + d*x])*Csc[c])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - C)*Csc[c/2]*(3*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)] + E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)])*Sec[c/2]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/(3*d*((1 + E^((2*I)*d*x))*Cos[c] + I*(-1 + E^((2*I)*d*x))*Sin[c]))))/(2*(A + 2*C + A*Cos[2*(c + d*x)]))","C",1
1080,1,43,48,0.1800584,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","\frac{2 \left((3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{C \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*((3*A + C)*EllipticF[(c + d*x)/2, 2] + (C*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
1081,1,73,80,0.3718775,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2),x]","\frac{(5 A+3 C) \sin (2 (c+d x))-2 (5 A+3 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 C \tan (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (5 A+3 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-2*(5*A + 3*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + (5*A + 3*C)*Sin[2*(c + d*x)] + 2*C*Tan[c + d*x])/(5*d*Cos[c + d*x]^(3/2))","A",1
1082,1,73,80,0.5549077,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/Cos[c + d*x]^(5/2),x]","\frac{(7 A+5 C) \sin (2 (c+d x))+2 (7 A+5 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 C \tan (c+d x)}{21 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(7*A + 5*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + (7*A + 5*C)*Sin[2*(c + d*x)] + 6*C*Tan[c + d*x])/(21*d*Cos[c + d*x]^(5/2))","A",1
1083,1,918,165,6.3120328,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(7 A+9 C) \cot (c)}{15 d}+\frac{(23 A+28 C) \cos (d x) \sin (c)}{84 d}+\frac{(19 A+18 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{A \cos (3 d x) \sin (3 c)}{28 d}+\frac{A \cos (4 d x) \sin (4 c)}{72 d}+\frac{(23 A+28 C) \cos (c) \sin (d x)}{84 d}+\frac{(19 A+18 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{A \cos (3 c) \sin (3 d x)}{28 d}+\frac{A \cos (4 c) \sin (4 d x)}{72 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{7 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (7 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (7 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (5 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/15*((7*A + 9*C)*Cot[c])/d + ((23*A + 28*C)*Cos[d*x]*Sin[c])/(84*d) + ((19*A + 18*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + (A*Cos[3*d*x]*Sin[3*c])/(28*d) + (A*Cos[4*d*x]*Sin[4*c])/(72*d) + ((23*A + 28*C)*Cos[c]*Sin[d*x])/(84*d) + ((19*A + 18*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + (A*Cos[3*c]*Sin[3*d*x])/(28*d) + (A*Cos[4*c]*Sin[4*d*x])/(72*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (7*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d) - (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1084,1,872,134,6.2665951,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(3 A+5 C) \cot (c)}{5 d}+\frac{(23 A+28 C) \cos (d x) \sin (c)}{84 d}+\frac{A \cos (2 d x) \sin (2 c)}{10 d}+\frac{A \cos (3 d x) \sin (3 c)}{28 d}+\frac{(23 A+28 C) \cos (c) \sin (d x)}{84 d}+\frac{A \cos (2 c) \sin (2 d x)}{10 d}+\frac{A \cos (3 c) \sin (3 d x)}{28 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((3*A + 5*C)*Cot[c])/d + ((23*A + 28*C)*Cos[d*x]*Sin[c])/(84*d) + (A*Cos[2*d*x]*Sin[2*c])/(10*d) + (A*Cos[3*d*x]*Sin[3*c])/(28*d) + ((23*A + 28*C)*Cos[c]*Sin[d*x])/(84*d) + (A*Cos[2*c]*Sin[2*d*x])/(10*d) + (A*Cos[3*c]*Sin[3*d*x])/(28*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1085,1,824,101,6.2991447,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(3 A+5 C) \cot (c)}{5 d}+\frac{A \cos (d x) \sin (c)}{3 d}+\frac{A \cos (2 d x) \sin (2 c)}{10 d}+\frac{A \cos (c) \sin (d x)}{3 d}+\frac{A \cos (2 c) \sin (2 d x)}{10 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((3*A + 5*C)*Cot[c])/d + (A*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[2*d*x]*Sin[2*c])/(10*d) + (A*Cos[c]*Sin[d*x])/(3*d) + (A*Cos[2*c]*Sin[2*d*x])/(10*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1086,1,813,95,6.3652291,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(\cos (2 c) A+A-2 C) \csc (c) \sec (c)}{2 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{A \cos (d x) \sin (c)}{3 d}+\frac{A \cos (c) \sin (d x)}{3 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((A - 2*C + A*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[c]*Sin[d*x])/(3*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1087,1,817,95,6.406225,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (C \sin (c)+3 C \sin (d x)) \sec (c+d x)}{3 d}-\frac{(\cos (2 c) A+A-2 C) \csc (c) \sec (c)}{2 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((A - 2*C + A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*C*Sin[d*x]))/(3*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1088,1,851,132,6.4811076,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 C \sin (c)+5 C \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 C \sin (c)+15 A \sin (d x)+9 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{(5 A+3 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+3 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((5*A + 3*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*C*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*C*Sin[c] + 15*A*Sin[d*x] + 9*C*Sin[d*x]))/(15*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1089,1,895,165,6.5634389,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{7 d}+\frac{\sec (c) (5 C \sin (c)+7 C \sin (d x)) \sec ^3(c+d x)}{35 d}+\frac{\sec (c) (21 C \sin (c)+35 A \sin (d x)+25 C \sin (d x)) \sec ^2(c+d x)}{105 d}+\frac{\sec (c) (35 A \sin (c)+25 C \sin (c)+105 A \sin (d x)+63 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{(5 A+3 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 A+3 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((5*A + 3*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (Sec[c]*Sec[c + d*x]^3*(5*C*Sin[c] + 7*C*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]^2*(21*C*Sin[c] + 35*A*Sin[d*x] + 25*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 25*C*Sin[c] + 105*A*Sin[d*x] + 63*C*Sin[d*x]))/(105*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (5*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1090,1,976,230,6.3430652,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","a^2 \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1)^2 \left(-\frac{(7 A+9 C) \cot (c)}{15 d}+\frac{(941 A+1122 C) \cos (d x) \sin (c)}{3696 d}+\frac{(19 A+18 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(101 A+44 C) \cos (3 d x) \sin (3 c)}{2464 d}+\frac{A \cos (4 d x) \sin (4 c)}{72 d}+\frac{A \cos (5 d x) \sin (5 c)}{352 d}+\frac{(941 A+1122 C) \cos (c) \sin (d x)}{3696 d}+\frac{(19 A+18 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(101 A+44 C) \cos (3 c) \sin (3 d x)}{2464 d}+\frac{A \cos (4 c) \sin (4 d x)}{72 d}+\frac{A \cos (5 c) \sin (5 d x)}{352 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{7 A (\cos (c+d x)+1)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{3 C (\cos (c+d x)+1)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{50 A (\cos (c+d x)+1)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{231 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x)+1)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}\right)","\frac{8 a^2 (25 A+33 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (7 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (89 A+99 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (7 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{8 a^2 (25 A+33 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{8 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}",1,"a^2*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/15*((7*A + 9*C)*Cot[c])/d + ((941*A + 1122*C)*Cos[d*x]*Sin[c])/(3696*d) + ((19*A + 18*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((101*A + 44*C)*Cos[3*d*x]*Sin[3*c])/(2464*d) + (A*Cos[4*d*x]*Sin[4*c])/(72*d) + (A*Cos[5*d*x]*Sin[5*c])/(352*d) + ((941*A + 1122*C)*Cos[c]*Sin[d*x])/(3696*d) + ((19*A + 18*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((101*A + 44*C)*Cos[3*c]*Sin[3*d*x])/(2464*d) + (A*Cos[4*c]*Sin[4*d*x])/(72*d) + (A*Cos[5*c]*Sin[5*d*x])/(352*d)) - (50*A*(1 + Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[c]^2]) - (2*C*(1 + Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (7*A*(1 + Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d) - (3*C*(1 + Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1091,1,1118,197,6.3463863,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(-\frac{8 (2 A+3 C) \cot (c)}{15 d}+\frac{(23 A+28 C) \cos (d x) \sin (c)}{42 d}+\frac{(37 A+18 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{A \cos (3 d x) \sin (3 c)}{14 d}+\frac{A \cos (4 d x) \sin (4 c)}{72 d}+\frac{(23 A+28 C) \cos (c) \sin (d x)}{42 d}+\frac{(37 A+18 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{A \cos (3 c) \sin (3 d x)}{14 d}+\frac{A \cos (4 c) \sin (4 d x)}{72 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}-\frac{8 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{15 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{4 C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{10 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{16 a^2 (2 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (19 A+21 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{8 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((-8*(2*A + 3*C)*Cot[c])/(15*d) + ((23*A + 28*C)*Cos[d*x]*Sin[c])/(42*d) + ((37*A + 18*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + (A*Cos[3*d*x]*Sin[3*c])/(14*d) + (A*Cos[4*d*x]*Sin[4*c])/(72*d) + ((23*A + 28*C)*Cos[c]*Sin[d*x])/(42*d) + ((37*A + 18*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + (A*Cos[3*c]*Sin[3*d*x])/(14*d) + (A*Cos[4*c]*Sin[4*d*x])/(72*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (10*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(A + 2*C + A*Cos[2*c + 2*d*x])) - (4*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1092,1,1070,164,6.4130058,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(-\frac{2 (3 A+5 C) \cot (c)}{5 d}+\frac{(51 A+28 C) \cos (d x) \sin (c)}{84 d}+\frac{A \cos (2 d x) \sin (2 c)}{5 d}+\frac{A \cos (3 d x) \sin (3 c)}{28 d}+\frac{(51 A+28 C) \cos (c) \sin (d x)}{84 d}+\frac{A \cos (2 c) \sin (2 d x)}{5 d}+\frac{A \cos (3 c) \sin (3 d x)}{28 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}-\frac{3 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C)}-\frac{4 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{7 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{8 a^2 (3 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{8 A \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((-2*(3*A + 5*C)*Cot[c])/(5*d) + ((51*A + 28*C)*Cos[d*x]*Sin[c])/(84*d) + (A*Cos[2*d*x]*Sin[2*c])/(5*d) + (A*Cos[3*d*x]*Sin[3*c])/(28*d) + ((51*A + 28*C)*Cos[c]*Sin[d*x])/(84*d) + (A*Cos[2*c]*Sin[2*d*x])/(5*d) + (A*Cos[3*c]*Sin[3*d*x])/(28*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (3*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + A*Cos[2*c + 2*d*x])) - (C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1093,1,799,158,6.5010226,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(-\frac{(8 \cos (2 c) A+8 A-5 C+5 C \cos (2 c)) \csc (c) \sec (c)}{10 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{2 A \cos (d x) \sin (c)}{3 d}+\frac{A \cos (2 d x) \sin (2 c)}{10 d}+\frac{2 A \cos (c) \sin (d x)}{3 d}+\frac{A \cos (2 c) \sin (2 d x)}{10 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}-\frac{4 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{2 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (7 A-15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}+\frac{16 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(-1/10*((8*A - 5*C + 8*A*Cos[2*c] + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + (2*A*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[2*d*x]*Sin[2*c])/(10*d) + (2*A*Cos[c]*Sin[d*x])/(3*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (A*Cos[2*c]*Sin[2*d*x])/(10*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1094,1,1040,154,6.5219456,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (C \sin (c)+6 C \sin (d x)) \sec (c+d x)}{3 d}-\frac{(\cos (2 c) A+A-2 C) \csc (c) \sec (c)}{d}+\frac{A \cos (d x) \sin (c)}{3 d}+\frac{A \cos (c) \sin (d x)}{3 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}-\frac{A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C)}+\frac{C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C)}-\frac{4 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{8 a^2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 (A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(-(((A - 2*C + A*Cos[2*c])*Csc[c]*Sec[c])/d) + (A*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[c]*Sin[d*x])/(3*d) + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(C*Sin[c] + 6*C*Sin[d*x]))/(3*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1095,1,800,156,6.6065321,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 C \sin (c)+10 C \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (10 C \sin (c)+15 A \sin (d x)+24 C \sin (d x)) \sec (c+d x)}{15 d}-\frac{(5 \cos (2 c) A-5 A-16 C) \csc (c) \sec (c)}{10 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}+\frac{4 C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{2 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (15 A+17 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{16 a^2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*(-1/10*((-5*A - 16*C + 5*A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 10*C*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(10*C*Sin[c] + 15*A*Sin[d*x] + 24*C*Sin[d*x]))/(15*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (4*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1096,1,1092,197,6.6994202,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{7 d}+\frac{\sec (c) (5 C \sin (c)+14 C \sin (d x)) \sec ^3(c+d x)}{35 d}+\frac{\sec (c) (42 C \sin (c)+35 A \sin (d x)+60 C \sin (d x)) \sec ^2(c+d x)}{105 d}+\frac{\sec (c) (35 A \sin (c)+60 C \sin (c)+210 A \sin (d x)+126 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{2 (5 A+3 C) \csc (c) \sec (c)}{5 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}+\frac{A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C)}+\frac{3 C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{4 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{7 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{8 a^2 (7 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+33 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+3 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((2*(5*A + 3*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (Sec[c]*Sec[c + d*x]^3*(5*C*Sin[c] + 14*C*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]^2*(42*C*Sin[c] + 35*A*Sin[d*x] + 60*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 60*C*Sin[c] + 210*A*Sin[d*x] + 126*C*Sin[d*x]))/(105*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (3*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1097,1,1137,230,6.7695875,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{9 d}+\frac{\sec (c) (7 C \sin (c)+18 C \sin (d x)) \sec ^4(c+d x)}{63 d}+\frac{\sec (c) (90 C \sin (c)+63 A \sin (d x)+112 C \sin (d x)) \sec ^3(c+d x)}{315 d}+\frac{\sec (c) (63 A \sin (c)+112 C \sin (c)+210 A \sin (d x)+150 C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac{2 \sec (c) (35 A \sin (c)+25 C \sin (c)+84 A \sin (d x)+56 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{8 (3 A+2 C) \csc (c) \sec (c)}{15 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C}+\frac{4 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{8 C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{15 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{2 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{10 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{16 a^2 (3 A+2 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (21 A+19 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{16 a^2 (3 A+2 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{8 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((8*(3*A + 2*C)*Csc[c]*Sec[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(9*d) + (Sec[c]*Sec[c + d*x]^4*(7*C*Sin[c] + 18*C*Sin[d*x]))/(63*d) + (2*Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 25*C*Sin[c] + 84*A*Sin[d*x] + 56*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]^3*(90*C*Sin[c] + 63*A*Sin[d*x] + 112*C*Sin[d*x]))/(315*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 112*C*Sin[c] + 210*A*Sin[d*x] + 150*C*Sin[d*x]))/(315*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (10*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (4*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (8*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1098,1,1022,279,6.398572,"\int \cos ^{\frac{13}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","a^3 \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1)^3 \left(-\frac{(175 A+221 C) \cot (c)}{390 d}+\frac{(1811 A+2134 C) \cos (d x) \sin (c)}{7392 d}+\frac{(7825 A+7592 C) \cos (2 d x) \sin (2 c)}{74880 d}+\frac{(215 A+132 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{(59 A+13 C) \cos (4 d x) \sin (4 c)}{3744 d}+\frac{3 A \cos (5 d x) \sin (5 c)}{704 d}+\frac{A \cos (6 d x) \sin (6 c)}{1664 d}+\frac{(1811 A+2134 C) \cos (c) \sin (d x)}{7392 d}+\frac{(7825 A+7592 C) \cos (2 c) \sin (2 d x)}{74880 d}+\frac{(215 A+132 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{(59 A+13 C) \cos (4 c) \sin (4 d x)}{3744 d}+\frac{3 A \cos (5 c) \sin (5 d x)}{704 d}+\frac{A \cos (6 c) \sin (6 d x)}{1664 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{35 A (\cos (c+d x)+1)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{156 d}-\frac{17 C (\cos (c+d x)+1)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{95 A (\cos (c+d x)+1)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{462 d \sqrt{\cot ^2(c)+1}}-\frac{11 C (\cos (c+d x)+1)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}\right)","\frac{4 a^3 (95 A+121 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (175 A+221 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{40 a^3 (118 A+143 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d}+\frac{4 a^3 (175 A+221 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{585 d}+\frac{2 (145 A+143 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{1287 d}+\frac{4 a^3 (95 A+121 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{12 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{143 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^3}{13 d}",1,"a^3*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/390*((175*A + 221*C)*Cot[c])/d + ((1811*A + 2134*C)*Cos[d*x]*Sin[c])/(7392*d) + ((7825*A + 7592*C)*Cos[2*d*x]*Sin[2*c])/(74880*d) + ((215*A + 132*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((59*A + 13*C)*Cos[4*d*x]*Sin[4*c])/(3744*d) + (3*A*Cos[5*d*x]*Sin[5*c])/(704*d) + (A*Cos[6*d*x]*Sin[6*c])/(1664*d) + ((1811*A + 2134*C)*Cos[c]*Sin[d*x])/(7392*d) + ((7825*A + 7592*C)*Cos[2*c]*Sin[2*d*x])/(74880*d) + ((215*A + 132*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((59*A + 13*C)*Cos[4*c]*Sin[4*d*x])/(3744*d) + (3*A*Cos[5*c]*Sin[5*d*x])/(704*d) + (A*Cos[6*c]*Sin[6*d*x])/(1664*d)) - (95*A*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(462*d*Sqrt[1 + Cot[c]^2]) - (11*C*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (35*A*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(156*d) - (17*C*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d))","C",0
1099,1,976,246,6.3545488,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","a^3 \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1)^3 \left(-\frac{(5 A+7 C) \cot (c)}{10 d}+\frac{(1953 A+2354 C) \cos (d x) \sin (c)}{7392 d}+\frac{(25 A+18 C) \cos (2 d x) \sin (2 c)}{240 d}+\frac{(189 A+44 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{A \cos (4 d x) \sin (4 c)}{96 d}+\frac{A \cos (5 d x) \sin (5 c)}{704 d}+\frac{(1953 A+2354 C) \cos (c) \sin (d x)}{7392 d}+\frac{(25 A+18 C) \cos (2 c) \sin (2 d x)}{240 d}+\frac{(189 A+44 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{A \cos (4 c) \sin (4 d x)}{96 d}+\frac{A \cos (5 c) \sin (5 d x)}{704 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{7 C (\cos (c+d x)+1)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x)+1)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{22 d \sqrt{\cot ^2(c)+1}}-\frac{13 C (\cos (c+d x)+1)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}\right)","\frac{4 a^3 (105 A+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^3 (35 A+44 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{385 d}+\frac{2 (35 A+33 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d}+\frac{4 a^3 (105 A+143 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{4 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"a^3*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/10*((5*A + 7*C)*Cot[c])/d + ((1953*A + 2354*C)*Cos[d*x]*Sin[c])/(7392*d) + ((25*A + 18*C)*Cos[2*d*x]*Sin[2*c])/(240*d) + ((189*A + 44*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + (A*Cos[4*d*x]*Sin[4*c])/(96*d) + (A*Cos[5*d*x]*Sin[5*c])/(704*d) + ((1953*A + 2354*C)*Cos[c]*Sin[d*x])/(7392*d) + ((25*A + 18*C)*Cos[2*c]*Sin[2*d*x])/(240*d) + ((189*A + 44*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + (A*Cos[4*c]*Sin[4*d*x])/(96*d) + (A*Cos[5*c]*Sin[5*d*x])/(704*d)) - (5*A*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (13*C*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (7*C*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d))","C",0
1100,1,1116,213,6.4673678,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(-\frac{(17 A+27 C) \cot (c)}{15 d}+\frac{(97 A+84 C) \cos (d x) \sin (c)}{168 d}+\frac{(73 A+18 C) \cos (2 d x) \sin (2 c)}{360 d}+\frac{3 A \cos (3 d x) \sin (3 c)}{56 d}+\frac{A \cos (4 d x) \sin (4 c)}{144 d}+\frac{(97 A+84 C) \cos (c) \sin (d x)}{168 d}+\frac{(73 A+18 C) \cos (2 c) \sin (2 d x)}{360 d}+\frac{3 A \cos (3 c) \sin (3 d x)}{56 d}+\frac{A \cos (4 c) \sin (4 d x)}{144 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C}-\frac{17 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{9 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{11 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (11 A+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (17 A+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (16 A+21 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (73 A+63 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{4 A \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(-1/15*((17*A + 27*C)*Cot[c])/d + ((97*A + 84*C)*Cos[d*x]*Sin[c])/(168*d) + ((73*A + 18*C)*Cos[2*d*x]*Sin[2*c])/(360*d) + (3*A*Cos[3*d*x]*Sin[3*c])/(56*d) + (A*Cos[4*d*x]*Sin[4*c])/(144*d) + ((97*A + 84*C)*Cos[c]*Sin[d*x])/(168*d) + ((73*A + 18*C)*Cos[2*c]*Sin[2*d*x])/(360*d) + (3*A*Cos[3*c]*Sin[3*d*x])/(56*d) + (A*Cos[4*c]*Sin[4*d*x])/(144*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (11*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + A*Cos[2*c + 2*d*x])) - (9*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1101,1,1108,215,6.6079507,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(-\frac{(14 \cos (2 c) A+14 A+5 C+15 C \cos (2 c)) \csc (c) \sec (c)}{20 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{2 d}+\frac{(107 A+28 C) \cos (d x) \sin (c)}{168 d}+\frac{3 A \cos (2 d x) \sin (2 c)}{20 d}+\frac{A \cos (3 d x) \sin (3 c)}{56 d}+\frac{(107 A+28 C) \cos (c) \sin (d x)}{168 d}+\frac{3 A \cos (2 c) \sin (2 d x)}{20 d}+\frac{A \cos (3 c) \sin (3 d x)}{56 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C}-\frac{7 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{13 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{5 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (13 A+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (41 A-35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (11 A-35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}+\frac{2 (A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(-1/20*((14*A + 5*C + 14*A*Cos[2*c] + 15*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((107*A + 28*C)*Cos[d*x]*Sin[c])/(168*d) + (3*A*Cos[2*d*x]*Sin[2*c])/(20*d) + (A*Cos[3*d*x]*Sin[3*c])/(56*d) + ((107*A + 28*C)*Cos[c]*Sin[d*x])/(168*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(2*d) + (3*A*Cos[2*c]*Sin[2*d*x])/(20*d) + (A*Cos[3*c]*Sin[3*d*x])/(56*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (13*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (7*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + A*Cos[2*c + 2*d*x])) - (C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1102,1,1089,211,6.6493419,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}+\frac{\sec (c) (C \sin (c)+9 C \sin (d x)) \sec (c+d x)}{6 d}-\frac{(18 \cos (2 c) A+18 A-25 C+5 C \cos (2 c)) \csc (c) \sec (c)}{20 d}+\frac{A \cos (d x) \sin (c)}{2 d}+\frac{A \cos (2 d x) \sin (2 c)}{20 d}+\frac{A \cos (c) \sin (d x)}{2 d}+\frac{A \cos (2 c) \sin (2 d x)}{20 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C}-\frac{9 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{5 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (9 A-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{8 a^3 (3 A-10 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (3 A-35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(-1/20*((18*A - 25*C + 18*A*Cos[2*c] + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(2*d) + (A*Cos[2*d*x]*Sin[2*c])/(20*d) + (A*Cos[c]*Sin[d*x])/(2*d) + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*(C*Sin[c] + 9*C*Sin[d*x]))/(6*d) + (A*Cos[2*c]*Sin[2*d*x])/(20*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (9*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1103,1,1085,213,6.6865622,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) (C \sin (c)+5 C \sin (d x)) \sec ^2(c+d x)}{10 d}+\frac{\sec (c) (5 C \sin (c)+5 A \sin (d x)+18 C \sin (d x)) \sec (c+d x)}{10 d}-\frac{(15 \cos (2 c) A+5 A-36 C) \csc (c) \sec (c)}{20 d}+\frac{A \cos (d x) \sin (c)}{6 d}+\frac{A \cos (c) \sin (d x)}{6 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C}-\frac{A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{9 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{5 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (5 A+21 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (5 A+11 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(-1/20*((5*A - 36*C + 15*A*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(6*d) + (A*Cos[c]*Sin[d*x])/(6*d) + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(10*d) + (Sec[c]*Sec[c + d*x]^2*(C*Sin[c] + 5*C*Sin[d*x]))/(10*d) + (Sec[c]*Sec[c + d*x]*(5*C*Sin[c] + 5*A*Sin[d*x] + 18*C*Sin[d*x]))/(10*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (5*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (9*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1104,1,1102,213,6.7590986,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{14 d}+\frac{\sec (c) (5 C \sin (c)+21 C \sin (d x)) \sec ^3(c+d x)}{70 d}+\frac{\sec (c) (63 C \sin (c)+35 A \sin (d x)+130 C \sin (d x)) \sec ^2(c+d x)}{210 d}+\frac{\sec (c) (35 A \sin (c)+130 C \sin (c)+315 A \sin (d x)+294 C \sin (d x)) \sec (c+d x)}{210 d}-\frac{(5 \cos (2 c) A-25 A-28 C) \csc (c) \sec (c)}{20 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C}+\frac{A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{7 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{5 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{13 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (35 A+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (5 A+7 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (70 A+53 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{12 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(-1/20*((-25*A - 28*C + 5*A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(14*d) + (Sec[c]*Sec[c + d*x]^3*(5*C*Sin[c] + 21*C*Sin[d*x]))/(70*d) + (Sec[c]*Sec[c + d*x]^2*(63*C*Sin[c] + 35*A*Sin[d*x] + 130*C*Sin[d*x]))/(210*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 130*C*Sin[c] + 315*A*Sin[d*x] + 294*C*Sin[d*x]))/(210*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (5*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (13*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (7*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1105,1,1135,246,6.8613428,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{18 d}+\frac{\sec (c) (7 C \sin (c)+27 C \sin (d x)) \sec ^4(c+d x)}{126 d}+\frac{\sec (c) (135 C \sin (c)+63 A \sin (d x)+238 C \sin (d x)) \sec ^3(c+d x)}{630 d}+\frac{\sec (c) (63 A \sin (c)+238 C \sin (c)+315 A \sin (d x)+330 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac{\sec (c) (105 A \sin (c)+110 C \sin (c)+378 A \sin (d x)+238 C \sin (d x)) \sec (c+d x)}{210 d}+\frac{(27 A+17 C) \csc (c) \sec (c)}{15 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C}+\frac{9 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{17 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{11 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (21 A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (27 A+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{8 a^3 (21 A+16 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (63 A+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (27 A+17 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(((27*A + 17*C)*Csc[c]*Sec[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^4*(7*C*Sin[c] + 27*C*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^3*(135*C*Sin[c] + 63*A*Sin[d*x] + 238*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(105*A*Sin[c] + 110*C*Sin[c] + 378*A*Sin[d*x] + 238*C*Sin[d*x]))/(210*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 238*C*Sin[c] + 315*A*Sin[d*x] + 330*C*Sin[d*x]))/(630*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (11*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (9*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (17*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1106,1,1179,279,6.9540642,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^6(c+d x)}{22 d}+\frac{\sec (c) (3 C \sin (c)+11 C \sin (d x)) \sec ^5(c+d x)}{66 d}+\frac{\sec (c) (77 C \sin (c)+33 A \sin (d x)+126 C \sin (d x)) \sec ^4(c+d x)}{462 d}+\frac{\sec (c) (165 A \sin (c)+630 C \sin (c)+693 A \sin (d x)+770 C \sin (d x)) \sec ^3(c+d x)}{2310 d}+\frac{\sec (c) (693 A \sin (c)+770 C \sin (c)+1430 A \sin (d x)+1050 C \sin (d x)) \sec ^2(c+d x)}{2310 d}+\frac{\sec (c) (715 A \sin (c)+525 C \sin (c)+1617 A \sin (d x)+1155 C \sin (d x)) \sec (c+d x)}{1155 d}+\frac{(7 A+5 C) \csc (c) \sec (c)}{5 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C}+\frac{7 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C)}+\frac{C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C)}-\frac{13 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}-\frac{5 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{11 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (143 A+105 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{4 a^3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (143 A+105 C) \sin (c+d x)}{231 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{8 a^3 (44 A+35 C) \sin (c+d x)}{385 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 (33 A+35 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (7 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{4 C \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{33 a d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*(((7*A + 5*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(22*d) + (Sec[c]*Sec[c + d*x]^5*(3*C*Sin[c] + 11*C*Sin[d*x]))/(66*d) + (Sec[c]*Sec[c + d*x]^4*(77*C*Sin[c] + 33*A*Sin[d*x] + 126*C*Sin[d*x]))/(462*d) + (Sec[c]*Sec[c + d*x]^3*(165*A*Sin[c] + 630*C*Sin[c] + 693*A*Sin[d*x] + 770*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]^2*(693*A*Sin[c] + 770*C*Sin[c] + 1430*A*Sin[d*x] + 1050*C*Sin[d*x]))/(2310*d) + (Sec[c]*Sec[c + d*x]*(715*A*Sin[c] + 525*C*Sin[c] + 1617*A*Sin[d*x] + 1155*C*Sin[d*x]))/(1155*d)))/(A + 2*C + A*Cos[2*c + 2*d*x]) - (13*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(11*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (7*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + A*Cos[2*c + 2*d*x]))","C",0
1107,1,1393,192,6.7641452,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\frac{21 i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{3 i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 (16 \cos (c) A+5 A+5 C+10 C \cos (c)) \csc (c)}{5 d}+\frac{2 (51 A+28 C) \cos (d x) \sin (c)}{21 d}-\frac{4 A \cos (2 d x) \sin (2 c)}{5 d}+\frac{2 A \cos (3 d x) \sin (3 c)}{7 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{2 (51 A+28 C) \cos (c) \sin (d x)}{21 d}-\frac{4 A \cos (2 c) \sin (2 d x)}{5 d}+\frac{2 A \cos (3 c) \sin (3 d x)}{7 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{30 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{10 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","\frac{5 (9 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(9 A+7 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(7 A+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (9 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}",1,"(((-21*I)/10)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((4*(5*A + 5*C + 16*A*Cos[c] + 10*C*Cos[c])*Csc[c])/(5*d) + (2*(51*A + 28*C)*Cos[d*x]*Sin[c])/(21*d) - (4*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (2*A*Cos[3*d*x]*Sin[3*c])/(7*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (2*(51*A + 28*C)*Cos[c]*Sin[d*x])/(21*d) - (4*A*Cos[2*c]*Sin[2*d*x])/(5*d) + (2*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (30*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (10*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1108,1,1345,159,6.6875111,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{21 i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{3 i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 (16 \cos (c) A+5 A+5 C+10 C \cos (c)) \csc (c)}{5 d}-\frac{8 A \cos (d x) \sin (c)}{3 d}+\frac{4 A \cos (2 d x) \sin (2 c)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}-\frac{8 A \cos (c) \sin (d x)}{3 d}+\frac{4 A \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{10 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}+\frac{2 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","-\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (7 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(5 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((21*I)/10)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((-4*(5*A + 5*C + 16*A*Cos[c] + 10*C*Cos[c])*Csc[c])/(5*d) - (8*A*Cos[d*x]*Sin[c])/(3*d) + (4*A*Cos[2*d*x]*Sin[2*c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (8*A*Cos[c]*Sin[d*x])/(3*d) + (4*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (10*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1109,1,1300,122,6.5981347,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\frac{3 i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 (2 \cos (c) A+A+C) \csc (c)}{d}+\frac{8 A \cos (d x) \sin (c)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{8 A \cos (c) \sin (d x)}{3 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{10 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{2 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","\frac{(5 A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((-3*I)/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - ((I/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((4*(A + C + 2*A*Cos[c])*Csc[c])/d + (8*A*Cos[d*x]*Sin[c])/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*A*Cos[c]*Sin[d*x])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (10*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1110,1,1270,84,6.5467672,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{3 i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 (2 \cos (c) A+A+C) \csc (c)}{d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{2 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{2 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","-\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"(((3*I)/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + ((I/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((-4*(A + C + 2*A*Cos[c])*Csc[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1111,1,1304,112,6.6677356,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\frac{i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{3 i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 (\cos (c) C+2 C+A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{8 C \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{2 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}+\frac{2 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","\frac{(A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A+3 C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"((-1/2*I)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((2*(2*C + A*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1112,1,1337,150,7.0980257,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{3 i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{8 C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{8 \sec (c) (C \sin (c)-3 C \sin (d x)) \sec (c+d x)}{3 d}-\frac{2 (\cos (c) C+2 C+A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{2 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{10 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A+5 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(A+3 C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"((I/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((-2*(2*C + A*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (8*Sec[c]*Sec[c + d*x]*(C*Sin[c] - 3*C*Sin[d*x]))/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (10*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1113,1,1382,192,7.4146035,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","-\frac{3 i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}-\frac{21 i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{8 C \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{8 \sec (c) (3 C \sin (c)-5 C \sin (d x)) \sec ^2(c+d x)}{15 d}-\frac{8 \sec (c) (5 C \sin (c)-15 A \sin (d x)-24 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{2 (5 \cos (c) A+10 A+16 C+5 C \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)}+\frac{2 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}+\frac{10 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","-\frac{(3 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A+5 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(5 A+7 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (5 A+7 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}",1,"(((-3*I)/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((21*I)/10)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2)*((2*(10*A + 16*C + 5*A*Cos[c] + 5*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) - (8*Sec[c]*Sec[c + d*x]*(5*C*Sin[c] - 15*A*Sin[d*x] - 24*C*Sin[d*x]))/(15*d) + (8*Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] - 5*C*Sin[d*x]))/(15*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (10*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1114,1,1398,196,6.8821903,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{56 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{4 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{16 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{16 (18 \cos (c) A+10 A+5 C+5 C \cos (c)) \csc (c)}{5 d}-\frac{32 A \cos (d x) \sin (c)}{3 d}+\frac{8 A \cos (2 d x) \sin (2 c)}{5 d}-\frac{32 A \cos (c) \sin (d x)}{3 d}+\frac{8 A \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{20 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}+\frac{20 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","-\frac{5 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{4 (14 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(3 A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{4 (14 A+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (3 A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(((56*I)/5)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + ((4*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*((-16*(10*A + 5*C + 18*A*Cos[c] + 5*C*Cos[c])*Csc[c])/(5*d) - (32*A*Cos[d*x]*Sin[c])/(3*d) + (8*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (32*A*Cos[c]*Sin[d*x])/(3*d) + (8*A*Cos[2*c]*Sin[2*d*x])/(5*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1115,1,1355,161,6.7769914,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\frac{7 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{8 (4 \cos (c) A+3 A+C) \csc (c)}{d}+\frac{16 A \cos (d x) \sin (c)}{3 d}+\frac{16 A \cos (c) \sin (d x)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}-\frac{40 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{8 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{2 (5 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(7 A+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{2 (5 A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((-7*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (I*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (40*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (8*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*((8*(3*A + C + 4*A*Cos[c])*Csc[c])/d + (16*A*Cos[d*x]*Sin[c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (16*A*Cos[c]*Sin[d*x])/(3*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1116,1,934,130,6.6081204,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{4 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{16 A \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{16 A \cot \left(\frac{c}{2}\right)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{20 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","-\frac{(5 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{4 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((4*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (4*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*((-16*A*Cot[c/2])/d - (16*A*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1117,1,1322,125,6.7233409,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{8 (A-C) \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}-\frac{8 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{8 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{2 (A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((-I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (I*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (8*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (8*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*((8*(A - C)*Csc[c])/d + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1118,1,954,151,6.738348,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{4 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{16 C \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{16 C \cot \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{16 C \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}+\frac{20 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{(A-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-5 C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}-\frac{4 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{4 C \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"((-4*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*((16*C*Cot[c/2]*Sec[c])/d + (16*C*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (16*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1119,1,1391,189,7.5058917,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{7 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+3 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (3 \cos (c) C+4 C+A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{16 C \sec (c) \sec ^2(c+d x) \sin (d x)}{3 d}+\frac{16 \sec (c) \sec (c+d x) (C \sin (c)-6 C \sin (d x))}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^2}-\frac{8 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{40 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{2 (A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A+7 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{2 (A+5 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(A+7 C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(I*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + ((7*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (8*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (40*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*((-4*(4*C + A*Cos[c] + 3*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/d + (16*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (16*Sec[c]*Sec[c + d*x]*(C*Sin[c] - 6*C*Sin[d*x]))/(3*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1120,1,1507,250,7.2504716,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{231 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{49 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(12 A \sin \left(\frac{d x}{2}\right)+7 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{16 (12 A+7 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(99 A \sin \left(\frac{d x}{2}\right)+29 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 (132 \cos (c) A+99 A+29 C+20 C \cos (c)) \csc (c)}{5 d}-\frac{32 A \cos (d x) \sin (c)}{d}+\frac{16 A \cos (2 d x) \sin (2 c)}{5 d}-\frac{32 A \cos (c) \sin (d x)}{d}+\frac{16 A \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{84 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}+\frac{52 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","-\frac{(63 A+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (33 A+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(63 A+13 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (33 A+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(63 A+13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 (6 A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(((231*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((49*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (84*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (52*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*((-8*(99*A + 29*C + 132*A*Cos[c] + 20*C*Cos[c])*Csc[c])/(5*d) - (32*A*Cos[d*x]*Sin[c])/d + (16*A*Cos[2*d*x]*Sin[2*c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(12*A*Sin[(d*x)/2] + 7*C*Sin[(d*x)/2]))/(15*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(99*A*Sin[(d*x)/2] + 29*C*Sin[(d*x)/2]))/(5*d) - (32*A*Cos[c]*Sin[d*x])/d + (16*A*Cos[2*c]*Sin[2*d*x])/(5*d) + (16*(12*A + 7*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1121,1,1470,209,7.0461885,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\frac{119 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{9 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(19 A \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (19 A+9 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(59 A \sin \left(\frac{d x}{2}\right)+9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 (60 \cos (c) A+59 A+9 C) \csc (c)}{5 d}+\frac{32 A \cos (d x) \sin (c)}{3 d}+\frac{32 A \cos (c) \sin (d x)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{44 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(11 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}-\frac{(119 A+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(11 A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a^3 d}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(((-119*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (((9*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (44*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*((8*(59*A + 9*C + 60*A*Cos[c])*Csc[c])/(5*d) + (32*A*Cos[d*x]*Sin[c])/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(19*A*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(15*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(59*A*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(5*d) + (32*A*Cos[c]*Sin[d*x])/(3*d) - (8*(19*A + 9*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1122,1,1446,186,8.0505591,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\frac{49 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(7 A \sin \left(\frac{d x}{2}\right)+2 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{16 (7 A+2 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(29 A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 (20 \cos (c) A+29 A-C) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{52 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","-\frac{(13 A-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(49 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 (4 A-C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(((49*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - ((I/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (52*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*((-8*(29*A - C + 20*A*Cos[c])*Csc[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(7*A*Sin[(d*x)/2] + 2*C*Sin[(d*x)/2]))/(15*d) + (16*(7*A + 2*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1123,1,1439,184,6.7633581,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 (9 A-C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 (9 A-C) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(9 A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{2 (3 A-2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"(((-9*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + ((I/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*((8*(9*A - C)*Csc[c])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(9*A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*(9*A - C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1124,1,1436,180,6.7948488,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{9 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)-3 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{16 (2 A-3 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-9 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 (A-9 C) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 (2 A-3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((-1/5*I)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((9*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*((8*(A - 9*C)*Csc[c])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(5*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(2*A*Sin[(d*x)/2] - 3*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (16*(2*A - 3*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1125,1,1473,209,7.0375038,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{49 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+11 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{8 (A+11 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-29 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (-29 \cos (c) C-20 C+A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{32 C \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}+\frac{52 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(A-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A-49 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}+\frac{(A-13 C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{2 (A-4 C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"((I/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (((49*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (52*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*((-4*(-20*C + A*Cos[c] - 29*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 29*C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(15*d) + (32*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (8*(A + 11*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1126,1,1505,242,7.6846427,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3),x]","\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{119 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}+\frac{\left(C \sec ^2(c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{16 \sec \left(\frac{c}{2}\right) \left(3 A \sin \left(\frac{d x}{2}\right)+8 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{16 (3 A+8 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)+59 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (59 \cos (c) C+60 C+9 A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{32 C \sec (c) \sec ^2(c+d x) \sin (d x)}{3 d}+\frac{32 \sec (c) \sec (c+d x) (C \sin (c)-9 C \sin (d x))}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (\cos (2 c+2 d x) A+A+2 C) (\sec (c+d x) a+a)^3}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{44 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) \left(C \sec ^2(c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(A+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{2 a^3 d}+\frac{(9 A+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A+119 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A+11 C) \sin (c+d x)}{2 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(9 A+119 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}-\frac{2 C \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(((9*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((119*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (44*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + C*Sec[c + d*x]^2)*((-4*(60*C + 9*A*Cos[c] + 59*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(3*A*Sin[(d*x)/2] + 8*C*Sin[(d*x)/2]))/(15*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] + 59*C*Sin[(d*x)/2]))/(5*d) + (32*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (32*Sec[c]*Sec[c + d*x]*(C*Sin[c] - 9*C*Sin[d*x]))/(3*d) - (16*(3*A + 8*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1127,1,109,213,0.284286,"\int \cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left((48 A+63 C) \cos ^2(c+d x)+(64 A+84 C) \cos (c+d x)+35 A \cos ^4(c+d x)+40 A \cos ^3(c+d x)+8 (16 A+21 C)\right)}{315 d (\cos (c+d x)+1)}","\frac{2 a (16 A+21 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a (16 A+21 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a (16 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[Cos[c + d*x]]*(8*(16*A + 21*C) + (64*A + 84*C)*Cos[c + d*x] + (48*A + 63*C)*Cos[c + d*x]^2 + 40*A*Cos[c + d*x]^3 + 35*A*Cos[c + d*x]^4)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(315*d*(1 + Cos[c + d*x]))","A",1
1128,1,90,168,0.2660843,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} ((141 A+140 C) \cos (c+d x)+36 A \cos (2 (c+d x))+15 A \cos (3 (c+d x))+228 A+280 C)}{210 d (\cos (c+d x)+1)}","\frac{2 a (24 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (24 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{7 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*(228*A + 280*C + (141*A + 140*C)*Cos[c + d*x] + 36*A*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(210*d*(1 + Cos[c + d*x]))","A",1
1129,1,68,122,0.2254295,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (8 A \cos (c+d x)+3 A \cos (2 (c+d x))+19 A+30 C)}{15 d}","\frac{2 a (7 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{5 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}",1,"(Sqrt[Cos[c + d*x]]*(19*A + 30*C + 8*A*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(15*d)","A",1
1130,1,92,136,0.5100478,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(A \left(3 \sin \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{3}{2} (c+d x)\right)\right)+3 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3 d}","\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + A*(3*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(3*d)","A",1
1131,1,90,135,0.5645945,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A+C \sec (c+d x))+\sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d}","\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}+\frac{\sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(2*A + C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
1132,1,105,144,0.5154625,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (8 A+3 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+C \left(\sin \left(\frac{1}{2} (c+d x)\right)+3 \sin \left(\frac{3}{2} (c+d x)\right)\right)\right)}{8 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{\sqrt{a} (8 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(8*A + 3*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + C*(Sin[(c + d*x)/2] + 3*Sin[(3*(c + d*x))/2])))/(8*d*Cos[c + d*x]^(3/2))","A",1
1133,1,125,189,0.8955665,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+5 C) \cos (2 (c+d x))+24 A+20 C \cos (c+d x)+31 C)+3 \sqrt{2} (8 A+5 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a (8 A+5 C) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(8*A + 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (24*A + 31*C + 20*C*Cos[c + d*x] + 3*(8*A + 5*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
1134,1,152,234,1.45871,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((432 A+539 C) \cos (c+d x)+4 (48 A+35 C) \cos (2 (c+d x))+144 A \cos (3 (c+d x))+192 A+105 C \cos (3 (c+d x))+332 C)+6 \sqrt{2} (48 A+35 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a (48 A+35 C) \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a (48 A+35 C) \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (48 A+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C \sin (c+d x)}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(48*A + 35*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (192*A + 332*C + (432*A + 539*C)*Cos[c + d*x] + 4*(48*A + 35*C)*Cos[2*(c + d*x)] + 144*A*Cos[3*(c + d*x)] + 105*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","A",1
1135,1,125,266,1.765825,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (5789 A+5566 C) \cos (c+d x)+8 (581 A+429 C) \cos (2 (c+d x))+1645 A \cos (3 (c+d x))+490 A \cos (4 (c+d x))+105 A \cos (5 (c+d x))+18494 A+660 C \cos (3 (c+d x))+21736 C)}{9240 d}","\frac{2 a^2 (28 A+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{231 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (112 A+143 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{385 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (112 A+143 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (112 A+143 C) \sin (c+d x)}{1155 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{33 d}",1,"(a*Sqrt[Cos[c + d*x]]*(18494*A + 21736*C + 2*(5789*A + 5566*C)*Cos[c + d*x] + 8*(581*A + 429*C)*Cos[2*(c + d*x)] + 1645*A*Cos[3*(c + d*x)] + 660*C*Cos[3*(c + d*x)] + 490*A*Cos[4*(c + d*x)] + 105*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(9240*d)","A",1
1136,1,103,219,1.1742456,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (799 A+756 C) \cos (c+d x)+4 (137 A+63 C) \cos (2 (c+d x))+170 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+2689 A+3276 C)}{1260 d}","\frac{2 a^2 (52 A+63 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+189 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (136 A+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}",1,"(a*Sqrt[Cos[c + d*x]]*(2689*A + 3276*C + 2*(799*A + 756*C)*Cos[c + d*x] + 4*(137*A + 63*C)*Cos[2*(c + d*x)] + 170*A*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d)","A",1
1137,1,85,169,0.7842101,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((253 A+140 C) \cos (c+d x)+78 A \cos (2 (c+d x))+15 A \cos (3 (c+d x))+494 A+700 C)}{210 d}","\frac{8 a^2 (19 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (19 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{6 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}",1,"(a*Sqrt[Cos[c + d*x]]*(494*A + 700*C + (253*A + 140*C)*Cos[c + d*x] + 78*A*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d)","A",1
1138,1,105,183,0.7648223,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (6 A \cos (c+d x)+A \cos (2 (c+d x))+13 A+10 C)+5 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{5 d}","\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (4 A+5 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{5 d}",1,"(a*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(5*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (13*A + 10*C + 6*A*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(5*d)","A",1
1139,1,110,189,0.8075247,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (10 A \cos (c+d x)+A \cos (2 (c+d x))+A+3 C)+9 \sqrt{2} C \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 d \sqrt{\cos (c+d x)}}","\frac{3 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 (8 A-3 C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{a (2 A-3 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(9*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(A + 3*C + 10*A*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d*Sqrt[Cos[c + d*x]])","A",1
1140,1,120,191,1.2304165,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (4 A \cos (2 (c+d x))+4 A+7 C \cos (c+d x)+2 C)+\sqrt{2} (8 A+7 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{3/2} (8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (8 A-5 C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{3 a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d \sqrt{\cos (c+d x)}}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(8*A + 7*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*(4*A + 2*C + 7*C*Cos[c + d*x] + 4*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d*Cos[c + d*x]^(3/2))","A",1
1141,1,126,191,1.6220982,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+11 C) \cos (2 (c+d x))+24 A+44 C \cos (c+d x)+49 C)+3 \sqrt{2} (24 A+11 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a^{3/2} (24 A+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+19 C) \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(24*A + 11*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (24*A + 49*C + 44*C*Cos[c + d*x] + 3*(8*A + 11*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
1142,1,154,238,2.7334593,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (7 (48 A+55 C) \cos (c+d x)+4 (16 A+25 C) \cos (2 (c+d x))+112 A \cos (3 (c+d x))+64 A+75 C \cos (3 (c+d x))+164 C)+2 \sqrt{2} (112 A+75 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{256 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a^{3/2} (112 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+75 C) \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (16 A+13 C) \sin (c+d x)}{32 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(2*Sqrt[2]*(112*A + 75*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (64*A + 164*C + 7*(48*A + 55*C)*Cos[c + d*x] + 4*(16*A + 25*C)*Cos[2*(c + d*x)] + 112*A*Cos[3*(c + d*x)] + 75*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(256*d*Cos[c + d*x]^(7/2))","A",1
1143,1,176,285,4.0723861,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (880 A+1273 C) \cos (c+d x)+4 (3280 A+3059 C) \cos (2 (c+d x))+3520 A \cos (3 (c+d x))+2640 A \cos (4 (c+d x))+10480 A+2660 C \cos (3 (c+d x))+1995 C \cos (4 (c+d x))+13313 C)+60 \sqrt{2} (176 A+133 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{a^{3/2} (176 A+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{3 a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(176*A + 133*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (10480*A + 13313*C + 12*(880*A + 1273*C)*Cos[c + d*x] + 4*(3280*A + 3059*C)*Cos[2*(c + d*x)] + 3520*A*Cos[3*(c + d*x)] + 2660*C*Cos[3*(c + d*x)] + 2640*A*Cos[4*(c + d*x)] + 1995*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d*Cos[c + d*x]^(9/2))","A",1
1144,1,148,313,3.1916962,"\int \cos ^{\frac{13}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (8 (226573 A+222794 C) \cos (c+d x)+(746519 A+581152 C) \cos (2 (c+d x))+287060 A \cos (3 (c+d x))+94010 A \cos (4 (c+d x))+23940 A \cos (5 (c+d x))+3465 A \cos (6 (c+d x))+2798182 A+148720 C \cos (3 (c+d x))+20020 C \cos (4 (c+d x))+3233516 C)}{720720 d}","\frac{2 a^3 (2224 A+2717 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (8368 A+10439 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15015 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^3 (8368 A+10439 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{45045 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^3 (8368 A+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+143 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{11}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d}+\frac{10 a A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(2798182*A + 3233516*C + 8*(226573*A + 222794*C)*Cos[c + d*x] + (746519*A + 581152*C)*Cos[2*(c + d*x)] + 287060*A*Cos[3*(c + d*x)] + 148720*C*Cos[3*(c + d*x)] + 94010*A*Cos[4*(c + d*x)] + 20020*C*Cos[4*(c + d*x)] + 23940*A*Cos[5*(c + d*x)] + 3465*A*Cos[6*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(720720*d)","A",1
1145,1,127,266,2.129582,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (6989 A+6666 C) \cos (c+d x)+16 (325 A+198 C) \cos (2 (c+d x))+1735 A \cos (3 (c+d x))+448 A \cos (4 (c+d x))+63 A \cos (5 (c+d x))+22928 A+396 C \cos (3 (c+d x))+27456 C)}{5544 d}","\frac{2 a^3 (232 A+297 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (568 A+759 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (568 A+759 C) \sin (c+d x)}{693 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (32 A+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{231 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d}+\frac{10 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(22928*A + 27456*C + 2*(6989*A + 6666*C)*Cos[c + d*x] + 16*(325*A + 198*C)*Cos[2*(c + d*x)] + 1735*A*Cos[3*(c + d*x)] + 396*C*Cos[3*(c + d*x)] + 448*A*Cos[4*(c + d*x)] + 63*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(5544*d)","A",1
1146,1,105,216,1.4046195,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (4 (779 A+588 C) \cos (c+d x)+4 (254 A+63 C) \cos (2 (c+d x))+260 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+5653 A+7476 C)}{1260 d}","\frac{64 a^3 (13 A+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (13 A+21 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a (13 A+21 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{9 d}+\frac{10 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(5653*A + 7476*C + 4*(779*A + 588*C)*Cos[c + d*x] + 4*(254*A + 63*C)*Cos[2*(c + d*x)] + 260*A*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d)","A",1
1147,1,125,230,1.3196372,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((101 A+28 C) \cos (c+d x)+24 A \cos (2 (c+d x))+3 A \cos (3 (c+d x))+208 A+224 C)+84 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{84 d}","\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (32 A+49 C) \sin (c+d x)}{21 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(84*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(208*A + 224*C + (101*A + 28*C)*Cos[c + d*x] + 24*A*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(84*d)","A",1
1148,1,131,230,1.379799,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((181 A+60 C) \cos (c+d x)+28 A \cos (2 (c+d x))+3 A \cos (3 (c+d x))+28 A+30 C)+150 \sqrt{2} C \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{60 d \sqrt{\cos (c+d x)}}","\frac{5 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 (64 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (16 A-15 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(150*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(28*A + 30*C + (181*A + 60*C)*Cos[c + d*x] + 28*A*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(60*d*Sqrt[Cos[c + d*x]])","A",1
1149,1,139,244,1.5054939,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((6 A+33 C) \cos (c+d x)+32 A \cos (2 (c+d x))+2 A \cos (3 (c+d x))+32 A+6 C)+6 \sqrt{2} (8 A+19 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{5/2} (8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (56 A-27 C) \sin (c+d x)}{12 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (8 A-21 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{12 d \sqrt{\cos (c+d x)}}-\frac{a (4 A-3 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{6 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(8*A + 19*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 4*(32*A + 6*C + (6*A + 33*C)*Cos[c + d*x] + 32*A*Cos[2*(c + d*x)] + 2*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(3/2))","A",1
1150,1,144,238,2.0489924,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((72 A+68 C) \cos (c+d x)+3 (8 A+25 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+24 A+91 C)+15 \sqrt{2} (8 A+5 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{5 a^{5/2} (8 A+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (24 A-49 C) \sin (c+d x)}{24 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+31 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{24 d \sqrt{\cos (c+d x)}}+\frac{5 a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt{\cos (c+d x)}}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*(8*A + 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (24*A + 91*C + (72*A + 68*C)*Cos[c + d*x] + 3*(8*A + 25*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
1151,1,155,238,3.1088558,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((1584 A+2203 C) \cos (c+d x)+4 (48 A+163 C) \cos (2 (c+d x))+528 A \cos (3 (c+d x))+192 A+489 C \cos (3 (c+d x))+844 C)+6 \sqrt{2} (304 A+163 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a^{5/2} (304 A+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+299 C) \sin (c+d x)}{192 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (16 A+17 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{5 a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(304*A + 163*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (192*A + 844*C + (1584*A + 2203*C)*Cos[c + d*x] + 4*(48*A + 163*C)*Cos[2*(c + d*x)] + 528*A*Cos[3*(c + d*x)] + 489*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","A",1
1152,1,178,285,3.0192569,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (1360 A+2343 C) \cos (c+d x)+4 (6640 A+6509 C) \cos (2 (c+d x))+5440 A \cos (3 (c+d x))+6000 A \cos (4 (c+d x))+20560 A+5660 C \cos (3 (c+d x))+4245 C \cos (4 (c+d x))+24863 C)+60 \sqrt{2} (400 A+283 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{a^{5/2} (400 A+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1040 A+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+79 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(400*A + 283*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (20560*A + 24863*C + 12*(1360*A + 2343*C)*Cos[c + d*x] + 4*(6640*A + 6509*C)*Cos[2*(c + d*x)] + 5440*A*Cos[3*(c + d*x)] + 5660*C*Cos[3*(c + d*x)] + 6000*A*Cos[4*(c + d*x)] + 4245*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d*Cos[c + d*x]^(9/2))","A",1
1153,1,200,332,4.2385267,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (14 (4056 A+4591 C) \cos (c+d x)+16 (1496 A+1711 C) \cos (2 (c+d x))+25448 A \cos (3 (c+d x))+5216 A \cos (4 (c+d x))+3912 A \cos (5 (c+d x))+18720 A+21721 C \cos (3 (c+d x))+4060 C \cos (4 (c+d x))+3045 C \cos (5 (c+d x))+27412 C)+24 \sqrt{2} (1304 A+1015 C) \cos ^6(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24576 d \cos ^{\frac{11}{2}}(c+d x)}","\frac{a^{5/2} (1304 A+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+23 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{96 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(24*Sqrt[2]*(1304*A + 1015*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^6 + (18720*A + 27412*C + 14*(4056*A + 4591*C)*Cos[c + d*x] + 16*(1496*A + 1711*C)*Cos[2*(c + d*x)] + 25448*A*Cos[3*(c + d*x)] + 21721*C*Cos[3*(c + d*x)] + 5216*A*Cos[4*(c + d*x)] + 4060*C*Cos[4*(c + d*x)] + 3912*A*Cos[5*(c + d*x)] + 3045*C*Cos[5*(c + d*x)])*Sin[(c + d*x)/2]))/(24576*d*Cos[c + d*x]^(11/2))","A",1
1154,1,166,244,1.5861413,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(105 \sqrt{2} (A+C) \sec ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 \sin ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\sec (c+d x)} \sec ^3(c+d x) (24 A \cos (c+d x)+15 A \cos (2 (c+d x))+101 A+70 C)\right)}{105 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (31 A+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}-\frac{2 (43 A+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}-\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}",1,"-1/105*(Cos[c + d*x]^(5/2)*(105*Sqrt[2]*(A + C)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(7/2) + 2*(101*A + 70*C + 24*A*Cos[c + d*x] + 15*A*Cos[2*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^3*Sin[(c + d*x)/2]^2)*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1155,1,153,201,0.3934266,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(\sqrt{1-\sec (c+d x)} \sec ^2(c+d x) (-2 A \cos (c+d x)+3 A \cos (2 (c+d x))+29 A+30 C)+15 \sqrt{2} (A+C) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{15 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (13 A+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}-\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^(3/2)*((29*A + 30*C - 2*A*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^2 + 15*Sqrt[2]*(A + C)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(5/2))*Sin[c + d*x])/(15*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1156,1,73,156,0.5203725,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(3 (A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 A \sin ^3\left(\frac{1}{2} (c+d x)\right)\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{2 A \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*(3*(A + C)*ArcTanh[Sin[(c + d*x)/2]] - 4*A*Sin[(c + d*x)/2]^3))/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1157,1,93,175,0.4454856,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-\left((A+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 A \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*Cos[(c + d*x)/2]*(-((A + C)*ArcTanh[Sin[(c + d*x)/2]]) + Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*A*Sin[(c + d*x)/2]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1158,1,105,173,0.3743325,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 (A+C) \cos (c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right)-\sqrt{2} C \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(Cos[(c + d*x)/2]*(2*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x] - Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*C*Sin[(c + d*x)/2]))/(d*Cos[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1159,1,130,223,0.8302707,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(8 (A+C) \cos ^2(c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} (8 A+7 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+C \left(\sin \left(\frac{3}{2} (c+d x)\right)-5 \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(8 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}-\frac{C \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"-1/4*(Cos[(c + d*x)/2]*(8*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x]^2 - Sqrt[2]*(8*A + 7*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + C*(-5*Sin[(c + d*x)/2] + Sin[(3*(c + d*x))/2])))/(d*Cos[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1160,1,149,266,1.3158365,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+7 C) \cos (2 (c+d x))+24 A-4 C \cos (c+d x)+37 C)+48 (A+C) \cos ^3(c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-3 \sqrt{2} (8 A+9 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{(8 A+7 C) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(8 A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{a} d}-\frac{C \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(48*(A + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x]^3 - 3*Sqrt[2]*(8*A + 9*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (24*A + 37*C - 4*C*Cos[c + d*x] + 3*(8*A + 7*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1161,1,118,268,1.9539409,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((39 A+20 C) \cos (c+d x)-2 A \cos (2 (c+d x))+A \cos (3 (c+d x))+47 A+25 C)-5 (15 A+7 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{(15 A+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(13 A+5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 a d \sqrt{a \sec (c+d x)+a}}+\frac{(49 A+25 C) \sin (c+d x)}{10 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(-5*(15*A + 7*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (47*A + 25*C + (39*A + 20*C)*Cos[c + d*x] - 2*A*Cos[2*(c + d*x)] + A*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1162,1,104,221,1.4292364,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{3 (11 A+3 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\tan \left(\frac{1}{2} (c+d x)\right) (12 A \cos (c+d x)-2 A \cos (2 (c+d x))+17 A+3 C)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(11 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{(19 A+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(3*(11*A + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] - (17*A + 3*C + 12*A*Cos[c + d*x] - 2*A*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1163,1,114,172,1.6756189,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (4 A \cos (c+d x)+5 A+C)-(7 A-C) (\cos (c+d x)+1) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d \left(\sin ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","-\frac{(7 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"-1/2*(Cos[(c + d*x)/2]^3*(-((7*A - C)*ArcTanh[Sin[(c + d*x)/2]]*(1 + Cos[c + d*x])) + 2*(5*A + C + 4*A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2)*(-1 + Sin[(c + d*x)/2]^2))","A",1
1164,1,114,185,1.3636989,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{-(A+C) \tan \left(\frac{1}{2} (c+d x)\right)+(3 A-5 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \sqrt{2} C \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(3 A-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"((3*A - 5*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + 4*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] - (A + C)*Tan[(c + d*x)/2])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1165,1,169,228,1.9951294,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right) \left(2 (A+9 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{12 \sqrt{2} C \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-2 \sin \left(\frac{1}{2} (c+d x)\right) (A+2 C \sec (c+d x)+3 C)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(A+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}-\frac{3 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(A+3 C) \sin (c+d x)}{2 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2)*(2*(A + 9*C)*ArcTanh[Sin[(c + d*x)/2]] + (12*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 - 2*(A + 3*C + 2*C*Sec[c + d*x])*Sin[(c + d*x)/2])/(-1 + Sin[(c + d*x)/2]^2)))/(d*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1166,1,213,285,3.2842564,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(A \cos ^2(c+d x)+C\right) \left(\sin \left(\frac{1}{2} (c+d x)\right) ((2 A+7 C) \cos (2 (c+d x))+2 A+6 C \cos (c+d x)+3 C)+(5 A+13 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{(8 A+19 C) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{3}{2} (c+d x)\right)\right)^2 \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{\sqrt{2}}\right)}{4 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)} (A \cos (2 (c+d x))+A+2 C)}","-\frac{(5 A+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(8 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(2 A+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"-1/4*((C + A*Cos[c + d*x]^2)*Sec[(c + d*x)/2]*((5*A + 13*C)*ArcTanh[Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])^2 - ((8*A + 19*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*(Cos[(c + d*x)/2] + Cos[(3*(c + d*x))/2])^2)/Sqrt[2] + (2*A + 3*C + 6*C*Cos[c + d*x] + (2*A + 7*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(a*d*Cos[c + d*x]^(5/2)*(A + 2*C + A*Cos[2*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1167,1,150,315,3.5480216,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (5 (887 A+255 C) \cos (c+d x)+16 (52 A+15 C) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+12 A \cos (4 (c+d x))+3491 A+975 C)-120 (283 A+75 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{960 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","-\frac{(283 A+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A+45 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{80 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(787 A+195 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{240 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(2671 A+735 C) \sin (c+d x)}{240 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(21 A+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(-120*(283*A + 75*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(3491*A + 975*C + 5*(887*A + 255*C)*Cos[c + d*x] + 16*(52*A + 15*C)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 12*A*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(960*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1168,1,132,266,2.7982428,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{6 (163 A+19 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\tan \left(\frac{1}{2} (c+d x)\right) ((479 A+39 C) \cos (c+d x)+80 A \cos (2 (c+d x))-8 A \cos (3 (c+d x))+379 A+27 C)}{48 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(163 A+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{5 (19 A+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(299 A+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(17 A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(6*(163*A + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 - (379*A + 27*C + (479*A + 39*C)*Cos[c + d*x] + 80*A*Cos[2*(c + d*x)] - 8*A*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(48*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1169,1,118,219,2.3633538,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (5 (17 A+C) \cos (c+d x)+16 A \cos (2 (c+d x))+65 A+C)-40 (15 A-C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{64 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","-\frac{5 (15 A-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(13 A-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(-40*(15*A - C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(65*A + C + 5*(17*A + C)*Cos[c + d*x] + 16*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(64*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1170,1,110,174,1.7029835,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((3 C-13 A) \cos (c+d x)-9 A+7 C)+8 (19 A+3 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{64 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","\frac{(19 A+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-7 C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(8*(19*A + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(-9*A + 7*C + (-13*A + 3*C)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(64*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1171,1,144,232,2.7934387,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((5 A-11 C) \cos (c+d x)+A-15 C)+2 (5 A-43 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+64 \sqrt{2} C \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(5 A-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(5 A-11 C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(2*(5*A - 43*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + 64*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (A - 15*C + (5*A - 11*C)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1172,1,187,277,3.2377987,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(A+C \sec ^2(c+d x)\right) \left((6 A+230 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sec ^3\left(\frac{1}{2} (c+d x)\right) (2 (7 A+55 C) \cos (c+d x)+(3 A+35 C) \cos (2 (c+d x))+3 A+67 C)-160 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \sqrt{\cos (c+d x)} (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 C)}","\frac{(3 A+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{5 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A+35 C) \sin (c+d x)}{16 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A-15 C) \sin (c+d x)}{16 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A+C) \sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(A + C*Sec[c + d*x]^2)*((6*A + 230*C)*ArcTanh[Sin[(c + d*x)/2]] - 160*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + ((3*A + 67*C + 2*(7*A + 55*C)*Cos[c + d*x] + (3*A + 35*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Sec[c + d*x]*Tan[(c + d*x)/2])/2))/(4*d*Sqrt[Cos[c + d*x]]*(A + 2*C + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
1173,1,77,111,0.4644648,"\int \cos ^{\frac{9}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (15 B \cos (2 (c+d x))+65 B+42 C \cos (c+d x))+50 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{10 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{10 B \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(126*C*EllipticE[(c + d*x)/2, 2] + 50*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*B + 42*C*Cos[c + d*x] + 15*B*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
1174,1,66,87,0.2113644,"\int \cos ^{\frac{7}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \left(\sin (c+d x) \sqrt{\cos (c+d x)} (3 B \cos (c+d x)+5 C)+9 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(9*B*EllipticE[(c + d*x)/2, 2] + 5*C*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*C + 3*B*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
1175,1,53,61,0.0986517,"\int \cos ^{\frac{5}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \left(B \left(F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)}\right)+3 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*(3*C*EllipticE[(c + d*x)/2, 2] + B*(EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*Sin[c + d*x])))/(3*d)","A",1
1176,1,35,35,0.0599679,"\int \cos ^{\frac{3}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}","\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(2*B*EllipticE[(c + d*x)/2, 2])/d + (2*C*EllipticF[(c + d*x)/2, 2])/d","A",1
1177,1,51,57,0.1293411,"\int \sqrt{\cos (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \left(B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{C \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}-\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*(-(C*EllipticE[(c + d*x)/2, 2]) + B*EllipticF[(c + d*x)/2, 2] + (C*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
1178,1,65,83,0.3807163,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","\frac{\frac{2 \sin (c+d x) (3 B \cos (c+d x)+C)}{\cos ^{\frac{3}{2}}(c+d x)}-6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*B*EllipticE[(c + d*x)/2, 2] + 2*C*EllipticF[(c + d*x)/2, 2] + (2*(C + 3*B*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))/(3*d)","A",1
1179,1,95,111,0.2820936,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2),x]","\frac{10 B \sin (c+d x)+10 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 C \sin (2 (c+d x))+6 C \tan (c+d x)-18 C \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{6 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 C \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 C \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}",1,"(-18*C*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*B*Sin[c + d*x] + 9*C*Sin[2*(c + d*x)] + 6*C*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
1180,1,86,123,0.5341496,"\int \cos ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (15 A \cos (2 (c+d x))+65 A+42 B \cos (c+d x)+70 C)+10 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{105 d}","\frac{2 (5 A+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (5 A+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(126*B*EllipticE[(c + d*x)/2, 2] + 10*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*A + 70*C + 42*B*Cos[c + d*x] + 15*A*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
1181,1,72,93,0.2640367,"\int \cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \left(\sin (c+d x) \sqrt{\cos (c+d x)} (3 A \cos (c+d x)+5 B)+3 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 (3 A+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}",1,"(2*(3*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*A*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
1182,1,682,65,6.2332224,"\int \cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\frac{2 B \csc (c) \cos ^2(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{d (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{4 A \csc (c) \cos ^2(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{4 C \csc (c) \cos ^2(c+d x) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 A \sin (c) \cos (d x)}{3 d}+\frac{4 A \cos (c) \sin (d x)}{3 d}-\frac{4 B \cot (c)}{d}\right)}{A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C}","\frac{2 (A+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*B*Cot[c])/d + (4*A*Cos[d*x]*Sin[c])/(3*d) + (4*A*Cos[c]*Sin[d*x])/(3*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^2*Csc[c]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1183,1,272,61,6.105226,"\int \sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-\frac{2 (A-C) \sec (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-4 B \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+\frac{\csc (c) \left(-2 \sqrt{\sec ^2(c)} (A \cos (2 c+d x)+(A-2 C) \cos (d x))+3 (A-C) \sec (c) \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+(A-C) \sec (c) \cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{d (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((Csc[c]*(3*(A - C)*Cos[c - d*x - ArcTan[Tan[c]]]*Sec[c] + (A - C)*Cos[c + d*x + ArcTan[Tan[c]]]*Sec[c] - 2*((A - 2*C)*Cos[d*x] + A*Cos[2*c + d*x])*Sqrt[Sec[c]^2]))/Sqrt[Sec[c]^2] - 4*B*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - (2*(A - C)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sec[c]*Sin[d*x + ArcTan[Tan[c]]])/(Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2])))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)]))","C",0
1184,1,69,87,0.5203385,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","\frac{2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{2 \sin (c+d x) (3 B \cos (c+d x)+C)}{\cos ^{\frac{3}{2}}(c+d x)}-6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}","\frac{2 (3 A+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(-6*B*EllipticE[(c + d*x)/2, 2] + 2*(3*A + C)*EllipticF[(c + d*x)/2, 2] + (2*(C + 3*B*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2))/(3*d)","A",1
1185,1,112,123,0.4571626,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2),x]","\frac{-6 (5 A+3 C) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 A \sin (2 (c+d x))+10 B \sin (c+d x)+10 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+9 C \sin (2 (c+d x))+6 C \tan (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","-\frac{2 (5 A+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (5 A+3 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(5*A + 3*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*B*Sin[c + d*x] + 15*A*Sin[2*(c + d*x)] + 9*C*Sin[2*(c + d*x)] + 6*C*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
1186,1,129,147,0.5289738,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(5/2),x]","\frac{10 (7 A+5 C) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+35 A \sin (2 (c+d x))+42 B \sin (c+d x)-126 B \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 B \sin (c+d x) \cos ^2(c+d x)+25 C \sin (2 (c+d x))+30 C \tan (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{2 (7 A+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 (7 A+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{6 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{6 B \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-126*B*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 10*(7*A + 5*C)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 42*B*Sin[c + d*x] + 126*B*Cos[c + d*x]^2*Sin[c + d*x] + 35*A*Sin[2*(c + d*x)] + 25*C*Sin[2*(c + d*x)] + 30*C*Tan[c + d*x])/(105*d*Cos[c + d*x]^(5/2))","A",1
1187,1,1292,175,6.3491451,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(7 A+9 B+9 C) \cot (c)}{15 d}+\frac{(23 A+23 B+28 C) \cos (d x) \sin (c)}{84 d}+\frac{(19 A+18 B+18 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(A+B) \cos (3 d x) \sin (3 c)}{28 d}+\frac{A \cos (4 d x) \sin (4 c)}{72 d}+\frac{(23 A+23 B+28 C) \cos (c) \sin (d x)}{84 d}+\frac{(19 A+18 B+18 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(A+B) \cos (3 c) \sin (3 d x)}{28 d}+\frac{A \cos (4 c) \sin (4 d x)}{72 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{7 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (5 (A+B)+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (7 A+9 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a (7 A+9 (B+C)) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{2 a (5 (A+B)+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/15*((7*A + 9*B + 9*C)*Cot[c])/d + ((23*A + 23*B + 28*C)*Cos[d*x]*Sin[c])/(84*d) + ((19*A + 18*B + 18*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((A + B)*Cos[3*d*x]*Sin[3*c])/(28*d) + (A*Cos[4*d*x]*Sin[4*c])/(72*d) + ((23*A + 23*B + 28*C)*Cos[c]*Sin[d*x])/(84*d) + ((19*A + 18*B + 18*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((A + B)*Cos[3*c]*Sin[3*d*x])/(28*d) + (A*Cos[4*c]*Sin[4*d*x])/(72*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (5*B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (7*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d) - (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1188,1,1240,142,6.3070372,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(3 A+3 B+5 C) \cot (c)}{5 d}+\frac{(23 A+28 B+28 C) \cos (d x) \sin (c)}{84 d}+\frac{(A+B) \cos (2 d x) \sin (2 c)}{10 d}+\frac{A \cos (3 d x) \sin (3 c)}{28 d}+\frac{(23 A+28 B+28 C) \cos (c) \sin (d x)}{84 d}+\frac{(A+B) \cos (2 c) \sin (2 d x)}{10 d}+\frac{A \cos (3 c) \sin (3 d x)}{28 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (5 A+7 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 a (3 (A+B)+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+7 (B+C)) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((3*A + 3*B + 5*C)*Cot[c])/d + ((23*A + 28*B + 28*C)*Cos[d*x]*Sin[c])/(84*d) + ((A + B)*Cos[2*d*x]*Sin[2*c])/(10*d) + (A*Cos[3*d*x]*Sin[3*c])/(28*d) + ((23*A + 28*B + 28*C)*Cos[c]*Sin[d*x])/(84*d) + ((A + B)*Cos[2*c]*Sin[2*d*x])/(10*d) + (A*Cos[3*c]*Sin[3*d*x])/(28*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1189,1,1186,106,6.3886683,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(3 A+5 B+5 C) \cot (c)}{5 d}+\frac{(A+B) \cos (d x) \sin (c)}{3 d}+\frac{A \cos (2 d x) \sin (2 c)}{10 d}+\frac{(A+B) \cos (c) \sin (d x)}{3 d}+\frac{A \cos (2 c) \sin (2 d x)}{10 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (3 A+5 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((3*A + 5*B + 5*C)*Cot[c])/d + ((A + B)*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[2*d*x]*Sin[2*c])/(10*d) + ((A + B)*Cos[c]*Sin[d*x])/(3*d) + (A*Cos[2*c]*Sin[2*d*x])/(10*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1190,1,1173,98,6.4738192,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(\cos (2 c) A+A+B-2 C+B \cos (2 c)) \csc (c) \sec (c)}{2 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{A \cos (d x) \sin (c)}{3 d}+\frac{A \cos (c) \sin (d x)}{3 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+3 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((A + B - 2*C + A*Cos[2*c] + B*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[c]*Sin[d*x])/(3*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) - (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1191,1,1180,103,6.548187,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (C \sin (c)+3 B \sin (d x)+3 C \sin (d x)) \sec (c+d x)}{3 d}-\frac{(\cos (2 c) A+A-2 B-2 C) \csc (c) \sec (c)}{2 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+3 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A-B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (B+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/2*((A - 2*B - 2*C + A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*B*Sin[d*x] + 3*C*Sin[d*x]))/(3*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
1192,1,1228,141,6.5998946,"\int \frac{(a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 C \sin (c)+5 B \sin (d x)+5 C \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 B \sin (c)+5 C \sin (c)+15 A \sin (d x)+15 B \sin (d x)+9 C \sin (d x)) \sec (c+d x)}{15 d}+\frac{(5 A+5 B+3 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (5 A+5 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 A+5 B+3 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (B+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((5*A + 5*B + 3*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*B*Sin[d*x] + 5*C*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*B*Sin[c] + 5*C*Sin[c] + 15*A*Sin[d*x] + 15*B*Sin[d*x] + 9*C*Sin[d*x]))/(15*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1193,1,1284,177,6.6352056,"\int \frac{(a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{7 d}+\frac{\sec (c) (5 C \sin (c)+7 B \sin (d x)+7 C \sin (d x)) \sec ^3(c+d x)}{35 d}+\frac{\sec (c) (21 B \sin (c)+21 C \sin (c)+35 A \sin (d x)+35 B \sin (d x)+25 C \sin (d x)) \sec ^2(c+d x)}{105 d}+\frac{\sec (c) (35 A \sin (c)+35 B \sin (c)+25 C \sin (c)+105 A \sin (d x)+63 B \sin (d x)+63 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{(5 A+3 B+3 C) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}+\frac{3 C (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{5 C (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (7 A+7 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 a (5 A+3 (B+C)) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (7 A+7 B+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 A+3 (B+C)) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (B+C) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((5*A + 3*B + 3*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (Sec[c]*Sec[c + d*x]^3*(5*C*Sin[c] + 7*B*Sin[d*x] + 7*C*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]^2*(21*B*Sin[c] + 21*C*Sin[c] + 35*A*Sin[d*x] + 35*B*Sin[d*x] + 25*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 35*B*Sin[c] + 25*C*Sin[c] + 105*A*Sin[d*x] + 63*B*Sin[d*x] + 63*C*Sin[d*x]))/(105*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (5*C*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) + (3*C*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1194,1,1364,251,6.410583,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a^2 \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1)^2 \left(-\frac{(7 A+8 B+9 C) \cot (c)}{15 d}+\frac{(941 A+1012 B+1122 C) \cos (d x) \sin (c)}{3696 d}+\frac{(38 A+37 B+36 C) \cos (2 d x) \sin (2 c)}{360 d}+\frac{(101 A+88 B+44 C) \cos (3 d x) \sin (3 c)}{2464 d}+\frac{(2 A+B) \cos (4 d x) \sin (4 c)}{144 d}+\frac{A \cos (5 d x) \sin (5 c)}{352 d}+\frac{(941 A+1012 B+1122 C) \cos (c) \sin (d x)}{3696 d}+\frac{(38 A+37 B+36 C) \cos (2 c) \sin (2 d x)}{360 d}+\frac{(101 A+88 B+44 C) \cos (3 c) \sin (3 d x)}{2464 d}+\frac{(2 A+B) \cos (4 c) \sin (4 d x)}{144 d}+\frac{A \cos (5 c) \sin (5 d x)}{352 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{7 A (\cos (c+d x)+1)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d}-\frac{4 B (\cos (c+d x)+1)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{3 C (\cos (c+d x)+1)^2 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{50 A (\cos (c+d x)+1)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{231 d \sqrt{\cot ^2(c)+1}}-\frac{5 B (\cos (c+d x)+1)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x)+1)^2 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}\right)","\frac{4 a^2 (50 A+55 B+66 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^2 (7 A+8 B+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (89 A+121 B+99 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d}+\frac{4 a^2 (7 A+8 B+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (50 A+55 B+66 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (4 A+11 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}",1,"a^2*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/15*((7*A + 8*B + 9*C)*Cot[c])/d + ((941*A + 1012*B + 1122*C)*Cos[d*x]*Sin[c])/(3696*d) + ((38*A + 37*B + 36*C)*Cos[2*d*x]*Sin[2*c])/(360*d) + ((101*A + 88*B + 44*C)*Cos[3*d*x]*Sin[3*c])/(2464*d) + ((2*A + B)*Cos[4*d*x]*Sin[4*c])/(144*d) + (A*Cos[5*d*x]*Sin[5*c])/(352*d) + ((941*A + 1012*B + 1122*C)*Cos[c]*Sin[d*x])/(3696*d) + ((38*A + 37*B + 36*C)*Cos[2*c]*Sin[2*d*x])/(360*d) + ((101*A + 88*B + 44*C)*Cos[3*c]*Sin[3*d*x])/(2464*d) + ((2*A + B)*Cos[4*c]*Sin[4*d*x])/(144*d) + (A*Cos[5*c]*Sin[5*d*x])/(352*d)) - (50*A*(1 + Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[c]^2]) - (5*B*(1 + Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (2*C*(1 + Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (7*A*(1 + Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d) - (4*B*(1 + Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d) - (3*C*(1 + Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
1195,1,1699,215,6.3868153,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 (8 A+9 B+12 C) \cot (c)}{15 d}+\frac{(46 A+51 B+56 C) \cos (d x) \sin (c)}{84 d}+\frac{(37 A+36 B+18 C) \cos (2 d x) \sin (2 c)}{180 d}+\frac{(2 A+B) \cos (3 d x) \sin (3 c)}{28 d}+\frac{A \cos (4 d x) \sin (4 c)}{72 d}+\frac{(46 A+51 B+56 C) \cos (c) \sin (d x)}{84 d}+\frac{(37 A+36 B+18 C) \cos (2 c) \sin (2 d x)}{180 d}+\frac{(2 A+B) \cos (3 c) \sin (3 d x)}{28 d}+\frac{A \cos (4 c) \sin (4 d x)}{72 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{8 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{3 B \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{10 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (5 A+6 B+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (8 A+9 B+12 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (19 A+27 B+21 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{4 a^2 (5 A+6 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 (4 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}{9 d}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(8*A + 9*B + 12*C)*Cot[c])/(15*d) + ((46*A + 51*B + 56*C)*Cos[d*x]*Sin[c])/(84*d) + ((37*A + 36*B + 18*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((2*A + B)*Cos[3*d*x]*Sin[3*c])/(28*d) + (A*Cos[4*d*x]*Sin[4*c])/(72*d) + ((46*A + 51*B + 56*C)*Cos[c]*Sin[d*x])/(84*d) + ((37*A + 36*B + 18*C)*Cos[2*c]*Sin[2*d*x])/(180*d) + ((2*A + B)*Cos[3*c]*Sin[3*d*x])/(28*d) + (A*Cos[4*c]*Sin[4*d*x])/(72*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (10*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (3*B*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1196,1,2001,179,6.7000266,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{4 a^2 (6 A+7 B+14 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (3 A+4 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (33 A+49 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (4 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{35 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}{7 d}",1,"(((3*I)/10)*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) + (((2*I)/5)*B*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) + ((I/2)*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) + (Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(3*A + 4*B + 5*C)*Cot[c])/(5*d) + ((51*A + 56*B + 28*C)*Cos[d*x]*Sin[c])/(84*d) + ((2*A + B)*Cos[2*d*x]*Sin[2*c])/(10*d) + (A*Cos[3*d*x]*Sin[3*c])/(28*d) + ((51*A + 56*B + 28*C)*Cos[c]*Sin[d*x])/(84*d) + ((2*A + B)*Cos[2*c]*Sin[2*d*x])/(10*d) + (A*Cos[3*c]*Sin[3*d*x])/(28*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1197,1,1356,170,6.584751,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{(8 \cos (2 c) A+8 A+10 B-5 C+10 B \cos (2 c)+5 C \cos (2 c)) \csc (c) \sec (c)}{10 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{(2 A+B) \cos (d x) \sin (c)}{3 d}+\frac{A \cos (2 d x) \sin (2 c)}{10 d}+\frac{(2 A+B) \cos (c) \sin (d x)}{3 d}+\frac{A \cos (2 c) \sin (2 d x)}{10 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{4 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{B \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (A+2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (7 A+5 B-15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{4 a^2 (4 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (A-5 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/10*((8*A + 10*B - 5*C + 8*A*Cos[2*c] + 10*B*Cos[2*c] + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((2*A + B)*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[2*d*x]*Sin[2*c])/(10*d) + ((2*A + B)*Cos[c]*Sin[d*x])/(3*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (A*Cos[2*c]*Sin[2*d*x])/(10*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (B*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1198,1,1583,170,6.7864516,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (C \sin (c)+3 B \sin (d x)+6 C \sin (d x)) \sec (c+d x)}{3 d}-\frac{(2 \cos (2 c) A+2 A-B-4 C+B \cos (2 c)) \csc (c) \sec (c)}{2 d}+\frac{A \cos (d x) \sin (c)}{3 d}+\frac{A \cos (c) \sin (d x)}{3 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}+\frac{i A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4(c+d x)}{2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{i C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4(c+d x)}{2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{2 B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (2 A+3 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (A-3 B-5 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{4 a^2 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (3 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"((I/2)*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - ((I/2)*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) + (Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/2*((2*A - B - 4*C + 2*A*Cos[2*c] + B*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[c]*Sin[d*x])/(3*d) + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*B*Sin[d*x] + 6*C*Sin[d*x]))/(3*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1199,1,1599,174,6.9137664,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 C \sin (c)+5 B \sin (d x)+10 C \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 B \sin (c)+10 C \sin (c)+15 A \sin (d x)+30 B \sin (d x)+24 C \sin (d x)) \sec (c+d x)}{15 d}-\frac{(5 \cos (2 c) A-5 A-20 B-16 C) \csc (c) \sec (c)}{10 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{i B \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4(c+d x)}{2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 i C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4(c+d x)}{5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{2 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (3 A+2 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (15 A+25 B+17 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 (5 B+4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (5 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"((-1/2*I)*B*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (((2*I)/5)*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) + (Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/10*((-5*A - 20*B - 16*C + 5*A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*B*Sin[d*x] + 10*C*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*B*Sin[c] + 10*C*Sin[c] + 15*A*Sin[d*x] + 30*B*Sin[d*x] + 24*C*Sin[d*x]))/(15*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1200,1,2041,215,7.0120057,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{4 a^2 (14 A+7 B+6 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (5 A+4 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (35 A+49 B+33 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+4 B+3 C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (7 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((-1/2*I)*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (((2*I)/5)*B*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (((3*I)/10)*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) + (Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(5*A + 4*B + 3*C)*Csc[c]*Sec[c])/(5*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (Sec[c]*Sec[c + d*x]^3*(5*C*Sin[c] + 7*B*Sin[d*x] + 14*C*Sin[d*x]))/(35*d) + (Sec[c]*Sec[c + d*x]^2*(21*B*Sin[c] + 42*C*Sin[c] + 35*A*Sin[d*x] + 70*B*Sin[d*x] + 60*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 70*B*Sin[c] + 60*C*Sin[c] + 210*A*Sin[d*x] + 168*B*Sin[d*x] + 126*C*Sin[d*x]))/(105*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1201,1,1741,251,6.9391061,"\int \frac{(a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{9 d}+\frac{\sec (c) (7 C \sin (c)+9 B \sin (d x)+18 C \sin (d x)) \sec ^4(c+d x)}{63 d}+\frac{\sec (c) (45 B \sin (c)+90 C \sin (c)+63 A \sin (d x)+126 B \sin (d x)+112 C \sin (d x)) \sec ^3(c+d x)}{315 d}+\frac{\sec (c) (63 A \sin (c)+126 B \sin (c)+112 C \sin (c)+210 A \sin (d x)+180 B \sin (d x)+150 C \sin (d x)) \sec ^2(c+d x)}{315 d}+\frac{2 \sec (c) (35 A \sin (c)+30 B \sin (c)+25 C \sin (c)+84 A \sin (d x)+63 B \sin (d x)+56 C \sin (d x)) \sec (c+d x)}{105 d}+\frac{2 (12 A+9 B+8 C) \csc (c) \sec (c)}{15 d}\right) \cos ^{\frac{9}{2}}(c+d x)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}+\frac{4 A \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{3 B \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{8 C \csc (c) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^4(c+d x)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{10 C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (\sec (c+d x) a+a)^2 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4(c+d x)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (7 A+6 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (12 A+9 B+8 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^2 (7 A+6 B+5 C) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (21 A+27 B+19 C) \sin (c+d x)}{105 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (12 A+9 B+8 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (9 B+4 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^2}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Cos[c + d*x]^(9/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(12*A + 9*B + 8*C)*Csc[c]*Sec[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(9*d) + (Sec[c]*Sec[c + d*x]^4*(7*C*Sin[c] + 9*B*Sin[d*x] + 18*C*Sin[d*x]))/(63*d) + (2*Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 30*B*Sin[c] + 25*C*Sin[c] + 84*A*Sin[d*x] + 63*B*Sin[d*x] + 56*C*Sin[d*x]))/(105*d) + (Sec[c]*Sec[c + d*x]^3*(45*B*Sin[c] + 90*C*Sin[c] + 63*A*Sin[d*x] + 126*B*Sin[d*x] + 112*C*Sin[d*x]))/(315*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 126*B*Sin[c] + 112*C*Sin[c] + 210*A*Sin[d*x] + 180*B*Sin[d*x] + 150*C*Sin[d*x]))/(315*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (10*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (4*A*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (3*B*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (8*C*Cos[c + d*x]^4*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1202,1,1364,267,6.4179576,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a^3 \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1)^3 \left(-\frac{(15 A+17 B+21 C) \cot (c)}{30 d}+\frac{(1953 A+2134 B+2354 C) \cos (d x) \sin (c)}{7392 d}+\frac{(75 A+73 B+54 C) \cos (2 d x) \sin (2 c)}{720 d}+\frac{(189 A+132 B+44 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{(3 A+B) \cos (4 d x) \sin (4 c)}{288 d}+\frac{A \cos (5 d x) \sin (5 c)}{704 d}+\frac{(1953 A+2134 B+2354 C) \cos (c) \sin (d x)}{7392 d}+\frac{(75 A+73 B+54 C) \cos (2 c) \sin (2 d x)}{720 d}+\frac{(189 A+132 B+44 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{(3 A+B) \cos (4 c) \sin (4 d x)}{288 d}+\frac{A \cos (5 c) \sin (5 d x)}{704 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{A (\cos (c+d x)+1)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 d}-\frac{17 B (\cos (c+d x)+1)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{7 C (\cos (c+d x)+1)^3 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 d}-\frac{5 A (\cos (c+d x)+1)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{22 d \sqrt{\cot ^2(c)+1}}-\frac{11 B (\cos (c+d x)+1)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}-\frac{13 C (\cos (c+d x)+1)^3 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{42 d \sqrt{\cot ^2(c)+1}}\right)","\frac{4 a^3 (105 A+121 B+143 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{4 a^3 (15 A+17 B+21 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d}+\frac{2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{693 d}+\frac{4 a^3 (105 A+121 B+143 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (6 A+11 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}",1,"a^3*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*((15*A + 17*B + 21*C)*Cot[c])/d + ((1953*A + 2134*B + 2354*C)*Cos[d*x]*Sin[c])/(7392*d) + ((75*A + 73*B + 54*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((189*A + 132*B + 44*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((3*A + B)*Cos[4*d*x]*Sin[4*c])/(288*d) + (A*Cos[5*d*x]*Sin[5*c])/(704*d) + ((1953*A + 2134*B + 2354*C)*Cos[c]*Sin[d*x])/(7392*d) + ((75*A + 73*B + 54*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((189*A + 132*B + 44*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((3*A + B)*Cos[4*c]*Sin[4*d*x])/(288*d) + (A*Cos[5*c]*Sin[5*d*x])/(704*d)) - (5*A*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot[c]^2]) - (11*B*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (13*C*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d*Sqrt[1 + Cot[c]^2]) - (A*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(4*d) - (17*B*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d) - (7*C*(1 + Cos[c + d*x])^3*Csc[c]*Sec[c/2 + (d*x)/2]^6*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(20*d))","C",0
1203,1,1697,231,6.5237523,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{(17 A+21 B+27 C) \cot (c)}{15 d}+\frac{(97 A+107 B+84 C) \cos (d x) \sin (c)}{168 d}+\frac{(73 A+54 B+18 C) \cos (2 d x) \sin (2 c)}{360 d}+\frac{(3 A+B) \cos (3 d x) \sin (3 c)}{56 d}+\frac{A \cos (4 d x) \sin (4 c)}{144 d}+\frac{(97 A+107 B+84 C) \cos (c) \sin (d x)}{168 d}+\frac{(73 A+54 B+18 C) \cos (2 c) \sin (2 d x)}{360 d}+\frac{(3 A+B) \cos (3 c) \sin (3 d x)}{56 d}+\frac{A \cos (4 c) \sin (4 d x)}{144 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{17 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{7 B \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{9 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{11 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{13 B \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (11 A+13 B+21 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (17 A+21 B+27 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (32 A+41 B+42 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (73 A+99 B+63 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{315 d}+\frac{2 (2 A+3 B) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/15*((17*A + 21*B + 27*C)*Cot[c])/d + ((97*A + 107*B + 84*C)*Cos[d*x]*Sin[c])/(168*d) + ((73*A + 54*B + 18*C)*Cos[2*d*x]*Sin[2*c])/(360*d) + ((3*A + B)*Cos[3*d*x]*Sin[3*c])/(56*d) + (A*Cos[4*d*x]*Sin[4*c])/(144*d) + ((97*A + 107*B + 84*C)*Cos[c]*Sin[d*x])/(168*d) + ((73*A + 54*B + 18*C)*Cos[2*c]*Sin[2*d*x])/(360*d) + ((3*A + B)*Cos[3*c]*Sin[3*d*x])/(56*d) + (A*Cos[4*c]*Sin[4*d*x])/(144*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (11*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (13*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (7*B*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (9*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1204,1,1688,227,6.7734024,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{(14 \cos (2 c) A+14 A+18 B+5 C+18 B \cos (2 c)+15 C \cos (2 c)) \csc (c) \sec (c)}{20 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{2 d}+\frac{(107 A+84 B+28 C) \cos (d x) \sin (c)}{168 d}+\frac{(3 A+B) \cos (2 d x) \sin (2 c)}{20 d}+\frac{A \cos (3 d x) \sin (3 c)}{56 d}+\frac{(107 A+84 B+28 C) \cos (c) \sin (d x)}{168 d}+\frac{(3 A+B) \cos (2 c) \sin (2 d x)}{20 d}+\frac{A \cos (3 c) \sin (3 d x)}{56 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{7 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{9 B \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{13 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{B \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{5 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (13 A+21 B+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^3 (7 A+9 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (41 A+42 B-35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{2 (11 A+7 B-35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{35 d}+\frac{2 (A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 a d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/20*((14*A + 18*B + 5*C + 14*A*Cos[2*c] + 18*B*Cos[2*c] + 15*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((107*A + 84*B + 28*C)*Cos[d*x]*Sin[c])/(168*d) + ((3*A + B)*Cos[2*d*x]*Sin[2*c])/(20*d) + (A*Cos[3*d*x]*Sin[3*c])/(56*d) + ((107*A + 84*B + 28*C)*Cos[c]*Sin[d*x])/(168*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(2*d) + ((3*A + B)*Cos[2*c]*Sin[2*d*x])/(20*d) + (A*Cos[3*c]*Sin[3*d*x])/(56*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (13*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (7*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (9*B*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1205,1,1672,226,6.7976238,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}+\frac{\sec (c) (C \sin (c)+3 B \sin (d x)+9 C \sin (d x)) \sec (c+d x)}{6 d}-\frac{(18 \cos (2 c) A+18 A+5 B-25 C+15 B \cos (2 c)+5 C \cos (2 c)) \csc (c) \sec (c)}{20 d}+\frac{(3 A+B) \cos (d x) \sin (c)}{6 d}+\frac{A \cos (2 d x) \sin (2 c)}{20 d}+\frac{(3 A+B) \cos (c) \sin (d x)}{6 d}+\frac{A \cos (2 c) \sin (2 d x)}{20 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{9 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{B \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{5 B \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{5 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (3 A+5 (B+C)) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (9 A+5 B-5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^3 (6 A-5 B-20 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (3 A-15 B-35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}{15 d}+\frac{2 (B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{a d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/20*((18*A + 5*B - 25*C + 18*A*Cos[2*c] + 15*B*Cos[2*c] + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((3*A + B)*Cos[d*x]*Sin[c])/(6*d) + (A*Cos[2*d*x]*Sin[2*c])/(20*d) + ((3*A + B)*Cos[c]*Sin[d*x])/(6*d) + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*B*Sin[d*x] + 9*C*Sin[d*x]))/(6*d) + (A*Cos[2*c]*Sin[2*d*x])/(20*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (9*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (B*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1206,1,1673,231,6.8455814,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) (3 C \sin (c)+5 B \sin (d x)+15 C \sin (d x)) \sec ^2(c+d x)}{30 d}+\frac{\sec (c) (5 B \sin (c)+15 C \sin (c)+15 A \sin (d x)+45 B \sin (d x)+54 C \sin (d x)) \sec (c+d x)}{30 d}-\frac{(15 \cos (2 c) A+5 A-25 B-36 C+5 B \cos (2 c)) \csc (c) \sec (c)}{20 d}+\frac{A \cos (d x) \sin (c)}{6 d}+\frac{A \cos (c) \sin (d x)}{6 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{B \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{9 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{5 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{5 B \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (5 A+5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^3 (5 A-5 B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}-\frac{4 a^3 (5 A+20 B+21 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 (15 A+35 B+33 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (5 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{15 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/20*((5*A - 25*B - 36*C + 15*A*Cos[2*c] + 5*B*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(6*d) + (A*Cos[c]*Sin[d*x])/(6*d) + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(10*d) + (Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*B*Sin[d*x] + 15*C*Sin[d*x]))/(30*d) + (Sec[c]*Sec[c + d*x]*(5*B*Sin[c] + 15*C*Sin[c] + 15*A*Sin[d*x] + 45*B*Sin[d*x] + 54*C*Sin[d*x]))/(30*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (5*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (B*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (9*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1207,1,1692,231,6.9581438,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{14 d}+\frac{\sec (c) (5 C \sin (c)+7 B \sin (d x)+21 C \sin (d x)) \sec ^3(c+d x)}{70 d}+\frac{\sec (c) (21 B \sin (c)+63 C \sin (c)+35 A \sin (d x)+105 B \sin (d x)+130 C \sin (d x)) \sec ^2(c+d x)}{210 d}+\frac{\sec (c) (35 A \sin (c)+105 B \sin (c)+130 C \sin (c)+315 A \sin (d x)+378 B \sin (d x)+294 C \sin (d x)) \sec (c+d x)}{210 d}-\frac{(5 \cos (2 c) A-25 A-36 B-28 C) \csc (c) \sec (c)}{20 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}+\frac{A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{9 B \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{7 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{5 A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{B \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{13 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (35 A+21 B+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (5 A+9 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (5 A+9 B+7 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (140 A+147 B+106 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (7 B+6 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{35 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/20*((-25*A - 36*B - 28*C + 5*A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(14*d) + (Sec[c]*Sec[c + d*x]^3*(5*C*Sin[c] + 7*B*Sin[d*x] + 21*C*Sin[d*x]))/(70*d) + (Sec[c]*Sec[c + d*x]^2*(21*B*Sin[c] + 63*C*Sin[c] + 35*A*Sin[d*x] + 105*B*Sin[d*x] + 130*C*Sin[d*x]))/(210*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 105*B*Sin[c] + 130*C*Sin[c] + 315*A*Sin[d*x] + 378*B*Sin[d*x] + 294*C*Sin[d*x]))/(210*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (5*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (13*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (9*B*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (7*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1208,1,1739,267,6.9815145,"\int \frac{(a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\cos ^{\frac{11}{2}}(c+d x) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{18 d}+\frac{\sec (c) (7 C \sin (c)+9 B \sin (d x)+27 C \sin (d x)) \sec ^4(c+d x)}{126 d}+\frac{\sec (c) (45 B \sin (c)+135 C \sin (c)+63 A \sin (d x)+189 B \sin (d x)+238 C \sin (d x)) \sec ^3(c+d x)}{630 d}+\frac{\sec (c) (63 A \sin (c)+189 B \sin (c)+238 C \sin (c)+315 A \sin (d x)+390 B \sin (d x)+330 C \sin (d x)) \sec ^2(c+d x)}{630 d}+\frac{\sec (c) (105 A \sin (c)+130 B \sin (c)+110 C \sin (c)+378 A \sin (d x)+294 B \sin (d x)+238 C \sin (d x)) \sec (c+d x)}{210 d}+\frac{(27 A+21 B+17 C) \csc (c) \sec (c)}{15 d}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}+\frac{9 A \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{7 B \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{17 C \cos ^5(c+d x) \csc (c) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{A \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{13 B \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{11 C \cos ^5(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^3 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^3 (21 A+13 B+11 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^3 (27 A+21 B+17 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^3 (42 A+41 B+32 C) \sin (c+d x)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (63 A+99 B+73 C) \sin (c+d x) \left(a^3 \cos (c+d x)+a^3\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (27 A+21 B+17 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (3 B+2 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{21 a d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((27*A + 21*B + 17*C)*Csc[c]*Sec[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(18*d) + (Sec[c]*Sec[c + d*x]^4*(7*C*Sin[c] + 9*B*Sin[d*x] + 27*C*Sin[d*x]))/(126*d) + (Sec[c]*Sec[c + d*x]^3*(45*B*Sin[c] + 135*C*Sin[c] + 63*A*Sin[d*x] + 189*B*Sin[d*x] + 238*C*Sin[d*x]))/(630*d) + (Sec[c]*Sec[c + d*x]*(105*A*Sin[c] + 130*B*Sin[c] + 110*C*Sin[c] + 378*A*Sin[d*x] + 294*B*Sin[d*x] + 238*C*Sin[d*x]))/(210*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 189*B*Sin[c] + 238*C*Sin[c] + 315*A*Sin[d*x] + 390*B*Sin[d*x] + 330*C*Sin[d*x]))/(630*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (13*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (11*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (9*A*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (7*B*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (17*C*Cos[c + d*x]^5*Csc[c]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1209,1,1416,310,6.5163032,"\int \cos ^{\frac{13}{2}}(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","a^4 \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1)^4 \left(-\frac{(185 A+208 B+247 C) \cot (c)}{390 d}+\frac{(3764 A+4087 B+4488 C) \cos (d x) \sin (c)}{14784 d}+\frac{(15625 A+15392 B+13208 C) \cos (2 d x) \sin (2 c)}{149760 d}+\frac{(404 A+321 B+176 C) \cos (3 d x) \sin (3 c)}{9856 d}+\frac{(98 A+52 B+13 C) \cos (4 d x) \sin (4 c)}{7488 d}+\frac{(4 A+B) \cos (5 d x) \sin (5 c)}{1408 d}+\frac{A \cos (6 d x) \sin (6 c)}{3328 d}+\frac{(3764 A+4087 B+4488 C) \cos (c) \sin (d x)}{14784 d}+\frac{(15625 A+15392 B+13208 C) \cos (2 c) \sin (2 d x)}{149760 d}+\frac{(404 A+321 B+176 C) \cos (3 c) \sin (3 d x)}{9856 d}+\frac{(98 A+52 B+13 C) \cos (4 c) \sin (4 d x)}{7488 d}+\frac{(4 A+B) \cos (5 c) \sin (5 d x)}{1408 d}+\frac{A \cos (6 c) \sin (6 d x)}{3328 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{37 A (\cos (c+d x)+1)^4 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{156 d}-\frac{4 B (\cos (c+d x)+1)^4 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{19 C (\cos (c+d x)+1)^4 \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{60 d}-\frac{50 A (\cos (c+d x)+1)^4 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{231 d \sqrt{\cot ^2(c)+1}}-\frac{113 B (\cos (c+d x)+1)^4 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{462 d \sqrt{\cot ^2(c)+1}}-\frac{2 C (\cos (c+d x)+1)^4 \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d \sqrt{\cot ^2(c)+1}}\right)","\frac{8 a^4 (100 A+113 B+132 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{8 a^4 (185 A+208 B+247 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{195 d}+\frac{4 a^4 (5255 A+6019 B+6721 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15015 d}+\frac{4 (1355 A+1612 B+1573 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{9009 d}+\frac{8 a^4 (100 A+113 B+132 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{231 d}+\frac{2 (13 A+17 B+11 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{99 d}+\frac{2 a (8 A+13 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{143 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^4}{13 d}",1,"a^4*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(-1/390*((185*A + 208*B + 247*C)*Cot[c])/d + ((3764*A + 4087*B + 4488*C)*Cos[d*x]*Sin[c])/(14784*d) + ((15625*A + 15392*B + 13208*C)*Cos[2*d*x]*Sin[2*c])/(149760*d) + ((404*A + 321*B + 176*C)*Cos[3*d*x]*Sin[3*c])/(9856*d) + ((98*A + 52*B + 13*C)*Cos[4*d*x]*Sin[4*c])/(7488*d) + ((4*A + B)*Cos[5*d*x]*Sin[5*c])/(1408*d) + (A*Cos[6*d*x]*Sin[6*c])/(3328*d) + ((3764*A + 4087*B + 4488*C)*Cos[c]*Sin[d*x])/(14784*d) + ((15625*A + 15392*B + 13208*C)*Cos[2*c]*Sin[2*d*x])/(149760*d) + ((404*A + 321*B + 176*C)*Cos[3*c]*Sin[3*d*x])/(9856*d) + ((98*A + 52*B + 13*C)*Cos[4*c]*Sin[4*d*x])/(7488*d) + ((4*A + B)*Cos[5*c]*Sin[5*d*x])/(1408*d) + (A*Cos[6*c]*Sin[6*d*x])/(3328*d)) - (50*A*(1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[c]^2]) - (113*B*(1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(462*d*Sqrt[1 + Cot[c]^2]) - (2*C*(1 + Cos[c + d*x])^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*Sqrt[1 + Cot[c]^2]) - (37*A*(1 + Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(156*d) - (4*B*(1 + Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d) - (19*C*(1 + Cos[c + d*x])^4*Csc[c]*Sec[c/2 + (d*x)/2]^8*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(60*d))","C",0
1210,1,1751,274,6.6542451,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{13}{2}}(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{(16 A+19 B+24 C) \cot (c)}{15 d}+\frac{(4087 A+4488 B+4202 C) \cos (d x) \sin (c)}{7392 d}+\frac{(148 A+127 B+72 C) \cos (2 d x) \sin (2 c)}{720 d}+\frac{(321 A+176 B+44 C) \cos (3 d x) \sin (3 c)}{4928 d}+\frac{(4 A+B) \cos (4 d x) \sin (4 c)}{288 d}+\frac{A \cos (5 d x) \sin (5 c)}{704 d}+\frac{(4087 A+4488 B+4202 C) \cos (c) \sin (d x)}{7392 d}+\frac{(148 A+127 B+72 C) \cos (2 c) \sin (2 d x)}{720 d}+\frac{(321 A+176 B+44 C) \cos (3 c) \sin (3 d x)}{4928 d}+\frac{(4 A+B) \cos (4 c) \sin (4 d x)}{288 d}+\frac{A \cos (5 c) \sin (5 d x)}{704 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{8 A \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{19 B \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 C \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{113 A \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{231 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{17 C \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{8 a^4 (113 A+132 B+187 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}+\frac{8 a^4 (16 A+19 B+24 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^4 (667 A+803 B+913 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{1155 d}+\frac{4 (769 A+946 B+891 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 \cos (c+d x)+a^4\right)}{3465 d}+\frac{2 (43 A+55 B+33 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{231 d}+\frac{2 a (8 A+11 B) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{99 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^4}{11 d}",1,"(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/15*((16*A + 19*B + 24*C)*Cot[c])/d + ((4087*A + 4488*B + 4202*C)*Cos[d*x]*Sin[c])/(7392*d) + ((148*A + 127*B + 72*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((321*A + 176*B + 44*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((4*A + B)*Cos[4*d*x]*Sin[4*c])/(288*d) + (A*Cos[5*d*x]*Sin[5*c])/(704*d) + ((4087*A + 4488*B + 4202*C)*Cos[c]*Sin[d*x])/(7392*d) + ((148*A + 127*B + 72*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((321*A + 176*B + 44*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((4*A + B)*Cos[4*c]*Sin[4*d*x])/(288*d) + (A*Cos[5*c]*Sin[5*d*x])/(704*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (113*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*A*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (19*B*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*C*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1211,1,1742,270,6.7963705,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{13}{2}}(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{(76 \cos (2 c) A+76 A+96 B+69 C+96 B \cos (2 c)+99 C \cos (2 c)) \csc (c) \sec (c)}{120 d}+\frac{C \sec (c+d x) \sin (d x) \sec (c)}{4 d}+\frac{(204 A+191 B+112 C) \cos (d x) \sin (c)}{336 d}+\frac{(127 A+72 B+18 C) \cos (2 d x) \sin (2 c)}{720 d}+\frac{(4 A+B) \cos (3 d x) \sin (3 c)}{112 d}+\frac{A \cos (4 d x) \sin (4 c)}{288 d}+\frac{(204 A+191 B+112 C) \cos (c) \sin (d x)}{336 d}+\frac{(127 A+72 B+18 C) \cos (2 c) \sin (2 d x)}{720 d}+\frac{(4 A+B) \cos (3 c) \sin (3 d x)}{112 d}+\frac{A \cos (4 c) \sin (4 d x)}{288 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{19 A \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 B \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{7 C \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 A \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{17 B \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 C \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{8 a^4 (12 A+17 B+28 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{8 a^4 (19 A+24 B+21 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^4 (73 A+83 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{4 (86 A+81 B-126 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 \cos (c+d x)+a^4\right)}{315 d}+\frac{2 (5 A+3 B-21 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{21 d}+\frac{2 a (A-9 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/120*((76*A + 96*B + 69*C + 76*A*Cos[2*c] + 96*B*Cos[2*c] + 99*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((204*A + 191*B + 112*C)*Cos[d*x]*Sin[c])/(336*d) + ((127*A + 72*B + 18*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((4*A + B)*Cos[3*d*x]*Sin[3*c])/(112*d) + (A*Cos[4*d*x]*Sin[4*c])/(288*d) + ((204*A + 191*B + 112*C)*Cos[c]*Sin[d*x])/(336*d) + (C*Sec[c]*Sec[c + d*x]*Sin[d*x])/(4*d) + ((127*A + 72*B + 18*C)*Cos[2*c]*Sin[2*d*x])/(720*d) + ((4*A + B)*Cos[3*c]*Sin[3*d*x])/(112*d) + (A*Cos[4*c]*Sin[4*d*x])/(288*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (19*A*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*B*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (7*C*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1212,1,1451,269,6.9181888,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{13}{2}}(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^2(c+d x)}{12 d}+\frac{\sec (c) (C \sin (c)+3 B \sin (d x)+12 C \sin (d x)) \sec (c+d x)}{12 d}-\frac{(32 \cos (2 c) A+32 A+23 B-20 C+33 B \cos (2 c)+20 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{(191 A+112 B+28 C) \cos (d x) \sin (c)}{336 d}+\frac{(4 A+B) \cos (2 d x) \sin (2 c)}{40 d}+\frac{A \cos (3 d x) \sin (3 c)}{112 d}+\frac{(191 A+112 B+28 C) \cos (c) \sin (d x)}{336 d}+\frac{(4 A+B) \cos (2 c) \sin (2 d x)}{40 d}+\frac{A \cos (3 c) \sin (3 d x)}{112 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{4 A \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{7 B \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{17 A \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{5 C \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{8 a^4 (17 A+28 B+35 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^4 (83 A+7 B-175 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{4 (27 A-42 B-175 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 \cos (c+d x)+a^4\right)}{105 d}+\frac{8 a^4 (8 A+7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (A-7 B-21 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)^2}{7 d}+\frac{2 a (3 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/40*((32*A + 23*B - 20*C + 32*A*Cos[2*c] + 33*B*Cos[2*c] + 20*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((191*A + 112*B + 28*C)*Cos[d*x]*Sin[c])/(336*d) + ((4*A + B)*Cos[2*d*x]*Sin[2*c])/(40*d) + (A*Cos[3*d*x]*Sin[3*c])/(112*d) + ((191*A + 112*B + 28*C)*Cos[c]*Sin[d*x])/(336*d) + (C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(12*d) + (Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*B*Sin[d*x] + 12*C*Sin[d*x]))/(12*d) + ((4*A + B)*Cos[2*c]*Sin[2*d*x])/(40*d) + (A*Cos[3*c]*Sin[3*d*x])/(112*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (17*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (7*B*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1213,1,1449,267,7.0218425,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{13}{2}}(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^3(c+d x)}{20 d}+\frac{\sec (c) (3 C \sin (c)+5 B \sin (d x)+20 C \sin (d x)) \sec ^2(c+d x)}{60 d}+\frac{\sec (c) (5 B \sin (c)+20 C \sin (c)+15 A \sin (d x)+60 B \sin (d x)+99 C \sin (d x)) \sec (c+d x)}{60 d}-\frac{(33 \cos (2 c) A+23 A-20 B-61 C+20 B \cos (2 c)+5 C \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{(4 A+B) \cos (d x) \sin (c)}{12 d}+\frac{A \cos (2 d x) \sin (2 c)}{40 d}+\frac{(4 A+B) \cos (c) \sin (d x)}{12 d}+\frac{A \cos (2 c) \sin (2 d x)}{40 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}-\frac{7 A \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{7 C \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 A \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{5 B \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 C \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{8 a^4 (4 A+5 B+4 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{4 a^4 (A-25 B-41 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}-\frac{4 (6 A+25 B+34 C) \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 \cos (c+d x)+a^4\right)}{15 d}+\frac{56 a^4 (A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (5 A+15 B+19 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a (5 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/40*((23*A - 20*B - 61*C + 33*A*Cos[2*c] + 20*B*Cos[2*c] + 5*C*Cos[2*c])*Csc[c]*Sec[c])/d + ((4*A + B)*Cos[d*x]*Sin[c])/(12*d) + (A*Cos[2*d*x]*Sin[2*c])/(40*d) + ((4*A + B)*Cos[c]*Sin[d*x])/(12*d) + (C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(20*d) + (Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*B*Sin[d*x] + 20*C*Sin[d*x]))/(60*d) + (Sec[c]*Sec[c + d*x]*(5*B*Sin[c] + 20*C*Sin[c] + 15*A*Sin[d*x] + 60*B*Sin[d*x] + 99*C*Sin[d*x]))/(60*d) + (A*Cos[2*c]*Sin[2*d*x])/(40*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (5*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (7*A*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (7*C*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1214,1,1454,271,7.161834,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{13}{2}}(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^4(c+d x)}{28 d}+\frac{\sec (c) (5 C \sin (c)+7 B \sin (d x)+28 C \sin (d x)) \sec ^3(c+d x)}{140 d}+\frac{\sec (c) (21 B \sin (c)+84 C \sin (c)+35 A \sin (d x)+140 B \sin (d x)+235 C \sin (d x)) \sec ^2(c+d x)}{420 d}+\frac{\sec (c) (35 A \sin (c)+140 B \sin (c)+235 C \sin (c)+420 A \sin (d x)+693 B \sin (d x)+672 C \sin (d x)) \sec (c+d x)}{420 d}-\frac{(20 \cos (2 c) A-20 A-61 B-64 C+5 B \cos (2 c)) \csc (c) \sec (c)}{40 d}+\frac{A \cos (d x) \sin (c)}{12 d}+\frac{A \cos (c) \sin (d x)}{12 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}+\frac{7 B \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 C \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{5 A \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{17 C \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{8 a^4 (35 A+28 B+17 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^4 (175 A+287 B+253 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d}+\frac{4 (175 A+238 B+197 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{105 d \sqrt{\cos (c+d x)}}-\frac{8 a^4 (7 B+8 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (35 A+77 B+73 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (7 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/40*((-20*A - 61*B - 64*C + 20*A*Cos[2*c] + 5*B*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(12*d) + (A*Cos[c]*Sin[d*x])/(12*d) + (C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(28*d) + (Sec[c]*Sec[c + d*x]^3*(5*C*Sin[c] + 7*B*Sin[d*x] + 28*C*Sin[d*x]))/(140*d) + (Sec[c]*Sec[c + d*x]^2*(21*B*Sin[c] + 84*C*Sin[c] + 35*A*Sin[d*x] + 140*B*Sin[d*x] + 235*C*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 140*B*Sin[c] + 235*C*Sin[c] + 420*A*Sin[d*x] + 693*B*Sin[d*x] + 672*C*Sin[d*x]))/(420*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (5*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (7*B*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*C*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1215,1,1748,274,7.2611433,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\cos ^{\frac{13}{2}}(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^5(c+d x)}{36 d}+\frac{\sec (c) (7 C \sin (c)+9 B \sin (d x)+36 C \sin (d x)) \sec ^4(c+d x)}{252 d}+\frac{\sec (c) (45 B \sin (c)+180 C \sin (c)+63 A \sin (d x)+252 B \sin (d x)+427 C \sin (d x)) \sec ^3(c+d x)}{1260 d}+\frac{\sec (c) (63 A \sin (c)+252 B \sin (c)+427 C \sin (c)+420 A \sin (d x)+705 B \sin (d x)+720 C \sin (d x)) \sec ^2(c+d x)}{1260 d}+\frac{\sec (c) (140 A \sin (c)+235 B \sin (c)+240 C \sin (c)+693 A \sin (d x)+672 B \sin (d x)+532 C \sin (d x)) \sec (c+d x)}{420 d}-\frac{(15 \cos (2 c) A-183 A-192 B-152 C) \csc (c) \sec (c)}{120 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}+\frac{7 A \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{4 B \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{19 C \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 A \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{17 B \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 C \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{8 a^4 (28 A+17 B+12 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{8 a^4 (21 A+24 B+19 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 (21 A+24 B+19 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{45 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^4 (287 A+253 B+193 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 (63 A+117 B+97 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 a (9 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(-1/120*((-183*A - 192*B - 152*C + 15*A*Cos[2*c])*Csc[c]*Sec[c])/d + (C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(36*d) + (Sec[c]*Sec[c + d*x]^4*(7*C*Sin[c] + 9*B*Sin[d*x] + 36*C*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^3*(45*B*Sin[c] + 180*C*Sin[c] + 63*A*Sin[d*x] + 252*B*Sin[d*x] + 427*C*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(140*A*Sin[c] + 235*B*Sin[c] + 240*C*Sin[c] + 693*A*Sin[d*x] + 672*B*Sin[d*x] + 532*C*Sin[d*x]))/(420*d) + (Sec[c]*Sec[c + d*x]^2*(63*A*Sin[c] + 252*B*Sin[c] + 427*C*Sin[c] + 420*A*Sin[d*x] + 705*B*Sin[d*x] + 720*C*Sin[d*x]))/(1260*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (4*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (17*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (7*A*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (4*B*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (19*C*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1216,1,1795,310,7.3545971,"\int \frac{(a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\cos ^{\frac{13}{2}}(c+d x) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{C \sec (c) \sin (d x) \sec ^6(c+d x)}{44 d}+\frac{\sec (c) (9 C \sin (c)+11 B \sin (d x)+44 C \sin (d x)) \sec ^5(c+d x)}{396 d}+\frac{\sec (c) (77 B \sin (c)+308 C \sin (c)+99 A \sin (d x)+396 B \sin (d x)+675 C \sin (d x)) \sec ^4(c+d x)}{2772 d}+\frac{\sec (c) (495 A \sin (c)+1980 B \sin (c)+3375 C \sin (c)+2772 A \sin (d x)+4697 B \sin (d x)+4928 C \sin (d x)) \sec ^3(c+d x)}{13860 d}+\frac{\sec (c) (2772 A \sin (c)+4697 B \sin (c)+4928 C \sin (c)+7755 A \sin (d x)+7920 B \sin (d x)+6780 C \sin (d x)) \sec ^2(c+d x)}{13860 d}+\frac{\sec (c) (2585 A \sin (c)+2640 B \sin (c)+2260 C \sin (c)+7392 A \sin (d x)+5852 B \sin (d x)+4928 C \sin (d x)) \sec (c+d x)}{4620 d}+\frac{(24 A+19 B+16 C) \csc (c) \sec (c)}{15 d}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)}+\frac{4 A \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{19 B \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{30 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{8 C \cos ^6(c+d x) \csc (c) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{17 A \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 B \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{113 C \cos ^6(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^4 \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{231 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{8 a^4 (187 A+132 B+113 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{231 d}-\frac{8 a^4 (24 A+19 B+16 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{4 a^4 (913 A+803 B+667 C) \sin (c+d x)}{1155 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 (891 A+946 B+769 C) \sin (c+d x) \left(a^4 \cos (c+d x)+a^4\right)}{3465 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{8 a^4 (24 A+19 B+16 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (33 A+55 B+43 C) \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)^2}{231 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 a (11 B+8 C) \sin (c+d x) (a \cos (c+d x)+a)^3}{99 d \cos ^{\frac{9}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{11 d \cos ^{\frac{11}{2}}(c+d x)}",1,"(Cos[c + d*x]^(13/2)*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((24*A + 19*B + 16*C)*Csc[c]*Sec[c])/(15*d) + (C*Sec[c]*Sec[c + d*x]^6*Sin[d*x])/(44*d) + (Sec[c]*Sec[c + d*x]^5*(9*C*Sin[c] + 11*B*Sin[d*x] + 44*C*Sin[d*x]))/(396*d) + (Sec[c]*Sec[c + d*x]^4*(77*B*Sin[c] + 308*C*Sin[c] + 99*A*Sin[d*x] + 396*B*Sin[d*x] + 675*C*Sin[d*x]))/(2772*d) + (Sec[c]*Sec[c + d*x]^3*(495*A*Sin[c] + 1980*B*Sin[c] + 3375*C*Sin[c] + 2772*A*Sin[d*x] + 4697*B*Sin[d*x] + 4928*C*Sin[d*x]))/(13860*d) + (Sec[c]*Sec[c + d*x]*(2585*A*Sin[c] + 2640*B*Sin[c] + 2260*C*Sin[c] + 7392*A*Sin[d*x] + 5852*B*Sin[d*x] + 4928*C*Sin[d*x]))/(4620*d) + (Sec[c]*Sec[c + d*x]^2*(2772*A*Sin[c] + 4697*B*Sin[c] + 4928*C*Sin[c] + 7755*A*Sin[d*x] + 7920*B*Sin[d*x] + 6780*C*Sin[d*x]))/(13860*d)))/(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]) - (17*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (113*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (4*A*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (19*B*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(30*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (8*C*Cos[c + d*x]^6*Csc[c]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1217,1,2117,210,6.8580299,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{5 (9 A-7 B+7 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a d}-\frac{3 (7 A-7 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(9 A-7 B+7 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a d}-\frac{(7 A-7 B+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}+\frac{5 (9 A-7 B+7 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a d}",1,"(((-21*I)/10)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (((21*I)/10)*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(5*A - 5*B + 5*C + 16*A*Cos[c] - 16*B*Cos[c] + 10*C*Cos[c])*Csc[c])/(5*d) + (2*(51*A - 28*B + 28*C)*Cos[d*x]*Sin[c])/(21*d) - (4*(A - B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (2*A*Cos[3*d*x]*Sin[3*c])/(7*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (2*(51*A - 28*B + 28*C)*Cos[c]*Sin[d*x])/(21*d) - (4*(A - B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (2*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (30*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (10*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (10*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1218,1,2063,174,6.7934187,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\text{Result too large to show}","-\frac{(5 A-5 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (7 A-5 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A-5 B+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{(5 A-5 B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((21*I)/10)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((3*I)/2)*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(5*A - 5*B + 5*C + 16*A*Cos[c] - 10*B*Cos[c] + 10*C*Cos[c])*Csc[c])/(5*d) - (8*(A - B)*Cos[d*x]*Sin[c])/(3*d) + (4*A*Cos[2*d*x]*Sin[2*c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (8*(A - B)*Cos[c]*Sin[d*x])/(3*d) + (4*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (10*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (10*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1219,1,2008,134,6.6807087,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{(5 A-3 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{(3 A-3 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-3 B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((-3*I)/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (((3*I)/2)*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - ((I/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(A - B + C + 2*A*Cos[c] - 2*B*Cos[c])*Csc[c])/d + (8*A*Cos[d*x]*Sin[c])/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*A*Cos[c]*Sin[d*x])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (10*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (2*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1220,1,1973,93,6.6535381,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{3 i A \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{i B \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{i C \cos (c+d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\cos ^{\frac{3}{2}}(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 (2 \cos (c) A+A-B+C) \csc (c)}{d}-\frac{4 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{2 A \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{2 B \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{2 C \cos (c+d x) \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","-\frac{(A-B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"(((3*I)/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - ((I/2)*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + ((I/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(A - B + C + 2*A*Cos[c])*Csc[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (2*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1221,1,2009,122,6.7374764,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])),x]","\text{Result too large to show}","\frac{(A+B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B+3 C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"((-1/2*I)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + ((I/2)*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(2*C + A*Cos[c] - B*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (2*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (2*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1222,1,2052,165,7.2310572,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\text{Result too large to show}","\frac{(3 A-3 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{(A-3 B+3 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}+\frac{(3 A-3 B+5 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(A-3 B+3 C) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"((I/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((3*I)/2)*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (((3*I)/2)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(-2*B + 2*C + A*Cos[c] - B*Cos[c] + C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (8*Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*B*Sin[d*x] - 3*C*Sin[d*x]))/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (2*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (10*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1223,1,2111,210,7.6483558,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","\text{Result too large to show}","-\frac{(3 A-5 B+5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (5 A-5 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B+C) \sin (c+d x)}{d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A-5 B+5 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(5 A-5 B+7 C) \sin (c+d x)}{5 a d \cos ^{\frac{5}{2}}(c+d x)}+\frac{3 (5 A-5 B+7 C) \sin (c+d x)}{5 a d \sqrt{\cos (c+d x)}}",1,"(((-3*I)/2)*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (((3*I)/2)*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) - (((21*I)/10)*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*(10*A - 10*B + 16*C + 5*A*Cos[c] - 5*B*Cos[c] + 5*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d + (8*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) - (8*Sec[c]*Sec[c + d*x]*(-5*B*Sin[c] + 5*C*Sin[c] - 15*A*Sin[d*x] + 15*B*Sin[d*x] - 24*C*Sin[d*x]))/(15*d) + (8*Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*B*Sin[d*x] - 5*C*Sin[d*x]))/(15*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])) + (2*A*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (10*B*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (10*C*Cos[c/2 + (d*x)/2]^2*Cos[c + d*x]*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
1224,1,2174,258,7.233778,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\text{Result too large to show}","\frac{5 (30 A-21 B+14 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 a^2 d}-\frac{7 (11 A-8 B+5 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(11 A-8 B+5 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(30 A-21 B+14 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 a^2 d}-\frac{7 (11 A-8 B+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}+\frac{5 (30 A-21 B+14 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(((-77*I)/5)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (((56*I)/5)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - ((7*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (200*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (40*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(25*A - 20*B + 15*C + 52*A*Cos[c] - 36*B*Cos[c] + 20*C*Cos[c])*Csc[c])/(5*d) + (4*(107*A - 56*B + 28*C)*Cos[d*x]*Sin[c])/(21*d) - (8*(2*A - B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*A*Cos[3*d*x]*Sin[3*c])/(7*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(5*A*Sin[(d*x)/2] - 4*B*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/d + (4*(107*A - 56*B + 28*C)*Cos[c]*Sin[d*x])/(21*d) - (8*(2*A - B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (4*A*Cos[3*c]*Sin[3*d*x])/(7*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1225,1,2120,214,7.0150702,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\text{Result too large to show}","-\frac{5 (3 A-2 B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(56 A-35 B+20 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(3 A-2 B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{(56 A-35 B+20 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (3 A-2 B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(((56*I)/5)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - ((7*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + ((4*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (40*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-8*(20*A - 15*B + 10*C + 36*A*Cos[c] - 20*B*Cos[c] + 10*C*Cos[c])*Csc[c])/(5*d) - (16*(2*A - B)*Cos[d*x]*Sin[c])/(3*d) + (8*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(4*A*Sin[(d*x)/2] - 3*B*Sin[(d*x)/2] + 2*C*Sin[(d*x)/2]))/d - (16*(2*A - B)*Cos[c]*Sin[d*x])/(3*d) + (8*A*Cos[2*c]*Sin[2*d*x])/(5*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1226,1,2064,180,6.9148994,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\text{Result too large to show}","\frac{(10 A-5 B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(7 A-4 B+C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{(10 A-5 B+2 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((-7*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + ((4*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (I*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (40*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (8*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(3*A - 2*B + C + 4*A*Cos[c] - 2*B*Cos[c])*Csc[c])/d + (16*A*Cos[d*x]*Sin[c])/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(3*A*Sin[(d*x)/2] - 2*B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (16*A*Cos[c]*Sin[d*x])/(3*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1227,1,1628,144,6.8366631,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{4 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{8 (2 \cos (c) A+2 A-B) \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{20 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{8 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{4 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","-\frac{(5 A-2 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(5 A-2 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}+\frac{(4 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((4*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (I*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (8*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (4*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-8*(2*A - B + 2*A*Cos[c])*Csc[c])/d - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - B*Sin[(d*x)/2]))/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1228,1,1620,133,6.7415577,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{8 (A-C) \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{8 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{4 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{8 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{(2 A+B+2 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((-I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (I*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (8*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (4*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (8*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(A - C)*Csc[c])/d + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1229,1,1660,167,6.8957871,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{4 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{\cos (c+d x)} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(2 C \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{4 (2 \cos (c) C+2 C-B \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{16 C \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{8 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}+\frac{20 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{(A+2 B-5 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A+2 B-5 C) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}+\frac{(B-4 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(B-4 C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"(I*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - ((4*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (8*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(2*C - B*Cos[c] + 2*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(B*Sin[(d*x)/2]) + 2*C*Sin[(d*x)/2]))/d + (16*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1230,1,2107,211,7.5464019,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","\text{Result too large to show}","\frac{(2 A-5 B+10 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-4 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-4 B+7 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}+\frac{(2 A-5 B+10 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(A-4 B+7 C) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(I*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - ((4*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + ((7*I)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (8*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (40*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(-2*B + 4*C + A*Cos[c] - 2*B*Cos[c] + 3*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - 2*B*Sin[(d*x)/2] + 3*C*Sin[(d*x)/2]))/d + (16*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (16*Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*B*Sin[d*x] - 6*C*Sin[d*x]))/(3*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1231,1,2164,250,8.1545621,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2),x]","\text{Result too large to show}","-\frac{5 (A-2 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(20 A-35 B+56 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(A-2 B+3 C) \sin (c+d x)}{a^2 d \cos ^{\frac{5}{2}}(c+d x) (\cos (c+d x)+1)}-\frac{5 (A-2 B+3 C) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(20 A-35 B+56 C) \sin (c+d x)}{15 a^2 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{(20 A-35 B+56 C) \sin (c+d x)}{5 a^2 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((-4*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + ((7*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) - (((56*I)/5)*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2) + (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (40*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (20*C*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(10*A - 20*B + 36*C + 10*A*Cos[c] - 15*B*Cos[c] + 20*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(3*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(2*A*Sin[(d*x)/2] - 3*B*Sin[(d*x)/2] + 4*C*Sin[(d*x)/2]))/d + (16*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) - (16*Sec[c]*Sec[c + d*x]*(-5*B*Sin[c] + 10*C*Sin[c] - 15*A*Sin[d*x] + 30*B*Sin[d*x] - 54*C*Sin[d*x]))/(15*d) + (16*Sec[c]*Sec[c + d*x]^2*(3*C*Sin[c] + 5*B*Sin[d*x] - 10*C*Sin[d*x]))/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^2)","C",0
1232,1,2257,273,7.3664026,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\text{Result too large to show}","-\frac{(63 A-33 B+13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{7 (33 A-17 B+7 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(63 A-33 B+13 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{7 (33 A-17 B+7 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^3 d}-\frac{(63 A-33 B+13 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(12 A-7 B+2 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(((231*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (((119*I)/5)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((49*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (84*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (44*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (52*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-8*(99*A - 59*B + 29*C + 132*A*Cos[c] - 60*B*Cos[c] + 20*C*Cos[c])*Csc[c])/(5*d) - (32*(3*A - B)*Cos[d*x]*Sin[c])/(3*d) + (16*A*Cos[2*d*x]*Sin[2*c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(24*A*Sin[(d*x)/2] - 19*B*Sin[(d*x)/2] + 14*C*Sin[(d*x)/2]))/(15*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(99*A*Sin[(d*x)/2] - 59*B*Sin[(d*x)/2] + 29*C*Sin[(d*x)/2]))/(5*d) - (32*(3*A - B)*Cos[c]*Sin[d*x])/(3*d) + (16*A*Cos[2*c]*Sin[2*d*x])/(5*d) + (8*(24*A - 19*B + 14*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1233,1,2206,234,7.1585914,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\text{Result too large to show}","\frac{(33 A-13 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(119 A-49 B+9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(119 A-49 B+9 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(33 A-13 B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(((-119*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((49*I)/5)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (((9*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (44*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (52*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(59*A - 29*B + 9*C + 60*A*Cos[c] - 20*B*Cos[c])*Csc[c])/(5*d) + (32*A*Cos[d*x]*Sin[c])/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(59*A*Sin[(d*x)/2] - 29*B*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(19*A*Sin[(d*x)/2] - 14*B*Sin[(d*x)/2] + 9*C*Sin[(d*x)/2]))/(15*d) + (32*A*Cos[c]*Sin[d*x])/(3*d) - (8*(19*A - 14*B + 9*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1234,1,2175,201,7.0587289,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\text{Result too large to show}","-\frac{(13 A-3 B-C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(49 A-9 B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A-3 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(8 A-3 B-2 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(((49*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (((9*I)/5)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - ((I/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (52*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-8*(29*A - 9*B - C + 20*A*Cos[c])*Csc[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(29*A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(14*A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2] + 4*C*Sin[(d*x)/2]))/(15*d) + (8*(14*A - 9*B + 4*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1235,1,2167,193,6.9406638,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","\text{Result too large to show}","\frac{(3 A+B+C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A+B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(9 A+B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(6 A-B-4 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"(((-9*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - ((I/5)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + ((I/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(9*A + B - C)*Csc[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(9*A*Sin[(d*x)/2] - 4*B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(15*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] + B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*(9*A - 4*B - C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1236,1,2164,191,6.9034372,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\text{Result too large to show}","\frac{(A+B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-B-9 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-B-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(4 A+B-6 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((-1/5*I)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + ((I/5)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((9*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(A - B - 9*C)*Csc[c])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(4*A*Sin[(d*x)/2] + B*Sin[(d*x)/2] - 6*C*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*(4*A + B - 6*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1237,1,2205,229,7.1863576,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","\text{Result too large to show}","\frac{(A+3 B-13 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A+9 B-49 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+9 B-49 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}+\frac{(A+3 B-13 C) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(2 A+3 B-8 C) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}-\frac{(A-B+C) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"((I/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((9*I)/5)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (((49*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (4*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (52*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(20*C - A*Cos[c] - 9*B*Cos[c] + 29*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2] - 29*C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] - 6*B*Sin[(d*x)/2] + 11*C*Sin[(d*x)/2]))/(15*d) + (32*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (8*(A - 6*B + 11*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1238,1,2248,268,8.1241906,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3),x]","\text{Result too large to show}","\frac{(3 A-13 B+33 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-49 B+119 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{30 d \cos ^{\frac{3}{2}}(c+d x) \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-13 B+33 C) \sin (c+d x)}{6 a^3 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{(9 A-49 B+119 C) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}-\frac{(A-B+C) \sin (c+d x)}{5 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^3}+\frac{(B-2 C) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"(((9*I)/5)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (((49*I)/5)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) + (((119*I)/5)*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3) - (4*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (52*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (44*C*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(-20*B + 60*C + 9*A*Cos[c] - 29*B*Cos[c] + 59*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(6*A*Sin[(d*x)/2] - 11*B*Sin[(d*x)/2] + 16*C*Sin[(d*x)/2]))/(15*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] - 29*B*Sin[(d*x)/2] + 59*C*Sin[(d*x)/2]))/(5*d) + (32*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (32*Sec[c]*Sec[c + d*x]*(C*Sin[c] + 3*B*Sin[d*x] - 9*C*Sin[d*x]))/(3*d) - (8*(6*A - 11*B + 16*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^3)","C",0
1239,1,2319,278,7.5239877,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\text{Result too large to show}","\frac{(339 A-108 B+17 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 a^4 d}-\frac{(176 A-57 B+8 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^4 d}-\frac{(43 A-15 B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{42 a^4 d (\cos (c+d x)+1)^2}-\frac{(176 A-57 B+8 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{30 a^4 d (\cos (c+d x)+1)}+\frac{(339 A-108 B+17 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{42 a^4 d}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(13 A-6 B-C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(((-352*I)/5)*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (((114*I)/5)*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (((16*I)/5)*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (904*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (288*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (136*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((16*(96*A - 37*B + 8*C + 80*A*Cos[c] - 20*B*Cos[c])*Csc[c])/(5*d) + (64*A*Cos[d*x]*Sin[c])/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(96*A*Sin[(d*x)/2] - 37*B*Sin[(d*x)/2] + 8*C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(33*A*Sin[(d*x)/2] - 26*B*Sin[(d*x)/2] + 19*C*Sin[(d*x)/2]))/(35*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(629*A*Sin[(d*x)/2] - 363*B*Sin[(d*x)/2] + 167*C*Sin[(d*x)/2]))/(105*d) + (64*A*Cos[c]*Sin[d*x])/(3*d) - (8*(629*A - 363*B + 167*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) + (8*(33*A - 26*B + 19*C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(35*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^6*Tan[c/2])/(7*d)))/(Cos[c + d*x]^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","C",0
1240,1,2286,244,7.3757109,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\text{Result too large to show}","-\frac{(108 A-17 B-4 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 a^4 d}+\frac{(57 A-8 B-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^4 d}-\frac{(141 A-29 B-13 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{210 a^4 d (\cos (c+d x)+1)^2}-\frac{(108 A-17 B-4 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{42 a^4 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(11 A-4 B-3 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(((114*I)/5)*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (((16*I)/5)*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (((2*I)/5)*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (288*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (136*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (32*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-16*(37*A - 8*B - C + 20*A*Cos[c])*Csc[c])/(5*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(37*A*Sin[(d*x)/2] - 8*B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(26*A*Sin[(d*x)/2] - 19*B*Sin[(d*x)/2] + 12*C*Sin[(d*x)/2]))/(35*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(363*A*Sin[(d*x)/2] - 167*B*Sin[(d*x)/2] + 41*C*Sin[(d*x)/2]))/(105*d) + (8*(363*A - 167*B + 41*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) - (8*(26*A - 19*B + 12*C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(35*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^6*Tan[c/2])/(7*d)))/(Cos[c + d*x]^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","C",0
1241,1,1862,232,7.1905306,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^4),x]","-\frac{16 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{2 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(19 A \sin \left(\frac{d x}{2}\right)-12 B \sin \left(\frac{d x}{2}\right)+5 C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d}+\frac{8 (19 A-12 B+5 C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(167 A \sin \left(\frac{d x}{2}\right)-41 B \sin \left(\frac{d x}{2}\right)-15 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d}-\frac{8 (167 A-41 B-15 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(8 A \sin \left(\frac{d x}{2}\right)+B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 (8 A+B) \csc (c)}{5 d}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{136 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}-\frac{32 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}-\frac{8 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}","\frac{(17 A+4 B+3 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 a^4 d}-\frac{(83 A+B-15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{210 a^4 d (\cos (c+d x)+1)^2}-\frac{(8 A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^4 d}+\frac{(8 A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 a^4 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(9 A-2 B-5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 a d (a \cos (c+d x)+a)^3}",1,"(((-16*I)/5)*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (((2*I)/5)*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (136*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (32*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (8*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((16*(8*A + B)*Csc[c])/(5*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(8*A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(167*A*Sin[(d*x)/2] - 41*B*Sin[(d*x)/2] - 15*C*Sin[(d*x)/2]))/(105*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(19*A*Sin[(d*x)/2] - 12*B*Sin[(d*x)/2] + 5*C*Sin[(d*x)/2]))/(35*d) - (8*(167*A - 41*B - 15*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) + (8*(19*A - 12*B + 5*C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(35*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^6*Tan[c/2])/(7*d)))/(Cos[c + d*x]^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","C",0
1242,1,1862,229,7.0963252,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^4),x]","-\frac{2 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{2 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d}+\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d}-\frac{8 \sec \left(\frac{c}{2}\right) \left(12 A \sin \left(\frac{d x}{2}\right)-5 B \sin \left(\frac{d x}{2}\right)-2 C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d}-\frac{8 (12 A-5 B-2 C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(41 A \sin \left(\frac{d x}{2}\right)+15 B \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d}+\frac{8 (41 A+15 B-C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d}+\frac{16 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{16 (A-C) \csc (c)}{5 d}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{32 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}-\frac{8 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}-\frac{32 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}","\frac{(4 A+3 B+4 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 a^4 d}+\frac{(41 A+15 B-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{210 a^4 d (\cos (c+d x)+1)^2}-\frac{(A-C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^4 d}+\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 a^4 d (\cos (c+d x)+1)}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{7 d (a \cos (c+d x)+a)^4}-\frac{(A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 a d (a \cos (c+d x)+a)^3}",1,"(((-2*I)/5)*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (((2*I)/5)*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (32*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (8*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (32*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((16*(A - C)*Csc[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(12*A*Sin[(d*x)/2] - 5*B*Sin[(d*x)/2] - 2*C*Sin[(d*x)/2]))/(35*d) + (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(41*A*Sin[(d*x)/2] + 15*B*Sin[(d*x)/2] - C*Sin[(d*x)/2]))/(105*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d) + (8*(41*A + 15*B - C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) - (8*(12*A - 5*B - 2*C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(35*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^6*Tan[c/2])/(7*d)))/(Cos[c + d*x]^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","C",0
1243,1,1862,234,7.1052619,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^4),x]","\frac{2 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{16 i C \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{\left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)+C \sin \left(\frac{d x}{2}\right)\right) \sec ^7\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d}-\frac{4 (A-B+C) \tan \left(\frac{c}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(5 A \sin \left(\frac{d x}{2}\right)+2 B \sin \left(\frac{d x}{2}\right)-9 C \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d}+\frac{8 (5 A+2 B-9 C) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{35 d}+\frac{8 \sec \left(\frac{c}{2}\right) \left(15 A \sin \left(\frac{d x}{2}\right)-B \sin \left(\frac{d x}{2}\right)-83 C \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d}+\frac{8 (15 A-B-83 C) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{105 d}-\frac{16 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)+8 C \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{16 (B+8 C) \csc (c)}{5 d}\right) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) (\sec (c+d x) a+a)^4}-\frac{8 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}-\frac{32 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}-\frac{136 C \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^4}","\frac{(3 A+4 B+17 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 a^4 d}+\frac{(15 A-B-83 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{210 a^4 d (\cos (c+d x)+1)^2}+\frac{(B+8 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^4 d}-\frac{(B+8 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 a^4 d (\cos (c+d x)+1)}+\frac{(5 A+2 B-9 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{35 a d (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{7 d (a \cos (c+d x)+a)^4}",1,"(((2*I)/5)*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (((16*I)/5)*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (8*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (32*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (136*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-16*(B + 8*C)*Csc[c])/(5*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(15*A*Sin[(d*x)/2] - B*Sin[(d*x)/2] - 83*C*Sin[(d*x)/2]))/(105*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(5*A*Sin[(d*x)/2] + 2*B*Sin[(d*x)/2] - 9*C*Sin[(d*x)/2]))/(35*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(B*Sin[(d*x)/2] + 8*C*Sin[(d*x)/2]))/(5*d) + (8*(15*A - B - 83*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) + (8*(5*A + 2*B - 9*C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(35*d) - (4*(A - B + C)*Sec[c/2 + (d*x)/2]^6*Tan[c/2])/(7*d)))/(Cos[c + d*x]^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","C",0
1244,1,2316,276,7.4811322,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^4),x]","\text{Result too large to show}","\frac{(4 A+17 B-108 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{42 a^4 d}+\frac{(A+8 B-57 C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^4 d}-\frac{(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt{\cos (c+d x)}}+\frac{(4 A+17 B-108 C) \sin (c+d x)}{42 a^4 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}+\frac{(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)^2}+\frac{(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}-\frac{(A-B+C) \sin (c+d x)}{7 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^4}",1,"(((2*I)/5)*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) + (((16*I)/5)*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (((114*I)/5)*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4) - (32*A*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) - (136*B*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (288*C*Cos[c/2 + (d*x)/2]^8*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]^8*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((8*(20*C - A*Cos[c] - 8*B*Cos[c] + 37*C*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) - (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(A*Sin[(d*x)/2] + 83*B*Sin[(d*x)/2] - 237*C*Sin[(d*x)/2]))/(105*d) - (16*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 8*B*Sin[(d*x)/2] - 37*C*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^7*(A*Sin[(d*x)/2] - B*Sin[(d*x)/2] + C*Sin[(d*x)/2]))/(7*d) + (8*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(2*A*Sin[(d*x)/2] - 9*B*Sin[(d*x)/2] + 16*C*Sin[(d*x)/2]))/(35*d) + (64*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d - (8*(A + 83*B - 237*C)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(105*d) + (8*(2*A - 9*B + 16*C)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(35*d) + (4*(A - B + C)*Sec[c/2 + (d*x)/2]^6*Tan[c/2])/(7*d)))/(Cos[c + d*x]^(3/2)*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + a*Sec[c + d*x])^4)","C",0
1245,1,127,226,0.5866108,"\int \cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} ((752 A+846 B+672 C) \cos (c+d x)+4 (83 A+54 B+63 C) \cos (2 (c+d x))+80 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+1321 A+90 B \cos (3 (c+d x))+1368 B+1596 C)}{1260 d (\cos (c+d x)+1)}","\frac{2 a (16 A+18 B+21 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a (16 A+18 B+21 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a (16 A+18 B+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}",1,"(Sqrt[Cos[c + d*x]]*(1321*A + 1368*B + 1596*C + (752*A + 846*B + 672*C)*Cos[c + d*x] + 4*(83*A + 54*B + 63*C)*Cos[2*(c + d*x)] + 80*A*Cos[3*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(1260*d*(1 + Cos[c + d*x]))","A",1
1246,1,105,178,0.3784734,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} ((141 A+28 (4 B+5 C)) \cos (c+d x)+6 (6 A+7 B) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+228 A+266 B+280 C)}{210 d (\cos (c+d x)+1)}","\frac{2 a (24 A+28 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (24 A+28 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{7 d}",1,"(Sqrt[Cos[c + d*x]]*(228*A + 266*B + 280*C + (141*A + 28*(4*B + 5*C))*Cos[c + d*x] + 6*(6*A + 7*B)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(210*d*(1 + Cos[c + d*x]))","A",1
1247,1,82,129,0.180293,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left((4 A+5 B) \cos (c+d x)+3 A \cos ^2(c+d x)+8 A+10 B+15 C\right)}{15 d (\cos (c+d x)+1)}","\frac{2 a (7 A+5 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 (A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{5 d}",1,"(2*Sqrt[Cos[c + d*x]]*(8*A + 10*B + 15*C + (4*A + 5*B)*Cos[c + d*x] + 3*A*Cos[c + d*x]^2)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(15*d*(1 + Cos[c + d*x]))","A",1
1248,1,94,140,0.5757368,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (A \cos (c+d x)+2 A+3 B)+3 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{3 d}","\frac{2 a (A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 \sqrt{a} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(2*A + 3*B + A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
1249,1,94,139,0.7445845,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 A+C \sec (c+d x))+\sqrt{2} (2 B+C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 d}","\frac{a (2 A-C) \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (2 B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(2*B + C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(2*A + C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
1250,1,109,151,0.6691993,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (8 A+4 B+3 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 B+2 C \sec (c+d x)+3 C)\right)}{8 d}","\frac{\sqrt{a} (8 A+4 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a (4 B+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(8*A + 4*B + 3*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*B + 3*C + 2*C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
1251,1,140,199,1.3828437,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+6 B+5 C) \cos (2 (c+d x))+24 A+4 (6 B+5 C) \cos (c+d x)+18 B+31 C)+3 \sqrt{2} (8 A+6 B+5 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a (8 A+6 B+5 C) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (8 A+6 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a (6 B+C) \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(8*A + 6*B + 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (24*A + 18*B + 31*C + 4*(6*B + 5*C)*Cos[c + d*x] + 3*(8*A + 6*B + 5*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
1252,1,178,247,2.1438982,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((432 A+77 (8 B+7 C)) \cos (c+d x)+4 (48 A+40 B+35 C) \cos (2 (c+d x))+144 A \cos (3 (c+d x))+192 A+120 B \cos (3 (c+d x))+160 B+105 C \cos (3 (c+d x))+332 C)+6 \sqrt{2} (48 A+40 B+35 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a (48 A+40 B+35 C) \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a (48 A+40 B+35 C) \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (48 A+40 B+35 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a (8 B+C) \sin (c+d x)}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(48*A + 40*B + 35*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (192*A + 160*B + 332*C + (432*A + 77*(8*B + 7*C))*Cos[c + d*x] + 4*(48*A + 40*B + 35*C)*Cos[2*(c + d*x)] + 144*A*Cos[3*(c + d*x)] + 120*B*Cos[3*(c + d*x)] + 105*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","A",1
1253,1,158,284,2.0483548,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((34734 A+44 (799 B+759 C)) \cos (c+d x)+8 (1743 A+1507 B+1287 C) \cos (2 (c+d x))+4935 A \cos (3 (c+d x))+1470 A \cos (4 (c+d x))+315 A \cos (5 (c+d x))+55482 A+3740 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+59158 B+1980 C \cos (3 (c+d x))+65208 C)}{27720 d}","\frac{2 a^2 (84 A+110 B+99 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (336 A+374 B+429 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (336 A+374 B+429 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (336 A+374 B+429 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+11 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}",1,"(a*Sqrt[Cos[c + d*x]]*(55482*A + 59158*B + 65208*C + (34734*A + 44*(799*B + 759*C))*Cos[c + d*x] + 8*(1743*A + 1507*B + 1287*C)*Cos[2*(c + d*x)] + 4935*A*Cos[3*(c + d*x)] + 3740*B*Cos[3*(c + d*x)] + 1980*C*Cos[3*(c + d*x)] + 1470*A*Cos[4*(c + d*x)] + 770*B*Cos[4*(c + d*x)] + 315*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(27720*d)","A",1
1254,1,123,232,1.3769322,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (2 (799 A+759 B+756 C) \cos (c+d x)+4 (137 A+117 B+63 C) \cos (2 (c+d x))+170 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+2689 A+90 B \cos (3 (c+d x))+2964 B+3276 C)}{1260 d}","\frac{2 a^2 (52 A+72 B+63 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+156 B+189 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (136 A+156 B+189 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (A+3 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{21 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}",1,"(a*Sqrt[Cos[c + d*x]]*(2689*A + 2964*B + 3276*C + 2*(799*A + 759*B + 756*C)*Cos[c + d*x] + 4*(137*A + 117*B + 63*C)*Cos[2*(c + d*x)] + 170*A*Cos[3*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d)","A",1
1255,1,100,181,0.9916734,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((253 A+28 (9 B+5 C)) \cos (c+d x)+6 (13 A+7 B) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+494 A+546 B+700 C)}{210 d}","\frac{8 a^2 (19 A+21 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (19 A+21 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 (3 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{7 d}",1,"(a*Sqrt[Cos[c + d*x]]*(494*A + 546*B + 700*C + (253*A + 28*(9*B + 5*C))*Cos[c + d*x] + 6*(13*A + 7*B)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(210*d)","A",1
1256,1,115,192,1.0920579,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (2 (9 A+5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+39 A+50 B+30 C)+15 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15 d}","\frac{2 a^{3/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (12 A+20 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(15*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (39*A + 50*B + 30*C + 2*(9*A + 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(15*d)","A",1
1257,1,122,197,1.3507098,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (2 (5 A+3 B) \cos (c+d x)+A \cos (2 (c+d x))+A+3 C)+3 \sqrt{2} (2 B+3 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 d}","\frac{a^{3/2} (2 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 (8 A+6 B-3 C) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{a (2 A-3 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(2*B + 3*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(A + 3*C + 2*(5*A + 3*B)*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sec[c + d*x]*Sin[(c + d*x)/2]))/(6*d)","A",1
1258,1,129,203,1.4373278,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 (2 A \cos (2 (c+d x))+2 A+C)+(4 B+7 C) \cos (c+d x))+\sqrt{2} (8 A+12 B+7 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{8 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{3/2} (8 A+12 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (8 A-4 B-5 C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a (4 B+3 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d \sqrt{\cos (c+d x)}}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(8*A + 12*B + 7*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*((4*B + 7*C)*Cos[c + d*x] + 2*(2*A + C + 2*A*Cos[2*(c + d*x)]))*Sin[(c + d*x)/2]))/(8*d*Cos[c + d*x]^(3/2))","A",1
1259,1,141,201,2.2722059,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (3 (8 A+14 B+11 C) \cos (2 (c+d x))+24 A+4 (6 B+11 C) \cos (c+d x)+42 B+49 C)+3 \sqrt{2} (24 A+14 B+11 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a^{3/2} (24 A+14 B+11 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (24 A+30 B+19 C) \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a (2 B+C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(24*A + 14*B + 11*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (24*A + 42*B + 49*C + 4*(6*B + 11*C)*Cos[c + d*x] + 3*(8*A + 14*B + 11*C)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
1260,1,176,253,3.6807492,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((1008 A+1048 B+1155 C) \cos (c+d x)+4 (48 A+88 B+75 C) \cos (2 (c+d x))+336 A \cos (3 (c+d x))+192 A+264 B \cos (3 (c+d x))+352 B+225 C \cos (3 (c+d x))+492 C)+6 \sqrt{2} (112 A+88 B+75 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a^{3/2} (112 A+88 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (112 A+88 B+75 C) \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (48 A+56 B+39 C) \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a (8 B+3 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{24 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(112*A + 88*B + 75*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (192*A + 352*B + 492*C + (1008*A + 1048*B + 1155*C)*Cos[c + d*x] + 4*(48*A + 88*B + 75*C)*Cos[2*(c + d*x)] + 336*A*Cos[3*(c + d*x)] + 264*B*Cos[3*(c + d*x)] + 225*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","A",1
1261,1,210,303,5.8052189,"\int \frac{(a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (880 A+1070 B+1273 C) \cos (c+d x)+4 (3280 A+3450 B+3059 C) \cos (2 (c+d x))+3520 A \cos (3 (c+d x))+2640 A \cos (4 (c+d x))+10480 A+3000 B \cos (3 (c+d x))+2250 B \cos (4 (c+d x))+11550 B+2660 C \cos (3 (c+d x))+1995 C \cos (4 (c+d x))+13313 C)+60 \sqrt{2} (176 A+150 B+133 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{a^{3/2} (176 A+150 B+133 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (176 A+150 B+133 C) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+90 B+67 C) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a (10 B+3 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(176*A + 150*B + 133*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (10480*A + 11550*B + 13313*C + 12*(880*A + 1070*B + 1273*C)*Cos[c + d*x] + 4*(3280*A + 3450*B + 3059*C)*Cos[2*(c + d*x)] + 3520*A*Cos[3*(c + d*x)] + 3000*B*Cos[3*(c + d*x)] + 2660*C*Cos[3*(c + d*x)] + 2640*A*Cos[4*(c + d*x)] + 2250*B*Cos[4*(c + d*x)] + 1995*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d*Cos[c + d*x]^(9/2))","A",1
1262,1,190,334,2.4884282,"\int \cos ^{\frac{13}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (4 (453146 A+454285 B+445588 C) \cos (c+d x)+(746519 A+676000 B+581152 C) \cos (2 (c+d x))+287060 A \cos (3 (c+d x))+94010 A \cos (4 (c+d x))+23940 A \cos (5 (c+d x))+3465 A \cos (6 (c+d x))+2798182 A+225550 B \cos (3 (c+d x))+58240 B \cos (4 (c+d x))+8190 B \cos (5 (c+d x))+2980640 B+148720 C \cos (3 (c+d x))+20020 C \cos (4 (c+d x))+3233516 C)}{720720 d}","\frac{2 a^3 (2224 A+2522 B+2717 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{9009 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15015 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^3 (8368 A+9230 B+10439 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{45045 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^3 (8368 A+9230 B+10439 C) \sin (c+d x)}{45045 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (136 A+182 B+143 C) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}+\frac{2 a (5 A+13 B) \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{143 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{11}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{13 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(2798182*A + 2980640*B + 3233516*C + 4*(453146*A + 454285*B + 445588*C)*Cos[c + d*x] + (746519*A + 676000*B + 581152*C)*Cos[2*(c + d*x)] + 287060*A*Cos[3*(c + d*x)] + 225550*B*Cos[3*(c + d*x)] + 148720*C*Cos[3*(c + d*x)] + 94010*A*Cos[4*(c + d*x)] + 58240*B*Cos[4*(c + d*x)] + 20020*C*Cos[4*(c + d*x)] + 23940*A*Cos[5*(c + d*x)] + 8190*B*Cos[5*(c + d*x)] + 3465*A*Cos[6*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(720720*d)","A",1
1263,1,157,284,2.2191833,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((69890 A+68552 B+66660 C) \cos (c+d x)+16 (1625 A+1397 B+990 C) \cos (2 (c+d x))+8675 A \cos (3 (c+d x))+2240 A \cos (4 (c+d x))+315 A \cos (5 (c+d x))+114640 A+5720 B \cos (3 (c+d x))+770 B \cos (4 (c+d x))+124366 B+1980 C \cos (3 (c+d x))+137280 C)}{27720 d}","\frac{2 a^3 (1160 A+1364 B+1485 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (2840 A+3212 B+3795 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (2840 A+3212 B+3795 C) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (32 A+44 B+33 C) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{231 d}+\frac{2 a (5 A+11 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{11 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(114640*A + 124366*B + 137280*C + (69890*A + 68552*B + 66660*C)*Cos[c + d*x] + 16*(1625*A + 1397*B + 990*C)*Cos[2*(c + d*x)] + 8675*A*Cos[3*(c + d*x)] + 5720*B*Cos[3*(c + d*x)] + 1980*C*Cos[3*(c + d*x)] + 2240*A*Cos[4*(c + d*x)] + 770*B*Cos[4*(c + d*x)] + 315*A*Cos[5*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(27720*d)","A",1
1264,1,124,231,1.538036,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} ((3116 A+3030 B+2352 C) \cos (c+d x)+4 (254 A+180 B+63 C) \cos (2 (c+d x))+260 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+5653 A+90 B \cos (3 (c+d x))+6240 B+7476 C)}{1260 d}","\frac{64 a^3 (13 A+15 B+21 C) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (13 A+15 B+21 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a (13 A+15 B+21 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 (5 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{9 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*(5653*A + 6240*B + 7476*C + (3116*A + 3030*B + 2352*C)*Cos[c + d*x] + 4*(254*A + 180*B + 63*C)*Cos[2*(c + d*x)] + 260*A*Cos[3*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(1260*d)","A",1
1265,1,137,242,1.7712627,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((505 A+392 B+140 C) \cos (c+d x)+6 (20 A+7 B) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+1040 A+1246 B+1120 C)+420 \sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 d}","\frac{2 a^{5/2} C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (160 A+224 B+245 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (40 A+56 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (5 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(a^2*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(420*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*(1040*A + 1246*B + 1120*C + (505*A + 392*B + 140*C)*Cos[c + d*x] + 6*(20*A + 7*B)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(420*d)","A",1
1266,1,149,243,1.4342278,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((181 A+160 B+60 C) \cos (c+d x)+2 (14 A+5 B) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+28 A+10 B+30 C)+30 \sqrt{2} (2 B+5 C) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{60 d \sqrt{\cos (c+d x)}}","\frac{a^{5/2} (2 B+5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 (64 A+70 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (16 A+10 B-15 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(30*Sqrt[2]*(2*B + 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*(28*A + 10*B + 30*C + (181*A + 160*B + 60*C)*Cos[c + d*x] + 2*(14*A + 5*B)*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(60*d*Sqrt[Cos[c + d*x]])","A",1
1267,1,155,253,1.8851528,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (3 (2 A+4 B+11 C) \cos (c+d x)+4 (8 A+3 B) \cos (2 (c+d x))+2 A \cos (3 (c+d x))+32 A+12 B+6 C)+6 \sqrt{2} (8 A+20 B+19 C) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{a^{5/2} (8 A+20 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (56 A+12 B-27 C) \sin (c+d x)}{12 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (8 A-12 B-21 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{12 d \sqrt{\cos (c+d x)}}-\frac{a (4 A-3 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{6 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(8*A + 20*B + 19*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 4*(32*A + 12*B + 6*C + 3*(2*A + 4*B + 11*C)*Cos[c + d*x] + 4*(8*A + 3*B)*Cos[2*(c + d*x)] + 2*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(3/2))","A",1
1268,1,157,253,2.5125138,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (4 (18 A+6 B+17 C) \cos (c+d x)+3 (8 A+22 B+25 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))+24 A+66 B+91 C)+3 \sqrt{2} (40 A+38 B+25 C) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a^{5/2} (40 A+38 B+25 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (24 A+42 B+31 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{24 d \sqrt{\cos (c+d x)}}+\frac{a (6 B+5 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt{\cos (c+d x)}}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(40*A + 38*B + 25*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (24*A + 66*B + 91*C + 4*(18*A + 6*B + 17*C)*Cos[c + d*x] + 3*(8*A + 22*B + 25*C)*Cos[2*(c + d*x)] + 24*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
1269,1,178,253,3.9146841,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((1584 A+2056 B+2203 C) \cos (c+d x)+4 (48 A+136 B+163 C) \cos (2 (c+d x))+528 A \cos (3 (c+d x))+192 A+600 B \cos (3 (c+d x))+544 B+489 C \cos (3 (c+d x))+844 C)+6 \sqrt{2} (304 A+200 B+163 C) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a^{5/2} (304 A+200 B+163 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (432 A+392 B+299 C) \sin (c+d x)}{192 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (16 A+24 B+17 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a (8 B+5 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(304*A + 200*B + 163*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (192*A + 544*B + 844*C + (1584*A + 2056*B + 2203*C)*Cos[c + d*x] + 4*(48*A + 136*B + 163*C)*Cos[2*(c + d*x)] + 528*A*Cos[3*(c + d*x)] + 600*B*Cos[3*(c + d*x)] + 489*C*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","A",1
1270,1,212,301,6.1115973,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (12 (1360 A+1950 B+2343 C) \cos (c+d x)+4 (6640 A+6730 B+6509 C) \cos (2 (c+d x))+5440 A \cos (3 (c+d x))+6000 A \cos (4 (c+d x))+20560 A+6520 B \cos (3 (c+d x))+4890 B \cos (4 (c+d x))+22030 B+5660 C \cos (3 (c+d x))+4245 C \cos (4 (c+d x))+24863 C)+60 \sqrt{2} (400 A+326 B+283 C) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{a^{5/2} (400 A+326 B+283 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (400 A+326 B+283 C) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1040 A+950 B+787 C) \sin (c+d x)}{960 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (80 A+110 B+79 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a (2 B+C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(400*A + 326*B + 283*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (20560*A + 22030*B + 24863*C + 12*(1360*A + 1950*B + 2343*C)*Cos[c + d*x] + 4*(6640*A + 6730*B + 6509*C)*Cos[2*(c + d*x)] + 5440*A*Cos[3*(c + d*x)] + 6520*B*Cos[3*(c + d*x)] + 5660*C*Cos[3*(c + d*x)] + 6000*A*Cos[4*(c + d*x)] + 4890*B*Cos[4*(c + d*x)] + 4245*C*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d*Cos[c + d*x]^(9/2))","A",1
1271,1,947,353,6.4885642,"\int \frac{(a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\frac{4 \sec ^5\left(\frac{1}{2} (c+d x)\right) (a (\sec (c+d x)+1))^{5/2} \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sqrt{\frac{1}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)} \left(\frac{A \sin \left(\frac{1}{2} (c+d x)\right)}{16 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{(2 A+B) \sin \left(\frac{1}{2} (c+d x)\right)}{24 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{(A+2 B+C) \sin \left(\frac{1}{2} (c+d x)\right)}{32 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^4}+\frac{(B+2 C) \sin \left(\frac{1}{2} (c+d x)\right)}{40 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^5}+\frac{C \sin \left(\frac{1}{2} (c+d x)\right)}{48 \left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^6}+\frac{3}{64} A \left(\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)+\frac{5}{384} (2 A+B) \left(\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^2}+3 \left(\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right)+\frac{7 (A+2 B+C) \left(\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^3}+5 \left(\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^2}+3 \left(\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right)\right)}{3072}+\frac{3 (B+2 C) \left(\frac{96 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^4}+7 \left(\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^3}+5 \left(\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^2}+3 \left(\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right)\right)\right)}{10240}+\frac{11 C \left(\frac{256 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^5}+3 \left(\frac{96 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^4}+7 \left(\frac{16 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^3}+5 \left(\frac{4 \sin \left(\frac{1}{2} (c+d x)\right)}{\left(1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)\right)^2}+3 \left(\sqrt{2} \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 \sin \left(\frac{1}{2} (c+d x)\right)}{1-2 \sin ^2\left(\frac{1}{2} (c+d x)\right)}\right)\right)\right)\right)\right)}{122880}\right)}{d (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sec ^{\frac{9}{2}}(c+d x)}","\frac{a^{5/2} (1304 A+1132 B+1015 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{512 d}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{512 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (1304 A+1132 B+1015 C) \sin (c+d x)}{768 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (680 A+628 B+545 C) \sin (c+d x)}{960 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (120 A+156 B+115 C) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{480 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a (12 B+5 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{60 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(4*Sec[(c + d*x)/2]^5*(a*(1 + Sec[c + d*x]))^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sqrt[(1 - 2*Sin[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - 2*Sin[(c + d*x)/2]^2]*((C*Sin[(c + d*x)/2])/(48*(1 - 2*Sin[(c + d*x)/2]^2)^6) + ((B + 2*C)*Sin[(c + d*x)/2])/(40*(1 - 2*Sin[(c + d*x)/2]^2)^5) + ((A + 2*B + C)*Sin[(c + d*x)/2])/(32*(1 - 2*Sin[(c + d*x)/2]^2)^4) + ((2*A + B)*Sin[(c + d*x)/2])/(24*(1 - 2*Sin[(c + d*x)/2]^2)^3) + (A*Sin[(c + d*x)/2])/(16*(1 - 2*Sin[(c + d*x)/2]^2)^2) + (3*A*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (2*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)))/64 + (5*(2*A + B)*((4*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^2 + 3*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (2*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2))))/384 + (7*(A + 2*B + C)*((16*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^3 + 5*((4*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^2 + 3*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (2*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)))))/3072 + (3*(B + 2*C)*((96*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^4 + 7*((16*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^3 + 5*((4*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^2 + 3*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (2*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2))))))/10240 + (11*C*((256*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^5 + 3*((96*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^4 + 7*((16*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^3 + 5*((4*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)^2 + 3*(Sqrt[2]*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (2*Sin[(c + d*x)/2])/(1 - 2*Sin[(c + d*x)/2]^2)))))))/122880))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2))","B",1
1272,1,178,257,0.8523542,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left((43 A-91 B+35 C) \sec ^3(c+d x)+(7 (B-5 C)-31 A) \sec ^2(c+d x)+3 (A-7 B) \sec (c+d x)-15 A\right)+105 \sqrt{2} (A-B+C) \sec ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{105 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (31 A-7 B+35 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}-\frac{2 (43 A-91 B+35 C) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"-1/105*(Cos[c + d*x]^(5/2)*(105*Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(7/2) + 2*Sqrt[1 - Sec[c + d*x]]*(-15*A + 3*(A - 7*B)*Sec[c + d*x] + (-31*A + 7*(B - 5*C))*Sec[c + d*x]^2 + (43*A - 91*B + 35*C)*Sec[c + d*x]^3))*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1273,1,163,211,0.5613016,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(\sqrt{1-\sec (c+d x)} \sec ^2(c+d x) (-2 (A-5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+29 A-10 B+30 C)+15 \sqrt{2} (A-B+C) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{15 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (13 A-5 B+15 C) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^(3/2)*((29*A - 10*B + 30*C - 2*(A - 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^2 + 15*Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(5/2))*Sin[c + d*x])/(15*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1274,1,88,163,0.6571294,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(3 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) (A \cos (c+d x)-A+3 B)\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{2 (A-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*(3*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + 2*(-A + 3*B + A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1275,1,96,178,0.5819449,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-\left((A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)+2 A \sin \left(\frac{1}{2} (c+d x)\right)+\sqrt{2} C \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*Cos[(c + d*x)/2]*(-((A - B + C)*ArcTanh[Sin[(c + d*x)/2]]) + Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*A*Sin[(c + d*x)/2]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1276,1,113,181,0.5406793,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 (A-B+C) \cos (c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (2 B-C) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 C \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(2 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{C \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(2*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x] + Sqrt[2]*(2*B - C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*C*Sin[(c + d*x)/2]))/(d*Cos[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1277,1,127,235,1.1047467,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(8 (A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} (8 A-4 B+7 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (-4 B-2 C \sec (c+d x)+C)\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{(8 A-4 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{(4 B-C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"-1/4*(Cos[(c + d*x)/2]*(8*(A - B + C)*ArcTanh[Sin[(c + d*x)/2]] - Sqrt[2]*(8*A - 4*B + 7*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(-4*B + C - 2*C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1278,1,154,281,1.136834,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((A-B+C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{48} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(3 (8 A-2 B+7 C)+2 (6 B-C) \sec (c+d x)+8 C \sec ^2(c+d x)\right)-3 \sqrt{2} (8 A-14 B+9 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(8 A-2 B+7 C) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (A-B+C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}-\frac{(8 A-14 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 \sqrt{a} d}+\frac{(6 B-C) \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{C \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(2*Cos[(c + d*x)/2]*((A - B + C)*ArcTanh[Sin[(c + d*x)/2]] + (-3*Sqrt[2]*(8*A - 14*B + 9*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(3*(8*A - 2*B + 7*C) + 2*(6*B - C)*Sec[c + d*x] + 8*C*Sec[c + d*x]^2)*Sin[(c + d*x)/2])/48))/(d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1279,1,137,184,0.4176352,"\int \frac{\sqrt{\cos (c+d x)} \left(a A+(A b+a B) \sec (c+d x)+b B \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \left(\sqrt{2} (a-b) (A-B) \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 a A \sqrt{1-\sec (c+d x)}-2 b B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (a-b) (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 b B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((2*a*A*Sqrt[1 - Sec[c + d*x]] - 2*b*B*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Sec[c + d*x]] + Sqrt[2]*(a - b)*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1280,1,135,283,2.906195,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (3 (39 A+20 (C-B)) \cos (c+d x)+(10 B-6 A) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+141 A-85 B+75 C)-15 (15 A-11 B+7 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{30 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{(15 A-11 B+7 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(39 A-35 B+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{30 a d \sqrt{a \sec (c+d x)+a}}+\frac{(147 A-95 B+75 C) \sin (c+d x)}{30 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(-15*(15*A - 11*B + 7*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (141*A - 85*B + 75*C + 3*(39*A + 20*(-B + C))*Cos[c + d*x] + (-6*A + 10*B)*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Tan[(c + d*x)/2])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1281,1,113,233,1.9727955,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (-12 (A-B) \cos (c+d x)+2 A \cos (2 (c+d x))-17 A+15 B-3 C)+3 (11 A-7 B+3 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(11 A-7 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B+3 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{(19 A-15 B+3 C) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(3*(11*A - 7*B + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (-17*A + 15*B - 3*C - 12*(A - B)*Cos[c + d*x] + 2*A*Cos[2*(c + d*x)])*Tan[(c + d*x)/2])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1282,1,96,181,1.5254817,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) (4 A \cos (c+d x)+5 A-B+C)-(7 A-3 B-C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{(7 A-3 B-C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B+C) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(-((7*A - 3*B - C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]) + (5*A - B + C + 4*A*Cos[c + d*x])*Tan[(c + d*x)/2])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1283,1,118,189,1.6225904,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{-(A-B+C) \tan \left(\frac{1}{2} (c+d x)\right)+(3 A+B-5 C) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \sqrt{2} C \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(3 A+B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"((3*A + B - 5*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + 4*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] - (A - B + C)*Tan[(c + d*x)/2])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1284,1,198,242,1.5127489,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 (A-5 B+9 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\frac{2 \left(\sin \left(\frac{1}{2} (c+d x)\right) (A-B+2 C \sec (c+d x)+3 C)+2 \sqrt{2} (2 B-3 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(A-5 B+9 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 B-3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{3/2} d}+\frac{(A-B+3 C) \sin (c+d x)}{2 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(A - 5*B + 9*C)*ArcTanh[Sin[(c + d*x)/2]] - (2*(2*Sqrt[2]*(2*B - 3*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 + (A - B + 3*C + 2*C*Sec[c + d*x])*Sin[(c + d*x)/2]))/(-1 + Sin[(c + d*x)/2]^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1285,1,239,300,2.2655301,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{\cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(2 (5 A-9 B+13 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{\sqrt{2} (8 A-12 B+19 C) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) ((-2 A+6 B-7 C) \cos (2 (c+d x))-2 A+(8 B-6 C) \cos (c+d x)+6 B-3 C)}{\sin ^2\left(\frac{1}{2} (c+d x)\right)-1}\right)}{d (a (\sec (c+d x)+1))^{3/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","-\frac{(5 A-9 B+13 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{2 \sqrt{2} a^{3/2} d}+\frac{(8 A-12 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 a^{3/2} d}-\frac{(2 A-6 B+7 C) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A-B+2 C) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"-((Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(2*(5*A - 9*B + 13*C)*ArcTanh[Sin[(c + d*x)/2]] + (Sqrt[2]*(8*A - 12*B + 19*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 + ((-2*A + 6*B - 3*C + (8*B - 6*C)*Cos[c + d*x] + (-2*A + 6*B - 7*C)*Cos[2*(c + d*x)])*Sec[c + d*x]^2*Sin[(c + d*x)/2])/2)/(-1 + Sin[(c + d*x)/2]^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(3/2)))","A",1
1286,1,173,333,4.2605891,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) (5 (887 A-479 B+255 C) \cos (c+d x)+16 (52 A-25 B+15 C) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+12 A \cos (4 (c+d x))+3491 A+40 B \cos (3 (c+d x))-1895 B+975 C)-120 (283 A-163 B+75 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{960 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","-\frac{(283 A-163 B+75 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A-85 B+45 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{80 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(787 A-475 B+195 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{240 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(2671 A-1495 B+735 C) \sin (c+d x)}{240 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(21 A-13 B+5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(-120*(283*A - 163*B + 75*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(3491*A - 1895*B + 975*C + 5*(887*A - 479*B + 255*C)*Cos[c + d*x] + 16*(52*A - 25*B + 15*C)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 40*B*Cos[3*(c + d*x)] + 12*A*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(960*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1287,1,146,281,3.2696671,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((-479 A+255 B-39 C) \cos (c+d x)+(48 B-80 A) \cos (2 (c+d x))+8 A \cos (3 (c+d x))-379 A+195 B-27 C)+24 (163 A-75 B+19 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{192 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","\frac{(163 A-75 B+19 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(95 A-39 B+15 C) \sin (c+d x) \sqrt{\cos (c+d x)}}{48 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(299 A-147 B+27 C) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(17 A-9 B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(24*(163*A - 75*B + 19*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(-379*A + 195*B - 27*C + (-479*A + 255*B - 39*C)*Cos[c + d*x] + (-80*A + 48*B)*Cos[2*(c + d*x)] + 8*A*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(192*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1288,1,128,231,2.7190238,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((85 A-13 B+5 C) \cos (c+d x)+16 A \cos (2 (c+d x))+65 A-9 B+C)-8 (75 A-19 B-5 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{64 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","-\frac{(75 A-19 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A-9 B+C) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(13 A-5 B-3 C) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(-8*(75*A - 19*B - 5*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(65*A - 9*B + C + (85*A - 13*B + 5*C)*Cos[c + d*x] + 16*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(64*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1289,1,119,183,1.9665491,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(8 (19 A+5 B+3 C) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 \sin \left(\frac{1}{2} (c+d x)\right) ((13 A-5 B-3 C) \cos (c+d x)+9 A-B-7 C)\right)}{64 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","\frac{(19 A+5 B+3 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B-7 C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(8*(19*A + 5*B + 3*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 - 4*(9*A - B - 7*C + (13*A - 5*B - 3*C)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(64*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
1290,1,153,241,3.2064888,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) ((5 A+3 B-11 C) \cos (c+d x)+A+7 B-15 C)+2 (5 A+3 B-43 C) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+64 \sqrt{2} C \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{16 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(5 A+3 B-43 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{2 C \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(5 A+3 B-11 C) \sin (c+d x)}{16 a d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(2*(5*A + 3*B - 43*C)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + 64*Sqrt[2]*C*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + (A + 7*B - 15*C + (5*A + 3*B - 11*C)*Cos[c + d*x])*Tan[(c + d*x)/2])/(16*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
1291,1,222,294,3.9365534,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\cos ^5\left(\frac{1}{2} (c+d x)\right) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left((6 A-86 B+230 C) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sec ^3\left(\frac{1}{2} (c+d x)\right) (2 (7 A-15 B+55 C) \cos (c+d x)+(3 A-11 B+35 C) \cos (2 (c+d x))+3 A-11 B+67 C)+32 \sqrt{2} (2 B-5 C) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \sqrt{\cos (c+d x)} (a (\sec (c+d x)+1))^{5/2} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{(3 A-43 B+115 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \tanh ^{-1}\left(\frac{\sqrt{a} \sin (c+d x) \sqrt{\sec (c+d x)}}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 B-5 C) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}+\frac{(3 A-11 B+35 C) \sin (c+d x)}{16 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A+7 B-15 C) \sin (c+d x)}{16 a d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B+C) \sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^5*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((6*A - 86*B + 230*C)*ArcTanh[Sin[(c + d*x)/2]] + 32*Sqrt[2]*(2*B - 5*C)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + ((3*A - 11*B + 67*C + 2*(7*A - 15*B + 55*C)*Cos[c + d*x] + (3*A - 11*B + 35*C)*Cos[2*(c + d*x)])*Sec[(c + d*x)/2]^3*Sec[c + d*x]*Tan[(c + d*x)/2])/2))/(4*d*Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
1292,1,143,190,1.0251256,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{60 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+7 b C)+84 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+9 a C+9 b B)+\sin (c+d x) \sqrt{\cos (c+d x)} (7 \cos (c+d x) (43 a A+36 a C+36 b B)+5 (18 (a B+A b) \cos (2 (c+d x))+7 a A \cos (3 (c+d x))+78 a B+78 A b+84 b C))}{630 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+7 b C)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a A+9 a C+9 b B)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (7 a A+9 a C+9 b B)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a B+5 A b+7 b C)}{21 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d}",1,"(84*(7*a*A + 9*b*B + 9*a*C)*EllipticE[(c + d*x)/2, 2] + 60*(5*A*b + 5*a*B + 7*b*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(43*a*A + 36*b*B + 36*a*C)*Cos[c + d*x] + 5*(78*A*b + 78*a*B + 84*b*C + 18*(A*b + a*B)*Cos[2*(c + d*x)] + 7*a*A*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
1293,1,117,154,0.8788194,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)+42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)+\sin (c+d x) \sqrt{\cos (c+d x)} (42 (a B+A b) \cos (c+d x)+15 a A \cos (2 (c+d x))+65 a A+70 a C+70 b B)}{105 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+7 a C+7 b B)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+5 b C)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a A+7 a C+7 b B)}{21 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d}",1,"(42*(3*A*b + 3*a*B + 5*b*C)*EllipticE[(c + d*x)/2, 2] + 10*(5*a*A + 7*b*B + 7*a*C)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*a*A + 70*b*B + 70*a*C + 42*(A*b + a*B)*Cos[c + d*x] + 15*a*A*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
1294,1,1569,116,6.5275198,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{4 (3 a A+5 b B+5 a C) \cot (c)}{5 d}+\frac{4 (A b+a B) \cos (d x) \sin (c)}{3 d}+\frac{2 a A \cos (2 d x) \sin (2 c)}{5 d}+\frac{4 (A b+a B) \cos (c) \sin (d x)}{3 d}+\frac{2 a A \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^{\frac{7}{2}}(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{6 a A \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^3(c+d x)}{5 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 b B \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{2 a C \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 A b \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 a B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 b C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b+3 b C)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a A+5 a C+5 b B)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(3*a*A + 5*b*B + 5*a*C)*Cot[c])/(5*d) + (4*(A*b + a*B)*Cos[d*x]*Sin[c])/(3*d) + (2*a*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*(A*b + a*B)*Cos[c]*Sin[d*x])/(3*d) + (2*a*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*A*b*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b*C*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (6*a*A*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (2*b*B*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (2*a*C*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1295,1,1904,106,6.9326824,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(-\frac{2 (A b-2 C b+A \cos (2 c) b+a B+a B \cos (2 c)) \csc (c) \sec (c)}{d}+\frac{4 b C \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{4 a A \cos (d x) \sin (c)}{3 d}+\frac{4 a A \cos (c) \sin (d x)}{3 d}\right) \cos ^{\frac{7}{2}}(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{i A b \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^3(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{i a B \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^3(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{i b C \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^3(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 a A \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 b B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 a C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (A+3 C)+3 b B)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a B+A b-b C)}{d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b C \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(I*A*b*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*a*B*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*b*C*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(A*b + a*B - 2*b*C + A*b*Cos[2*c] + a*B*Cos[2*c])*Csc[c]*Sec[c])/d + (4*a*A*Cos[d*x]*Sin[c])/(3*d) + (4*a*A*Cos[c]*Sin[d*x])/(3*d) + (4*b*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a*A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b*B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*C*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1296,1,1909,112,6.986821,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{4 b C \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{4 \sec (c) (b C \sin (c)+3 b B \sin (d x)+3 a C \sin (d x)) \sec (c+d x)}{3 d}-\frac{2 (a A+a \cos (2 c) A-2 b B-2 a C) \csc (c) \sec (c)}{d}\right) \cos ^{\frac{7}{2}}(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}+\frac{i a A \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^3(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{i b B \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^3(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{i a C \csc (c) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^3(c+d x)}{(b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x))}-\frac{4 A b \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 a B \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{4 b C \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (a+b \sec (c+d x)) \left(C \sec ^2(c+d x)+B \sec (c+d x)+A\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^3(c+d x)}{3 d (b+a \cos (c+d x)) (\cos (2 c+2 d x) A+A+2 C+2 B \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (3 a B+3 A b+b C)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (b B-a (A-C))}{d}+\frac{2 (a C+b B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(I*a*A*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*b*B*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*a*C*Cos[c + d*x]^3*Csc[c]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(a*A - 2*b*B - 2*a*C + a*A*Cos[2*c])*Csc[c]*Sec[c])/d + (4*b*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(b*C*Sin[c] + 3*b*B*Sin[d*x] + 3*a*C*Sin[d*x]))/(3*d)))/((b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*A*b*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b*C*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1297,1,136,152,1.4844835,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{3 \sin (2 (c+d x)) (5 a B+5 A b+3 b C)+10 \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)-6 \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)+10 (a C+b B) \sin (c+d x)+6 b C \tan (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a (3 A+C)+b B)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a B+5 A b+3 b C)}{5 d}+\frac{2 \sin (c+d x) (5 a B+5 A b+3 b C)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a C+b B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(5*A*b + 5*a*B + 3*b*C)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(b*B + a*(3*A + C))*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*(b*B + a*C)*Sin[c + d*x] + 3*(5*A*b + 5*a*B + 3*b*C)*Sin[2*(c + d*x)] + 6*b*C*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
1298,1,173,190,4.0473798,"\int \frac{(a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)-42 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)+\frac{\sin (c+d x) (21 \cos (c+d x) (15 a A+13 a C+13 b B)+10 \cos (2 (c+d x)) (7 a B+7 A b+5 b C)+105 a A \cos (3 (c+d x))+70 a B+63 a C \cos (3 (c+d x))+70 A b+63 b B \cos (3 (c+d x))+110 b C)}{2 \cos ^{\frac{7}{2}}(c+d x)}}{105 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (7 a B+7 A b+5 b C)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) (5 a A+3 a C+3 b B)}{5 d}+\frac{2 \sin (c+d x) (7 a B+7 A b+5 b C)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) (5 a A+3 a C+3 b B)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 (a C+b B) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b C \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(-42*(5*a*A + 3*b*B + 3*a*C)*EllipticE[(c + d*x)/2, 2] + 10*(7*A*b + 7*a*B + 5*b*C)*EllipticF[(c + d*x)/2, 2] + ((70*A*b + 70*a*B + 110*b*C + 21*(15*a*A + 13*b*B + 13*a*C)*Cos[c + d*x] + 10*(7*A*b + 7*a*B + 5*b*C)*Cos[2*(c + d*x)] + 105*a*A*Cos[3*(c + d*x)] + 63*b*B*Cos[3*(c + d*x)] + 63*a*C*Cos[3*(c + d*x)])*Sin[c + d*x])/(2*Cos[c + d*x]^(7/2)))/(105*d)","A",1
1299,1,194,250,1.2742536,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{60 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+2 a b (5 A+7 C)+7 b^2 B\right)+84 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(7 \cos (c+d x) \left(a^2 (43 A+36 C)+72 a b B+36 A b^2\right)+5 \left(7 a^2 A \cos (3 (c+d x))+78 a^2 B+18 a (a B+2 A b) \cos (2 (c+d x))+156 a A b+168 a b C+84 b^2 B\right)\right)}{630 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (7 A+9 C)+18 a b B+3 b^2 (3 A+5 C)\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 (7 A+9 C)+18 a b B+4 A b^2\right)}{45 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 B+10 a A b+14 a b C+7 b^2 B\right)}{21 d}+\frac{2 a (9 a B+4 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{9 d}",1,"(84*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 60*(5*a^2*B + 7*b^2*B + 2*a*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(7*(36*A*b^2 + 72*a*b*B + a^2*(43*A + 36*C))*Cos[c + d*x] + 5*(156*a*A*b + 78*a^2*B + 84*b^2*B + 168*a*b*C + 18*a*(2*A*b + a*B)*Cos[2*(c + d*x)] + 7*a^2*A*Cos[3*(c + d*x)]))*Sin[c + d*x])/(630*d)","A",1
1300,1,2361,202,6.8424523,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (5 A+7 C)+14 a b B+7 b^2 (A+3 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+6 a A b+10 a b C+5 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 (5 A+7 C)+14 a b B+4 A b^2\right)}{21 d}+\frac{2 a (7 a B+4 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}",1,"(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Cot[c])/(5*d) + ((23*a^2*A + 28*A*b^2 + 56*a*b*B + 28*a^2*C)*Cos[d*x]*Sin[c])/(21*d) + (2*a*(2*A*b + a*B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (a^2*A*Cos[3*d*x]*Sin[3*c])/(7*d) + ((23*a^2*A + 28*A*b^2 + 56*a*b*B + 28*a^2*C)*Cos[c]*Sin[d*x])/(21*d) + (2*a*(2*A*b + a*B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (a^2*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^2*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a*b*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (12*a*A*b*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (6*a^2*B*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (2*b^2*B*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a*b*C*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1301,1,3011,186,7.4603206,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A+3 C)+3 b^2 B\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)+10 a b B+5 b^2 (A-C)\right)}{5 d}+\frac{2 a^2 (A-5 C) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} (a B+2 A b-6 b C)}{3 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^2}{d \sqrt{\cos (c+d x)}}",1,"(((3*I)/5)*a^2*A*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*A*b^2*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((2*I)*a*b*B*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*a^2*C*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*b^2*C*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(3*a^2*A + 5*A*b^2 + 10*a*b*B + 5*a^2*C - 10*b^2*C + 3*a^2*A*Cos[2*c] + 5*A*b^2*Cos[2*c] + 10*a*b*B*Cos[2*c] + 5*a^2*C*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*a*(2*A*b + a*B)*Cos[d*x]*Sin[c])/(3*d) + (2*a^2*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*a*(2*A*b + a*B)*Cos[c]*Sin[d*x])/(3*d) + (4*b^2*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*a^2*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (8*a*A*b*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^2*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a*b*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1302,1,2779,180,7.5134684,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)+6 a b B+b^2 (3 A+C)\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 B+2 a b (A-C)-b^2 B\right)}{d}+\frac{2 a^2 (A-C) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b (4 a C+3 b B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^2}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*I)*a*A*b*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*a^2*B*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*b^2*B*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((2*I)*a*b*C*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(2*a*A*b + a^2*B - 2*b^2*B - 4*a*b*C + 2*a*A*b*Cos[2*c] + a^2*B*Cos[2*c])*Csc[c]*Sec[c])/d + (4*a^2*A*Cos[d*x]*Sin[c])/(3*d) + (4*a^2*A*Cos[c]*Sin[d*x])/(3*d) + (4*b^2*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(b^2*C*Sin[c] + 3*b^2*B*Sin[d*x] + 6*a*b*C*Sin[d*x]))/(3*d)))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a^2*A*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^2*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a*b*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^2*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1303,1,3017,201,7.6289922,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 B+2 a b (3 A+C)+b^2 B\right)}{3 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^2 (A-C)+10 a b B+b^2 (5 A+3 C)\right)}{5 d}+\frac{2 \sin (c+d x) \left(4 a^2 C+10 a b B+5 A b^2+3 b^2 C\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b (4 a C+5 b B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^2}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(I*a^2*A*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*A*b^2*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((2*I)*a*b*B*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*a^2*C*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (((3*I)/5)*b^2*C*Cos[c + d*x]^4*Csc[c]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(5*a^2*A - 10*A*b^2 - 20*a*b*B - 10*a^2*C - 6*b^2*C + 5*a^2*A*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*b^2*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]^2*(3*b^2*C*Sin[c] + 5*b^2*B*Sin[d*x] + 10*a*b*C*Sin[d*x]))/(15*d) + (4*Sec[c]*Sec[c + d*x]*(5*b^2*B*Sin[c] + 10*a*b*C*Sin[c] + 15*A*b^2*Sin[d*x] + 30*a*b*B*Sin[d*x] + 15*a^2*C*Sin[d*x] + 9*b^2*C*Sin[d*x]))/(15*d)))/((b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (8*a*A*b*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^2*B*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a*b*C*Cos[c + d*x]^4*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1304,1,218,249,4.2895514,"\int \frac{(a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left(5 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)-21 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+2 a b (5 A+3 C)+3 b^2 B\right)+\frac{5 \sin (c+d x) \left(7 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 \sin (c+d x) \left(5 a^2 B+2 a b (5 A+3 C)+3 b^2 B\right)}{\sqrt{\cos (c+d x)}}+\frac{21 b (2 a C+b B) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{15 b^2 C \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}\right)}{105 d}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 (3 A+C)+14 a b B+b^2 (7 A+5 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d}+\frac{2 \sin (c+d x) \left(4 a^2 C+14 a b B+7 A b^2+5 b^2 C\right)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(5 a^2 B+10 a A b+6 a b C+3 b^2 B\right)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b (4 a C+7 b B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^2}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(-21*(5*a^2*B + 3*b^2*B + 2*a*b*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2] + 5*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2] + (15*b^2*C*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (21*b*(b*B + 2*a*C)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (5*(7*A*b^2 + 14*a*b*B + 7*a^2*C + 5*b^2*C)*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(5*a^2*B + 3*b^2*B + 2*a*b*(5*A + 3*C))*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(105*d)","A",1
1305,1,286,361,1.870489,"\int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right)+154 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right)+\frac{1}{12} \sin (c+d x) \sqrt{\cos (c+d x)} \left(154 \cos (c+d x) \left(43 a^3 B+3 a^2 b (43 A+36 C)+108 a b^2 B+36 A b^3\right)+5 \left(36 a \cos (2 (c+d x)) \left(a^2 (16 A+11 C)+33 a b B+33 A b^2\right)+154 a^2 (a B+3 A b) \cos (3 (c+d x))+3 \left(21 a^3 A \cos (4 (c+d x))+a^3 (531 A+572 C)+1716 a^2 b B+132 a b^2 (13 A+14 C)+616 b^3 B\right)\right)\right)}{1155 d}","\frac{2 a \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(9 a^2 (9 A+11 C)+143 a b B+24 A b^2\right)}{693 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 B+3 a^2 b (7 A+9 C)+27 a b^2 B+3 b^3 (3 A+5 C)\right)}{15 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(77 a^3 B+33 a^2 b (7 A+9 C)+242 a b^2 B+24 A b^3\right)}{495 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^3 (9 A+11 C)+165 a^2 b B+33 a b^2 (5 A+7 C)+77 b^3 B\right)}{231 d}+\frac{2 (11 a B+6 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^3}{11 d}",1,"(154*(7*a^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154*(36*A*b^3 + 43*a^3*B + 108*a*b^2*B + 3*a^2*b*(43*A + 36*C))*Cos[c + d*x] + 5*(36*a*(33*A*b^2 + 33*a*b*B + a^2*(16*A + 11*C))*Cos[2*(c + d*x)] + 154*a^2*(3*A*b + a*B)*Cos[3*(c + d*x)] + 3*(1716*a^2*b*B + 616*b^3*B + 132*a*b^2*(13*A + 14*C) + a^3*(531*A + 572*C) + 21*a^3*A*Cos[4*(c + d*x)])))*Sin[c + d*x])/12)/(1155*d)","A",1
1306,1,3237,296,7.2211517,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+99 a b B+24 A b^2\right)}{315 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+3 a^2 b (5 A+7 C)+21 a b^2 B+7 b^3 (A+3 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (7 A+9 C)+27 a^2 b B+9 a b^2 (3 A+5 C)+15 b^3 B\right)}{15 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(15 a^3 B+9 a^2 b (5 A+7 C)+54 a b^2 B+8 A b^3\right)}{63 d}+\frac{2 (3 a B+2 A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^2}{21 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}",1,"(Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B + 9*a^3*C + 45*a*b^2*C)*Cot[c])/(15*d) + ((69*a^2*A*b + 28*A*b^3 + 23*a^3*B + 84*a*b^2*B + 84*a^2*b*C)*Cos[d*x]*Sin[c])/(21*d) + (a*(19*a^2*A + 54*A*b^2 + 54*a*b*B + 18*a^2*C)*Cos[2*d*x]*Sin[2*c])/(45*d) + (a^2*(3*A*b + a*B)*Cos[3*d*x]*Sin[3*c])/(7*d) + (a^3*A*Cos[4*d*x]*Sin[4*c])/(18*d) + ((69*a^2*A*b + 28*A*b^3 + 23*a^3*B + 84*a*b^2*B + 84*a^2*b*C)*Cos[c]*Sin[d*x])/(21*d) + (a*(19*a^2*A + 54*A*b^2 + 54*a*b*B + 18*a^2*C)*Cos[2*c]*Sin[2*d*x])/(45*d) + (a^2*(3*A*b + a*B)*Cos[3*c]*Sin[3*d*x])/(7*d) + (a^3*A*Cos[4*c]*Sin[4*d*x])/(18*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^2*A*b*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^3*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*a^3*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*b^2*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*b*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^3*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (14*a^3*A*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (18*a*A*b^2*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (18*a^2*b*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (2*b^3*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (6*a^3*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (6*a*b^2*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1307,1,3915,277,7.990742,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 (5 A+7 C)+21 a b B+6 b^2 (3 A-7 C)\right)}{21 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (7 a B+11 A b-35 b C)}{35 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (5 A+7 C)+21 a^2 b B+21 a b^2 (A+3 C)+21 b^3 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^3 B+3 a^2 b (3 A+5 C)+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\frac{2 a (A-7 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^2}{7 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{d \sqrt{\cos (c+d x)}}",1,"(((9*I)/5)*a^2*A*b*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*A*b^3*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (((3*I)/5)*a^3*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((3*I)*a*b^2*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((3*I)*a^2*b*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*b^3*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B + 15*a^2*b*C - 10*b^3*C + 9*a^2*A*b*Cos[2*c] + 5*A*b^3*Cos[2*c] + 3*a^3*B*Cos[2*c] + 15*a*b^2*B*Cos[2*c] + 15*a^2*b*C*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (a*(23*a^2*A + 84*A*b^2 + 84*a*b*B + 28*a^2*C)*Cos[d*x]*Sin[c])/(21*d) + (2*a^2*(3*A*b + a*B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (a^3*A*Cos[3*d*x]*Sin[3*c])/(7*d) + (a*(23*a^2*A + 84*A*b^2 + 84*a*b*B + 28*a^2*C)*Cos[c]*Sin[d*x])/(21*d) + (4*b^3*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*a^2*(3*A*b + a*B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (a^3*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^3*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*A*b^2*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*b*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^3*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^3*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (12*a*b^2*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1308,1,3868,267,8.2343862,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 B+3 a b (A-5 C)-6 b^2 B\right)}{3 d}+\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (3 a A-35 a C-15 b B)}{15 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 B+3 a^2 b (A+3 C)+9 a b^2 B+b^3 (3 A+C)\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (3 A+5 C)+15 a^2 b B+15 a b^2 (A-C)-5 b^3 B\right)}{5 d}+\frac{2 (2 a C+b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(((3*I)/5)*a^3*A*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((3*I)*a*A*b^2*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((3*I)*a^2*b*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*b^3*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*a^3*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((3*I)*a*b^2*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 10*b^3*B + 5*a^3*C - 30*a*b^2*C + 3*a^3*A*Cos[2*c] + 15*a*A*b^2*Cos[2*c] + 15*a^2*b*B*Cos[2*c] + 5*a^3*C*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*a^2*(3*A*b + a*B)*Cos[d*x]*Sin[c])/(3*d) + (2*a^3*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*a^2*(3*A*b + a*B)*Cos[c]*Sin[d*x])/(3*d) + (4*b^3*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(b^3*C*Sin[c] + 3*b^3*B*Sin[d*x] + 9*a*b^2*C*Sin[d*x]))/(3*d) + (2*a^3*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a^2*A*b*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^3*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^3*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (12*a*b^2*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (12*a^2*b*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^3*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1309,1,3871,274,8.3675839,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 b \sin (c+d x) \left(24 a^2 C+35 a b B+15 A b^2+9 b^2 C\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (5 a A-9 a C-5 b B)}{15 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (A+3 C)+9 a^2 b B+3 a b^2 (3 A+C)+b^3 B\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^3 B+15 a^2 b (A-C)-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}+\frac{2 (6 a C+5 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"((3*I)*a^2*A*b*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*A*b^3*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*a^3*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((3*I)*a*b^2*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((3*I)*a^2*b*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (((3*I)/5)*b^3*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(15*a^2*A*b - 10*A*b^3 + 5*a^3*B - 30*a*b^2*B - 30*a^2*b*C - 6*b^3*C + 15*a^2*A*b*Cos[2*c] + 5*a^3*B*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*a^3*A*Cos[d*x]*Sin[c])/(3*d) + (4*a^3*A*Cos[c]*Sin[d*x])/(3*d) + (4*b^3*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]^2*(3*b^3*C*Sin[c] + 5*b^3*B*Sin[d*x] + 15*a*b^2*C*Sin[d*x]))/(15*d) + (4*Sec[c]*Sec[c + d*x]*(5*b^3*B*Sin[c] + 15*a*b^2*C*Sin[c] + 15*A*b^3*Sin[d*x] + 45*a*b^2*B*Sin[d*x] + 45*a^2*b*C*Sin[d*x] + 9*b^3*C*Sin[d*x]))/(15*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a^3*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (12*a*A*b^2*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (12*a^2*b*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^3*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^3*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*b^2*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1310,1,3933,294,8.4430694,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 b \sin (c+d x) \left(24 a^2 C+63 a b B+35 A b^2+25 b^2 C\right)}{105 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^3 B+21 a^2 b (3 A+C)+21 a b^2 B+b^3 (7 A+5 C)\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 (A-C)+15 a^2 b B+3 a b^2 (5 A+3 C)+3 b^3 B\right)}{5 d}+\frac{2 \sin (c+d x) \left(24 a^3 C+98 a^2 b B+21 a b^2 (5 A+3 C)+21 b^3 B\right)}{35 d \sqrt{\cos (c+d x)}}+\frac{2 (6 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(I*a^3*A*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((3*I)*a*A*b^2*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((3*I)*a^2*b*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (((3*I)/5)*b^3*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*a^3*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (((9*I)/5)*a*b^2*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(5*a^3*A - 30*a*A*b^2 - 30*a^2*b*B - 6*b^3*B - 10*a^3*C - 18*a*b^2*C + 5*a^3*A*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*b^3*C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (4*Sec[c]*Sec[c + d*x]^3*(5*b^3*C*Sin[c] + 7*b^3*B*Sin[d*x] + 21*a*b^2*C*Sin[d*x]))/(35*d) + (4*Sec[c]*Sec[c + d*x]*(35*A*b^3*Sin[c] + 105*a*b^2*B*Sin[c] + 105*a^2*b*C*Sin[c] + 25*b^3*C*Sin[c] + 315*a*A*b^2*Sin[d*x] + 315*a^2*b*B*Sin[d*x] + 63*b^3*B*Sin[d*x] + 105*a^3*C*Sin[d*x] + 189*a*b^2*C*Sin[d*x]))/(105*d) + (4*Sec[c]*Sec[c + d*x]^2*(21*b^3*B*Sin[c] + 63*a*b^2*C*Sin[c] + 35*A*b^3*Sin[d*x] + 105*a*b^2*B*Sin[d*x] + 105*a^2*b*C*Sin[d*x] + 25*b^3*C*Sin[d*x]))/(105*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (12*a^2*A*b*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^3*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^3*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*b^2*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*b*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*b^3*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1311,1,3345,357,8.1706973,"\int \frac{(a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{2 b \sin (c+d x) \left(24 a^2 C+99 a b B+63 A b^2+49 b^2 C\right)}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^3 (3 A+C)+21 a^2 b B+3 a b^2 (7 A+5 C)+5 b^3 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(8 a^3 C+54 a^2 b B+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(15 a^3 B+9 a^2 b (5 A+3 C)+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (2 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^2}{21 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^3}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(45*a^2*A*b + 9*A*b^3 + 15*a^3*B + 27*a*b^2*B + 27*a^2*b*C + 7*b^3*C)*Csc[c]*Sec[c])/(15*d) + (4*b^3*C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(9*d) + (4*Sec[c]*Sec[c + d*x]^4*(7*b^3*C*Sin[c] + 9*b^3*B*Sin[d*x] + 27*a*b^2*C*Sin[d*x]))/(63*d) + (4*Sec[c]*Sec[c + d*x]^2*(63*A*b^3*Sin[c] + 189*a*b^2*B*Sin[c] + 189*a^2*b*C*Sin[c] + 49*b^3*C*Sin[c] + 315*a*A*b^2*Sin[d*x] + 315*a^2*b*B*Sin[d*x] + 75*b^3*B*Sin[d*x] + 105*a^3*C*Sin[d*x] + 225*a*b^2*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]^3*(45*b^3*B*Sin[c] + 135*a*b^2*C*Sin[c] + 63*A*b^3*Sin[d*x] + 189*a*b^2*B*Sin[d*x] + 189*a^2*b*C*Sin[d*x] + 49*b^3*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]*(105*a*A*b^2*Sin[c] + 105*a^2*b*B*Sin[c] + 25*b^3*B*Sin[c] + 35*a^3*C*Sin[c] + 75*a*b^2*C*Sin[c] + 315*a^2*A*b*Sin[d*x] + 63*A*b^3*Sin[d*x] + 105*a^3*B*Sin[d*x] + 189*a*b^2*B*Sin[d*x] + 189*a^2*b*C*Sin[d*x] + 49*b^3*C*Sin[d*x]))/(105*d)))/((b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a^3*A*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a*A*b^2*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^2*b*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*b^3*B*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^3*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*a*b^2*C*Cos[c + d*x]^5*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) + (6*a^2*A*b*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (6*A*b^3*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*a^3*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (18*a*b^2*B*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (18*a^2*b*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (14*b^3*C*Cos[c + d*x]^5*Csc[c]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1312,1,320,404,2.3457844,"\int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{10 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right)+154 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)+\frac{1}{12} \sin (c+d x) \sqrt{\cos (c+d x)} \left(154 a \cos (c+d x) \left(43 a^3 B+4 a^2 b (43 A+36 C)+216 a b^2 B+144 A b^3\right)+5 \left(154 a^3 (a B+4 A b) \cos (3 (c+d x))+36 a^2 \cos (2 (c+d x)) \left(a^2 (16 A+11 C)+44 a b B+66 A b^2\right)+3 \left(21 a^4 A \cos (4 (c+d x))+a^4 (531 A+572 C)+2288 a^3 b B+264 a^2 b^2 (13 A+14 C)+2464 a b^3 B+616 A b^4\right)\right)\right)}{1155 d}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right) (a \cos (c+d x)+b)^2}{231 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right)}{3465 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right)}{231 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right)}{693 d}+\frac{2 (11 a B+8 A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^3}{99 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^4}{11 d}",1,"(154*(7*a^4*B + 54*a^2*b^2*B + 15*b^4*B + 12*a*b^3*(3*A + 5*C) + 4*a^3*b*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2] + 10*(220*a^3*b*B + 308*a*b^3*B + 77*b^4*(A + 3*C) + 66*a^2*b^2*(5*A + 7*C) + 5*a^4*(9*A + 11*C))*EllipticF[(c + d*x)/2, 2] + (Sqrt[Cos[c + d*x]]*(154*a*(144*A*b^3 + 43*a^3*B + 216*a*b^2*B + 4*a^2*b*(43*A + 36*C))*Cos[c + d*x] + 5*(36*a^2*(66*A*b^2 + 44*a*b*B + a^2*(16*A + 11*C))*Cos[2*(c + d*x)] + 154*a^3*(4*A*b + a*B)*Cos[3*(c + d*x)] + 3*(616*A*b^4 + 2288*a^3*b*B + 2464*a*b^3*B + 264*a^2*b^2*(13*A + 14*C) + a^4*(531*A + 572*C) + 21*a^4*A*Cos[4*(c + d*x)])))*Sin[c + d*x])/12)/(1155*d)","A",1
1313,1,4114,377,8.2872463,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+162 a b B+3 b^2 (41 A-105 C)\right)}{315 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(15 a^3 B+12 a^2 b (5 A+7 C)+117 a b^2 B+2 b^3 (31 A-63 C)\right)}{63 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+4 a^3 b (5 A+7 C)+42 a^2 b^2 B+28 a b^3 (A+3 C)+21 b^4 B\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (7 A+9 C)+36 a^3 b B+18 a^2 b^2 (3 A+5 C)+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} (3 a B+5 A b-21 b C) (a \cos (c+d x)+b)^2}{21 d}+\frac{2 a (A-9 C) \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^3}{9 d}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(7*a^4*A + 54*a^2*A*b^2 + 15*A*b^4 + 36*a^3*b*B + 60*a*b^3*B + 9*a^4*C + 90*a^2*b^2*C - 30*b^4*C + 7*a^4*A*Cos[2*c] + 54*a^2*A*b^2*Cos[2*c] + 15*A*b^4*Cos[2*c] + 36*a^3*b*B*Cos[2*c] + 60*a*b^3*B*Cos[2*c] + 9*a^4*C*Cos[2*c] + 90*a^2*b^2*C*Cos[2*c])*Csc[c]*Sec[c])/(15*d) + (a*(92*a^2*A*b + 112*A*b^3 + 23*a^3*B + 168*a*b^2*B + 112*a^2*b*C)*Cos[d*x]*Sin[c])/(21*d) + (a^2*(19*a^2*A + 108*A*b^2 + 72*a*b*B + 18*a^2*C)*Cos[2*d*x]*Sin[2*c])/(45*d) + (a^3*(4*A*b + a*B)*Cos[3*d*x]*Sin[3*c])/(7*d) + (a^4*A*Cos[4*d*x]*Sin[4*c])/(18*d) + (a*(92*a^2*A*b + 112*A*b^3 + 23*a^3*B + 168*a*b^2*B + 112*a^2*b*C)*Cos[c]*Sin[d*x])/(21*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (a^2*(19*a^2*A + 108*A*b^2 + 72*a*b*B + 18*a^2*C)*Cos[2*c]*Sin[2*d*x])/(45*d) + (a^3*(4*A*b + a*B)*Cos[3*c]*Sin[3*d*x])/(7*d) + (a^4*A*Cos[4*c]*Sin[4*d*x])/(18*d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (80*a^3*A*b*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a*A*b^3*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*a^4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a^2*b^2*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a^3*b*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a*b^3*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (14*a^4*A*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (36*a^2*A*b^2*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (2*A*b^4*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (24*a^3*b*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (8*a*b^3*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (6*a^4*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (12*a^2*b^2*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*b^4*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1314,1,4776,371,8.9405078,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(21 a^2 B+54 a A b-350 a b C-105 b^2 B\right)}{105 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^3 (5 A+7 C)+28 a^2 b B+3 a b^2 (13 A-49 C)-42 b^3 B\right)}{21 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (5 A+7 C)+28 a^3 b B+42 a^2 b^2 (A+3 C)+84 a b^3 B+7 b^4 (3 A+C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^4 B+4 a^3 b (3 A+5 C)+30 a^2 b^2 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} (a A-21 a C-7 b B) (a \cos (c+d x)+b)^2}{7 d}+\frac{2 (8 a C+3 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{3 d \sqrt{\cos (c+d x)}}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(((12*I)/5)*a^3*A*b*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((4*I)*a*A*b^3*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (((3*I)/5)*a^4*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((6*I)*a^2*b^2*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*b^4*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((4*I)*a^3*b*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((4*I)*a*b^3*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(12*a^3*A*b + 20*a*A*b^3 + 3*a^4*B + 30*a^2*b^2*B - 10*b^4*B + 20*a^3*b*C - 40*a*b^3*C + 12*a^3*A*b*Cos[2*c] + 20*a*A*b^3*Cos[2*c] + 3*a^4*B*Cos[2*c] + 30*a^2*b^2*B*Cos[2*c] + 20*a^3*b*C*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (a^2*(23*a^2*A + 168*A*b^2 + 112*a*b*B + 28*a^2*C)*Cos[d*x]*Sin[c])/(21*d) + (2*a^3*(4*A*b + a*B)*Cos[2*d*x]*Sin[2*c])/(5*d) + (a^4*A*Cos[3*d*x]*Sin[3*c])/(7*d) + (a^2*(23*a^2*A + 168*A*b^2 + 112*a*b*B + 28*a^2*C)*Cos[c]*Sin[d*x])/(21*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(b^4*C*Sin[c] + 3*b^4*B*Sin[d*x] + 12*a*b^3*C*Sin[d*x]))/(3*d) + (2*a^3*(4*A*b + a*B)*Cos[2*c]*Sin[2*d*x])/(5*d) + (a^4*A*Cos[3*c]*Sin[3*d*x])/(7*d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (20*a^4*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a^2*A*b^2*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^4*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a^3*b*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a*b^3*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (24*a^2*b^2*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1315,1,4960,388,9.060439,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-\left(a^2 (3 A-59 C)\right)+50 a b B+3 b^2 (5 A+3 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(16 a^2 C+15 a b B+5 A b^2+3 b^2 C\right) (a \cos (c+d x)+b)^2}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^3 B+4 a^2 b (5 A-33 C)-105 a b^2 B-6 b^3 (5 A+3 C)\right)}{15 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 B+4 a^3 b (A+3 C)+18 a^2 b^2 B+4 a b^3 (3 A+C)+b^4 B\right)}{3 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (3 A+5 C)+20 a^3 b B+30 a^2 b^2 (A-C)-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}+\frac{2 (8 a C+5 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(((3*I)/5)*a^4*A*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((6*I)*a^2*A*b^2*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (I*A*b^4*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + ((4*I)*a^3*b*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((4*I)*a*b^3*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*a^4*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((6*I)*a^2*b^2*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (((3*I)/5)*b^4*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(3*a^4*A + 30*a^2*A*b^2 - 10*A*b^4 + 20*a^3*b*B - 40*a*b^3*B + 5*a^4*C - 60*a^2*b^2*C - 6*b^4*C + 3*a^4*A*Cos[2*c] + 30*a^2*A*b^2*Cos[2*c] + 20*a^3*b*B*Cos[2*c] + 5*a^4*C*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*a^3*(4*A*b + a*B)*Cos[d*x]*Sin[c])/(3*d) + (2*a^4*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (4*a^3*(4*A*b + a*B)*Cos[c]*Sin[d*x])/(3*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]^2*(3*b^4*C*Sin[c] + 5*b^4*B*Sin[d*x] + 20*a*b^3*C*Sin[d*x]))/(15*d) + (4*Sec[c]*Sec[c + d*x]*(5*b^4*B*Sin[c] + 20*a*b^3*C*Sin[c] + 15*A*b^4*Sin[d*x] + 60*a*b^3*B*Sin[d*x] + 90*a^2*b^2*C*Sin[d*x] + 9*b^4*C*Sin[d*x]))/(15*d) + (2*a^4*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (16*a^3*A*b*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a*A*b^3*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (24*a^2*b^2*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*b^4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a^3*b*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a*b^3*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1316,1,4791,384,9.3018255,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(48 a^2 C+77 a b B+35 A b^2+25 b^2 C\right) (a \cos (c+d x)+b)^2}{105 d \cos ^{\frac{3}{2}}(c+d x)}-\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-\left(a^2 (35 A-87 C)\right)+98 a b B+5 b^2 (7 A+5 C)\right)}{105 d}+\frac{2 b \sin (c+d x) \left(192 a^3 C+413 a^2 b B+2 a b^2 (175 A+101 C)+63 b^3 B\right)}{105 d \sqrt{\cos (c+d x)}}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^4 (A+3 C)+84 a^3 b B+42 a^2 b^2 (3 A+C)+28 a b^3 B+b^4 (7 A+5 C)\right)}{21 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^4 B+20 a^3 b (A-C)-30 a^2 b^2 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}+\frac{2 (8 a C+7 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"((4*I)*a^3*A*b*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((4*I)*a*A*b^3*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (I*a^4*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((6*I)*a^2*b^2*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (((3*I)/5)*b^4*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - ((4*I)*a^3*b*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (((12*I)/5)*a*b^3*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(20*a^3*A*b - 40*a*A*b^3 + 5*a^4*B - 60*a^2*b^2*B - 6*b^4*B - 40*a^3*b*C - 24*a*b^3*C + 20*a^3*A*b*Cos[2*c] + 5*a^4*B*Cos[2*c])*Csc[c]*Sec[c])/(5*d) + (4*a^4*A*Cos[d*x]*Sin[c])/(3*d) + (4*a^4*A*Cos[c]*Sin[d*x])/(3*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(7*d) + (4*Sec[c]*Sec[c + d*x]^3*(5*b^4*C*Sin[c] + 7*b^4*B*Sin[d*x] + 28*a*b^3*C*Sin[d*x]))/(35*d) + (4*Sec[c]*Sec[c + d*x]*(35*A*b^4*Sin[c] + 140*a*b^3*B*Sin[c] + 210*a^2*b^2*C*Sin[c] + 25*b^4*C*Sin[c] + 420*a*A*b^3*Sin[d*x] + 630*a^2*b^2*B*Sin[d*x] + 63*b^4*B*Sin[d*x] + 420*a^3*b*C*Sin[d*x] + 252*a*b^3*C*Sin[d*x]))/(105*d) + (4*Sec[c]*Sec[c + d*x]^2*(21*b^4*B*Sin[c] + 84*a*b^3*C*Sin[c] + 35*A*b^4*Sin[d*x] + 140*a*b^3*B*Sin[d*x] + 210*a^2*b^2*C*Sin[d*x] + 25*b^4*C*Sin[d*x]))/(105*d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*a^4*A*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (24*a^2*A*b^2*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*b^4*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a^3*b*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a*b^3*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a^2*b^2*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*b^4*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2])","C",0
1317,1,4150,401,8.9457647,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(48 a^2 C+117 a b B+63 A b^2+49 b^2 C\right) (a \cos (c+d x)+b)^2}{315 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b \sin (c+d x) \left(64 a^3 C+261 a^2 b B+2 a b^2 (147 A+101 C)+75 b^3 B\right)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^4 B+28 a^3 b (3 A+C)+42 a^2 b^2 B+4 a b^3 (7 A+5 C)+5 b^4 B\right)}{21 d}-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 (A-C)+60 a^3 b B+18 a^2 b^2 (5 A+3 C)+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(192 a^4 C+1098 a^3 b B+7 a^2 b^2 (261 A+155 C)+756 a b^3 B+21 b^4 (9 A+7 C)\right)}{315 d \sqrt{\cos (c+d x)}}+\frac{2 (8 a C+9 b B) \sin (c+d x) (a \cos (c+d x)+b)^3}{63 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{2 C \sin (c+d x) (a \cos (c+d x)+b)^4}{9 d \cos ^{\frac{9}{2}}(c+d x)}",1,"(Cos[c + d*x]^(13/2)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-2*(15*a^4*A - 180*a^2*A*b^2 - 18*A*b^4 - 120*a^3*b*B - 72*a*b^3*B - 30*a^4*C - 108*a^2*b^2*C - 14*b^4*C + 15*a^4*A*Cos[2*c])*Csc[c]*Sec[c])/(15*d) + (4*b^4*C*Sec[c]*Sec[c + d*x]^5*Sin[d*x])/(9*d) + (4*Sec[c]*Sec[c + d*x]^4*(7*b^4*C*Sin[c] + 9*b^4*B*Sin[d*x] + 36*a*b^3*C*Sin[d*x]))/(63*d) + (4*Sec[c]*Sec[c + d*x]^2*(63*A*b^4*Sin[c] + 252*a*b^3*B*Sin[c] + 378*a^2*b^2*C*Sin[c] + 49*b^4*C*Sin[c] + 420*a*A*b^3*Sin[d*x] + 630*a^2*b^2*B*Sin[d*x] + 75*b^4*B*Sin[d*x] + 420*a^3*b*C*Sin[d*x] + 300*a*b^3*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]^3*(45*b^4*B*Sin[c] + 180*a*b^3*C*Sin[c] + 63*A*b^4*Sin[d*x] + 252*a*b^3*B*Sin[d*x] + 378*a^2*b^2*C*Sin[d*x] + 49*b^4*C*Sin[d*x]))/(315*d) + (4*Sec[c]*Sec[c + d*x]*(140*a*A*b^3*Sin[c] + 210*a^2*b^2*B*Sin[c] + 25*b^4*B*Sin[c] + 140*a^3*b*C*Sin[c] + 100*a*b^3*C*Sin[c] + 630*a^2*A*b^2*Sin[d*x] + 63*A*b^4*Sin[d*x] + 420*a^3*b*B*Sin[d*x] + 252*a*b^3*B*Sin[d*x] + 105*a^4*C*Sin[d*x] + 378*a^2*b^2*C*Sin[d*x] + 49*b^4*C*Sin[d*x]))/(105*d)))/((b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (16*a^3*A*b*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a*A*b^3*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (4*a^4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (8*a^2*b^2*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (20*b^4*B*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (16*a^3*b*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (80*a*b^3*C*Cos[c + d*x]^6*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[1 + Cot[c]^2]) - (2*a^4*A*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (12*a^2*A*b^2*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (6*A*b^4*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (8*a^3*b*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (24*a*b^3*B*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (2*a^4*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (36*a^2*b^2*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) + (14*b^4*C*Cos[c + d*x]^6*Csc[c]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(b + a*Cos[c + d*x])^4*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x]))","C",0
1318,1,271,209,2.2804059,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 a^2 \left(3 a^2 (3 A+5 C)-5 a b B+5 A b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}+4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (3 a A \cos (c+d x)+5 a B-5 A b)+2 a^2 (5 a B+4 A b) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{30 a^4 d}","\frac{2 b^2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a+b)}-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}+\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (3 A+5 C)-5 a b B+5 A b^2\right)}{5 a^3 d}-\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^3 (-B)+a^2 b (A+3 C)-3 a b^2 B+3 A b^3\right)}{3 a^4 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}",1,"((2*a^2*(5*A*b^2 - 5*a*b*B + 3*a^2*(3*A + 5*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 2*a^2*(4*A*b + 5*a*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + 4*a^2*Sqrt[Cos[c + d*x]]*(-5*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x] + (6*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2]))/(30*a^4*d)","A",1
1319,1,215,147,1.217047,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\frac{6 (a B-A b) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+\frac{2 (3 a B-A b) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{4 (A+3 C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+4 A \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d}","-\frac{2 b \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a+b)}-\frac{2 (A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (A+3 C)-3 a b B+3 A b^2\right)}{3 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"((2*(-(A*b) + 3*a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*(A + 3*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + 4*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x] + (6*(-(A*b) + a*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/(6*a*d)","A",1
1320,0,0,97,64.1922406,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a+b)}-\frac{2 (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]), x]","F",-1
1321,1,210,118,2.5951029,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{-\frac{2 C \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 b (A-C) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{2 (2 b B-3 a C) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{4 C \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{2 b d}","-\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a+b)}+\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 C E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}+\frac{2 C \sin (c+d x)}{b d \sqrt{\cos (c+d x)}}",1,"((2*(2*b*B - 3*a*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (2*b*(A - C)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (4*C*Sin[c + d*x])/Sqrt[Cos[c + d*x]] - (2*C*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/(2*b*d)","A",1
1322,1,266,158,2.2821251,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\frac{\frac{2 b \left(9 a^2 C-9 a b B+6 A b^2+2 b^2 C\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 (a C-b B) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{b \left(8 a b C-6 b^2 B\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{12 b (b B-a C) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{4 b^2 C \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}}{6 b^3 d}","\frac{2 \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}-\frac{2 (b B-a C) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}+\frac{2 (b B-a C) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 C F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}+\frac{2 C \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*b*(6*A*b^2 - 9*a*b*B + 9*a^2*C + 2*b^2*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (b*(-6*b^2*B + 8*a*b*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (4*b^2*C*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (12*b*(b*B - a*C)*Sin[c + d*x])/Sqrt[Cos[c + d*x]] + (6*(-(b*B) + a*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/(6*b^3*d)","A",1
1323,1,332,236,4.2889635,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),x]","-\frac{\frac{2 b \left(20 a^2 C-20 a b B+15 A b^2+9 b^2 C\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}-\frac{2 \left(3 \sin (2 (c+d x)) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)+10 b (b B-a C) \sin (c+d x)+6 b^2 C \tan (c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{6 \sin (c+d x) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(45 a^3 C-45 a^2 b B+a b^2 (45 A+19 C)-10 b^3 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{30 b^3 d}","-\frac{2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d}+\frac{2 \sin (c+d x) \left(5 a^2 C-5 a b B+5 A b^2+3 b^2 C\right)}{5 b^3 d \sqrt{\cos (c+d x)}}-\frac{2 a \left(A b^2-a (b B-a C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 (b B-a C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d}+\frac{2 (b B-a C) \sin (c+d x)}{3 b^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 C \sin (c+d x)}{5 b d \cos ^{\frac{5}{2}}(c+d x)}",1,"-1/30*((2*(-45*a^2*b*B - 10*b^3*B + 45*a^3*C + a*b^2*(45*A + 19*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (2*b*(15*A*b^2 - 20*a*b*B + 20*a^2*C + 9*b^2*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (6*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]) - (2*(10*b*(b*B - a*C)*Sin[c + d*x] + 3*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sin[2*(c + d*x)] + 6*b^2*C*Tan[c + d*x]))/Cos[c + d*x]^(3/2))/(b^3*d)","A",1
1324,1,337,346,3.5039934,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(\frac{3 b \left(a (a C-b B)+A b^2\right)}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}+2 A\right)-\frac{\frac{8 \left(a^2 (A+3 C)-3 a b B+2 A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(6 a^3 B-a^2 b (8 A+3 C)-3 a b^2 B+5 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(2 a^3 B+a^2 b (C-4 A)-3 a b^2 B+5 A b^3\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(b-a) (a+b)}}{12 a^2 d}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-\left(a^2 (2 A-3 C)\right)-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2 a^3 B-a^2 b (4 A-C)-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-2 a^4 (A+3 C)+12 a^3 b B-a^2 b^2 (16 A-3 C)-9 a b^3 B+15 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}+\frac{b \left(-3 a^4 C+5 a^3 b B-a^2 b^2 (7 A-C)-3 a b^3 B+5 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}",1,"(4*Sqrt[Cos[c + d*x]]*(2*A + (3*b*(A*b^2 + a*(-(b*B) + a*C)))/((-a^2 + b^2)*(b + a*Cos[c + d*x])))*Sin[c + d*x] - ((2*(5*A*b^3 + 6*a^3*B - 3*a*b^2*B - a^2*b*(8*A + 3*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(2*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(5*A*b^3 + 2*a^3*B - 3*a*b^2*B + a^2*b*(-4*A + C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(12*a^2*d)","A",1
1325,1,299,257,2.9590256,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\frac{2 \left(a^2 (2 A+C)-a b B-A b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \sin (c+d x) \left(a^2 (2 A-C)+a b B-3 A b^2\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}-\frac{8 (-a B+A b+b C) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}}{(a-b) (a+b)}}{4 a d}","-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A-C)\right)-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(2 a^3 B-a^2 b (4 A+C)-a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\left(a^4 (-C)+3 a^3 b B-a^2 b^2 (5 A+C)-a b^3 B+3 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((4*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + ((2*(-(A*b^2) - a*b*B + a^2*(2*A + C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*(A*b - a*B + b*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (2*(-3*A*b^2 + a*b*B + a^2*(2*A - C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
1326,1,298,239,3.8647991,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","-\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\frac{2 \left(3 a^2 C+a b B-A b^2-4 b^2 C\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \sin (c+d x) \left(a (a C-b B)+A b^2\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+\frac{8 b (a (A+C)-b B) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}}{(b-a) (a+b)}}{4 b d}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-\left(a^2 (2 A+C)\right)+a b B+A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\left(a^4 C+a^3 b B-3 a^2 b^2 (A+C)+a b^3 B+A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 b d (a-b) (a+b)^2}",1,"-1/4*((4*(A*b^2 + a*(-(b*B) + a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + ((2*(-(A*b^2) + a*b*B + 3*a^2*C - 4*b^2*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-(b*B) + a*(A + C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + (2*(A*b^2 + a*(-(b*B) + a*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(b*d)","A",1
1327,1,337,307,4.8940086,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{a \sin (c+d x) \left(a (a C-b B)+A b^2\right)}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}+2 C \tan (c+d x)\right)-\frac{\frac{4 b \left(2 a^2 C-a b B+A b^2-b^2 C\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{2 \sin (c+d x) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(9 a^3 C-3 a^2 b B-a b^2 (A+10 C)+4 b^3 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b) (a+b)}}{4 b^2 d}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(A b^2-a (b B-a C)\right)}{a b d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(3 a^2 C-a b B+A b^2-2 b^2 C\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}+\frac{\left(-3 a^4 C+a^3 b B+a^2 b^2 (A+5 C)-3 a b^3 B+A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b^2 d (a+b) \left(a^2-b^2\right)}",1,"(-(((2*(-3*a^2*b*B + 4*b^3*B + 9*a^3*C - a*b^2*(A + 10*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*b*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*((a*(A*b^2 + a*(-(b*B) + a*C))*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + 2*C*Tan[c + d*x]))/(4*b^2*d)","A",1
1328,1,423,387,6.7732592,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{8 b \left(b^2-a^2\right) (3 b B-5 a C) \sin (c+d x)+8 b^2 C \left(b^2-a^2\right) \tan (c+d x)+6 a \sin (2 (c+d x)) \left(5 a^3 C-3 a^2 b B+a b^2 (A-4 C)+2 b^3 B\right)}{\left(b^2-a^2\right) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}+\frac{\frac{8 b \left(10 a^3 C-6 a^2 b B+a b^2 (3 A-7 C)+3 b^3 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}+\frac{6 \sin (c+d x) \left(5 a^3 C-3 a^2 b B+a b^2 (A-4 C)+2 b^3 B\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(45 a^4 C-27 a^3 b B+a^2 b^2 (9 A-44 C)+30 a b^3 B-4 b^4 (3 A+C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b) (a+b)}}{12 b^3 d}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}+\frac{\sin (c+d x) \left(5 a^2 C-3 a b B+3 A b^2-2 b^2 C\right)}{3 b^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}+\frac{\sin (c+d x) \left(-5 a^3 C+3 a^2 b B-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(-5 a^4 C+3 a^3 b B-a^2 b^2 (A-7 C)-5 a b^3 B+3 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}",1,"(((2*(-27*a^3*b*B + 30*a*b^3*B + a^2*b^2*(9*A - 44*C) + 45*a^4*C - 4*b^4*(3*A + C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-6*a^2*b*B + 3*b^3*B + a*b^2*(3*A - 7*C) + 10*a^3*C)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + (6*(-3*a^2*b*B + 2*b^3*B + a*b^2*(A - 4*C) + 5*a^3*C)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)) + (8*b*(-a^2 + b^2)*(3*b*B - 5*a*C)*Sin[c + d*x] + 6*a*(-3*a^2*b*B + 2*b^3*B + a*b^2*(A - 4*C) + 5*a^3*C)*Sin[2*(c + d*x)] + 8*b^2*(-a^2 + b^2)*C*Tan[c + d*x])/((-a^2 + b^2)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])))/(12*b^3*d)","A",1
1329,1,520,538,7.1736406,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(4 a^6 A-21 a^4 b^2 C+4 A \left(a^3-a b^2\right)^2 \cos (2 (c+d x))+33 a^3 b^3 B-57 a^2 A b^4+3 a^2 b^4 C+a b \cos (c+d x) \left(a^4 (16 A-27 C)+39 a^3 b B+a^2 b^2 (9 C-83 A)-21 a b^3 B+49 A b^4\right)-15 a b^5 B+35 A b^6\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{\frac{16 \left(2 a^4 (A+3 C)-12 a^3 b B+a^2 b^2 (14 A+3 C)+3 a b^3 B-7 A b^4\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(24 a^5 B-a^4 b (56 A+15 C)-21 a^3 b^2 B+a^2 b^3 (73 A-3 C)+15 a b^4 B-35 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \sin (c+d x) \left(8 a^5 B+a^4 (9 b C-24 A b)-29 a^3 b^2 B+a^2 b^3 (65 A-3 C)+15 a b^4 B-35 A b^5\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 a^3 d}","\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}-\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-5 a^4 C+9 a^3 b B-a^2 b^2 (13 A+C)-3 a b^3 B+7 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 (8 A-21 C)+33 a^3 b B-a^2 b^2 (61 A-3 C)-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-8 a^5 B+a^4 (24 A b-9 b C)+29 a^3 b^2 B-a^2 b^3 (65 A-3 C)-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(8 a^6 (A+3 C)-72 a^5 b B+a^4 b^2 (128 A-15 C)+99 a^3 b^3 B-a^2 b^4 (223 A-9 C)-45 a b^5 B+105 A b^6\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{b \left(15 a^6 C-35 a^5 b B+3 a^4 b^2 (21 A-2 C)+38 a^3 b^3 B-a^2 b^4 (86 A-3 C)-15 a b^5 B+35 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(4*a^6*A - 57*a^2*A*b^4 + 35*A*b^6 + 33*a^3*b^3*B - 15*a*b^5*B - 21*a^4*b^2*C + 3*a^2*b^4*C + a*b*(49*A*b^4 + 39*a^3*b*B - 21*a*b^3*B + a^4*(16*A - 27*C) + a^2*b^2*(-83*A + 9*C))*Cos[c + d*x] + 4*A*(a^3 - a*b^2)^2*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + ((2*(-35*A*b^5 + 24*a^5*B - 21*a^3*b^2*B + 15*a*b^4*B + a^2*b^3*(73*A - 3*C) - a^4*b*(56*A + 15*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(-7*A*b^4 - 12*a^3*b*B + 3*a*b^3*B + 2*a^4*(A + 3*C) + a^2*b^2*(14*A + 3*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-35*A*b^5 + 8*a^5*B - 29*a^3*b^2*B + 15*a*b^4*B + a^2*b^3*(65*A - 3*C) + a^4*(-24*A*b + 9*b*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*a^3*d)","A",1
1330,1,437,426,5.8920377,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \cos (c+d x) \left(5 a^4 C-9 a^3 b B+a^2 b^2 (13 A+C)+3 a b^3 B-7 A b^4\right)+b \left(3 a^4 C-7 a^3 b B+a^2 b^2 (11 A+3 C)+a b^3 B-5 A b^4\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{\frac{8 \left(2 a^3 B-a^2 b (4 A+3 C)+a b^2 B+A b^3\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(a^4 (8 A+C)-5 a^3 b B+a^2 b^2 (5 C-7 A)-a b^3 B+5 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\sin (c+d x) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 a^2 d}","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A-5 C)+9 a^3 b B-a^2 b^2 (29 A+C)-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-3 a^4 C+7 a^3 b B-a^2 b^2 (11 A+3 C)-a b^3 B+5 A b^4\right)}{4 a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-8 a^5 B+a^4 b (24 A+7 C)+5 a^3 b^2 B-a^2 b^3 (33 A+C)-3 a b^4 B+15 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^6 C-15 a^5 b B+5 a^4 b^2 (7 A+2 C)+6 a^3 b^3 B-a^2 b^4 (38 A+C)-3 a b^5 B+15 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(b*(-5*A*b^4 - 7*a^3*b*B + a*b^3*B + 3*a^4*C + a^2*b^2*(11*A + 3*C)) + a*(-7*A*b^4 - 9*a^3*b*B + 3*a*b^3*B + 5*a^4*C + a^2*b^2*(13*A + C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (((5*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*(8*A + C) + a^2*b^2*(-7*A + 5*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(A*b^3 + 2*a^3*B + a*b^2*B - a^2*b*(4*A + 3*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B + a^4*(8*A - 5*C) - a^2*b^2*(29*A + C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*a^2*d)","A",1
1331,1,424,423,5.6933864,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{\frac{8 b \left(a^2 (2 A+C)-3 a b B+b^2 (A+2 C)\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(3 a^4 C+a^3 b B-a^2 b^2 (5 A+9 C)+5 a b^3 B-A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\sin (c+d x) \left(a^4 C-5 a^3 b B+a^2 b^2 (9 A+5 C)-a b^3 B-3 A b^4\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \cos (c+d x) \left(a^4 C-5 a^3 b B+a^2 b^2 (9 A+5 C)-a b^3 B-3 A b^4\right)-b \left(a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{8 a b d}","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 a d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (8 A+3 C)-7 a^3 b B-a^2 b^2 (5 A-3 C)+a b^3 B+3 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 (-C)+5 a^3 b B-a^2 b^2 (9 A+5 C)+a b^3 B+3 A b^4\right)}{4 a b d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{\left(a^6 (-C)-3 a^5 b B+5 a^4 b^2 (3 A+2 C)-10 a^3 b^3 B-3 a^2 b^4 (2 A-C)+a b^5 B+3 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 b d (a-b)^2 (a+b)^3}",1,"((-2*Sqrt[Cos[c + d*x]]*(-(b*(A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))) + a*(-3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + a^2*b^2*(9*A + 5*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (((-(A*b^4) + a^3*b*B + 5*a*b^3*B + 3*a^4*C - a^2*b^2*(5*A + 9*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-3*a*b*B + a^2*(2*A + C) + b^2*(A + 2*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((-3*A*b^4 - 5*a^3*b*B - a*b^3*B + a^4*C + a^2*b^2*(9*A + 5*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*a*b*d)","A",1
1332,1,440,409,6.2105193,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)-a \cos (c+d x) \left(3 a^4 C+a^3 b B-a^2 b^2 (5 A+9 C)+5 a b^3 B-A b^4\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{\frac{8 b \left(a^3 C+a^2 b B-a b^2 (3 A+4 C)+2 b^3 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}+\frac{\left(9 a^4 C+3 a^3 b B+a^2 b^2 (A-19 C)-9 a b^3 B+b^4 (5 A+16 C)\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\sin (c+d x) \left(3 a^4 C+a^3 b B-a^2 b^2 (5 A+9 C)+5 a b^3 B-A b^4\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 b^2 d}","-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 C+3 a^3 b B-7 a^2 b^2 (A+C)+3 a b^3 B+A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-3 a^4 C-a^3 b B+a^2 b^2 (5 A+9 C)-5 a b^3 B+A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}-\frac{\left(-3 a^6 C-a^5 b B-3 a^4 b^2 (A-2 C)+10 a^3 b^3 B-5 a^2 b^4 (2 A+3 C)+3 a b^5 B+A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b^2 d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(b*(3*A*b^4 + a^3*b*B - 7*a*b^3*B - 5*a^4*C + a^2*b^2*(3*A + 11*C)) - a*(-(A*b^4) + a^3*b*B + 5*a*b^3*B + 3*a^4*C - a^2*b^2*(5*A + 9*C))*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (((3*a^3*b*B - 9*a*b^3*B + a^2*b^2*(A - 19*C) + 9*a^4*C + b^4*(5*A + 16*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + ((-(A*b^4) + a^3*b*B + 5*a*b^3*B + 3*a^4*C - a^2*b^2*(5*A + 9*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b^2*d)","A",1
1333,1,501,496,6.9129527,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{\sqrt{\cos (c+d x)} \left(16 C \left(b^3-a^2 b\right)^2 \tan (c+d x)+a^2 \sin (2 (c+d x)) \left(15 a^4 C-3 a^3 b B-a^2 b^2 (A+29 C)+9 a b^3 B+b^4 (8 C-5 A)\right)+2 a b \sin (c+d x) \left(25 a^4 C-5 a^3 b B+a^2 b^2 (A-47 C)+11 a b^3 B+b^4 (16 C-7 A)\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{-\frac{8 b \left(-5 a^4 C+a^3 b B+a^2 b^2 (A+10 C)-4 a b^3 B+2 b^4 (A-C)\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}+\frac{\sin (c+d x) \left(15 a^4 C-3 a^3 b B-a^2 b^2 (A+29 C)+9 a b^3 B+b^4 (8 C-5 A)\right) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{\left(45 a^5 C-9 a^4 b B-a^3 b^2 (3 A+95 C)+19 a^2 b^3 B+a b^4 (9 A+56 C)-16 b^5 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b)^2 (a+b)^2}}{8 b^3 d}","-\frac{\sin (c+d x) \left(A b^2-a (b B-a C)\right)}{2 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^2}+\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\sin (c+d x) \left(-15 a^4 C+3 a^3 b B+a^2 b^2 (A+29 C)-9 a b^3 B+b^4 (5 A-8 C)\right)}{4 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\sin (c+d x) \left(-5 a^4 C+a^3 b B+a^2 b^2 (3 A+11 C)-7 a b^3 B+3 A b^4\right)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}-\frac{\left(15 a^6 C-3 a^5 b B-a^4 b^2 (A+38 C)+6 a^3 b^3 B+5 a^2 b^4 (2 A+7 C)-15 a b^5 B+3 A b^6\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^3 d (a-b)^2 (a+b)^3}",1,"(-((((-9*a^4*b*B + 19*a^2*b^3*B - 16*b^5*B + 45*a^5*C + a*b^4*(9*A + 56*C) - a^3*b^2*(3*A + 95*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*b*(a^3*b*B - 4*a*b^3*B + 2*b^4*(A - C) - 5*a^4*C + a^2*b^2*(A + 10*C))*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + ((-3*a^3*b*B + 9*a*b^3*B + 15*a^4*C + b^4*(-5*A + 8*C) - a^2*b^2*(A + 29*C))*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + (Sqrt[Cos[c + d*x]]*(2*a*b*(-5*a^3*b*B + 11*a*b^3*B + a^2*b^2*(A - 47*C) + 25*a^4*C + b^4*(-7*A + 16*C))*Sin[c + d*x] + a^2*(-3*a^3*b*B + 9*a*b^3*B + 15*a^4*C + b^4*(-5*A + 8*C) - a^2*b^2*(A + 29*C))*Sin[2*(c + d*x)] + 16*(-(a^2*b) + b^3)^2*C*Tan[c + d*x]))/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(8*b^3*d)","A",1
1334,1,3595,457,24.0755307,"\int \cos ^{\frac{9}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-7 a^2 (7 A+9 C)-9 a b B+6 A b^2\right) \sqrt{a+b \sec (c+d x)}}{315 a^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(75 a^3 B+a^2 b (13 A+21 C)-12 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)}}{315 a^3 d}-\frac{2 \left(a^2-b^2\right) \left(-75 a^3 B+6 a^2 b (6 A+7 C)-24 a b^2 B+16 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(-21 a^4 (7 A+9 C)-57 a^3 b B+6 a^2 b^2 (4 A+7 C)-24 a b^3 B+16 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (9 a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{63 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(((57*a^2*A*b + 32*A*b^3 + 345*a^3*B - 48*a*b^2*B + 84*a^2*b*C)*Sin[c + d*x])/(630*a^3) + ((133*a^2*A - 12*A*b^2 + 18*a*b*B + 126*a^2*C)*Sin[2*(c + d*x)])/(630*a^2) + ((A*b + 9*a*B)*Sin[3*(c + d*x)])/(126*a) + (A*Sin[4*(c + d*x)])/36))/d - (2*Cos[c + d*x]^(3/2)*((7*a*A*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*A*b^2*Sqrt[Cos[c + d*x]])/(105*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*A*b^4*Sqrt[Cos[c + d*x]])/(315*a^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (19*b*B*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*b^3*B*Sqrt[Cos[c + d*x]])/(105*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*a*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^2*C*Sqrt[Cos[c + d*x]])/(15*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (37*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (4*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(315*a^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[b + a*Cos[c + d*x]]) + (7*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((-I)*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^4*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*(-1/315*(Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^3*(b + a*Cos[c + d*x])^(3/2)) + (Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^4*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(((16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2))/2 - I*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - a*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] + (3*(16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(315*a^4*Sqrt[b + a*Cos[c + d*x]]) - (Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-16*A*b^4 + 57*a^3*b*B + 24*a*b^3*B - 6*a^2*b^2*(4*A + 7*C) + 21*a^4*(7*A + 9*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^3 + 12*a*b^2*(A + 2*B) - 6*a^2*b*(6*A + 3*B + 7*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^4 - 57*a^3*b*B - 24*a*b^3*B + 6*a^2*b^2*(4*A + 7*C) - 21*a^4*(7*A + 9*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*a^4*Sqrt[b + a*Cos[c + d*x]])))","C",0
1335,1,3071,360,22.6164463,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-5 a^2 (5 A+7 C)-7 a b B+4 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 a^2 d}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A+35 a^2 C-14 a b B+8 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left(63 a^3 B+a^2 b (19 A+35 C)-14 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(((115*a^2*A - 16*A*b^2 + 28*a*b*B + 140*a^2*C)*Sin[c + d*x])/(210*a^2) + ((A*b + 7*a*B)*Sin[2*(c + d*x)])/(35*a) + (A*Sin[3*(c + d*x)])/14))/d - (2*Cos[c + d*x]^(3/2)*((19*A*b*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*A*b^3*Sqrt[Cos[c + d*x]])/(105*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (3*a*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^2*B*Sqrt[Cos[c + d*x]])/(15*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (b*C*Sqrt[Cos[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (2*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*a*Sqrt[b + a*Cos[c + d*x]]) + (7*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (a*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((-I)*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^3*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*(-1/105*(Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^2*(b + a*Cos[c + d*x])^(3/2)) + (Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(35*a^3*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(105*a^3*Sqrt[b + a*Cos[c + d*x]]) - (Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(35*a^3*Sqrt[b + a*Cos[c + d*x]])))","C",0
1336,1,404,273,17.153269,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(\frac{2 (5 a B+A b) \sin (c+d x)}{15 a}+\frac{1}{5} A \sin (2 (c+d x))\right)}{d}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \sqrt{a+b \sec (c+d x)} \left(-\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \left(3 a^2 (3 A+5 C)+5 a b B-2 A b^2\right) (a \cos (c+d x)+b)-i (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(3 a^2 (3 A+5 C)+5 a b B-2 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i a (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) (9 a A+5 a (B+3 C)-2 A b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{15 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}","-\frac{2 \sqrt{\cos (c+d x)} \left(-3 a^2 (3 A+5 C)-5 a b B+2 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \left(a^2-b^2\right) (2 A b-5 a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 (5 a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*((2*(A*b + 5*a*B)*Sin[c + d*x])/(15*a) + (A*Sin[2*(c + d*x)])/5))/d - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((-I)*(a + b)*(-2*A*b^2 + 5*a*b*B + 3*a^2*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(9*a*A - 2*A*b + 5*a*(B + 3*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-2*A*b^2 + 5*a*b*B + 3*a^2*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^2*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","C",0
1337,1,43023,277,33.3205665,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","-\frac{2 \left(A b^2-a^2 (A+3 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 (3 a B+A b) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1338,1,64644,258,32.6368491,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{(2 A-C) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 a B+b C) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"Result too large to show","C",0
1339,1,100266,346,33.3612671,"\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{\left(a^2 (-C)+4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(8 a A+3 a C+4 b B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a C+4 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}-\frac{(a C+4 b B) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{C \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
1340,1,131249,447,33.9210432,"\int \frac{\sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{24 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left(a^2 (-C)+18 a b B+24 A b^2+16 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left(-3 a^2 C+6 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{\left(a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a C+6 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{12 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Result too large to show","C",0
1341,1,3703,455,24.1641618,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+72 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)}}{315 a d}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-75 a^3 B-2 a^2 b (44 A+63 C)-9 a b^2 B+4 A b^3\right) \sqrt{a+b \sec (c+d x)}}{315 a^2 d}+\frac{2 \left(a^2-b^2\right) \left(75 a^3 B+a^2 (39 A b+63 b C)-18 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left(21 a^4 (7 A+9 C)+246 a^3 b B+3 a^2 b^2 (11 A+21 C)-18 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (3 a B+A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{9 d}",1,"(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((402*a^2*A*b - 16*A*b^3 + 345*a^3*B + 36*a*b^2*B + 504*a^2*b*C)*Sin[c + d*x])/(315*a^2) + ((133*a^2*A + 6*A*b^2 + 144*a*b*B + 126*a^2*C)*Sin[2*(c + d*x)])/(315*a) + ((10*A*b + 9*a*B)*Sin[3*(c + d*x)])/63 + (a*A*Sin[4*(c + d*x)])/18))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((14*a^2*A*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (22*A*b^2*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4*Sqrt[Cos[c + d*x]])/(315*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (164*a*b*B*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^3*B*Sqrt[Cos[c + d*x]])/(35*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^2*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (124*a*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(315*a*Sqrt[b + a*Cos[c + d*x]]) + (10*a^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (34*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (8*a*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^3*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^2*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^3*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(315*a^3*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^2*b*(13*A + 57*B + 21*C) + 3*a^3*(49*A + 25*B + 63*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(11*A + 21*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*a^3*Sqrt[b + a*Cos[c + d*x]])))","C",0
1342,1,3261,359,22.8637104,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 (5 A+7 C)+42 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 a d}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A+35 a^2 C+21 a b B-6 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(-63 a^3 B-2 a^2 b (41 A+70 C)-21 a b^2 B+6 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+3 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{7 d}",1,"(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((115*a^2*A + 12*A*b^2 + 168*a*b*B + 140*a^2*C)*Sin[c + d*x])/(105*a) + (2*(8*A*b + 7*a*B)*Sin[2*(c + d*x)])/35 + (a*A*Sin[3*(c + d*x)])/7))/(d*(b + a*Cos[c + d*x])*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((164*a*A*b*Sqrt[Cos[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b^3*Sqrt[Cos[c + d*x]])/(35*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^2*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*a*b*C*Sqrt[Cos[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (10*a^2*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (34*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (8*a*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^2*d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(7/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(35*a^2*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(105*a^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + a^2*(25*A + 63*B + 35*C) + 3*a*b*(19*A + 7*(B + 5*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-6*A*b^3 + 63*a^3*B + 21*a*b^2*B + 2*a^2*b*(41*A + 70*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(35*a^2*Sqrt[b + a*Cos[c + d*x]])))","C",0
1343,1,56321,356,35.1502585,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sqrt{\cos (c+d x)} \left(3 a^2 (3 A+5 C)+20 a b B+3 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \left(-5 a^3 B-3 a^2 b (A+5 C)+5 a b^2 B+3 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 (5 a B+3 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}+\frac{2 b^2 C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1344,1,79958,340,33.3372758,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\left(2 a^2 (A+3 C)+6 a b B-b^2 (2 A-3 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\cos (c+d x)} (6 a B+8 A b-3 b C) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{b (2 A-3 C) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}{3 d}+\frac{b (3 a C+2 b B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1345,1,120732,353,34.4149837,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\left(8 a^2 B+a b (8 A+7 C)+4 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2 C+12 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{\cos (c+d x)} (8 a A-5 a C-4 b B) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(3 a C+4 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d \sqrt{\cos (c+d x)}}",1,"Result too large to show","C",0
1346,1,132839,446,34.237911,"\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\left(a^2 (48 A+17 C)+42 a b B+8 b^2 (3 A+2 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left(3 a^2 C+30 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(a^3 (-C)+6 a^2 b B+12 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
1347,1,179293,551,35.4671652,"\int \frac{(a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(3 a^2 C+56 a b B+48 A b^2+36 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{96 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{192 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left(-3 a^3 C+136 a^2 b B+12 a b^2 (28 A+19 C)+128 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left(-9 a^3 C+24 a^2 b B+12 a b^2 (20 A+13 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{\left(-3 a^4 C+8 a^3 b B-24 a^2 b^2 (2 A+C)-96 a b^3 B-16 b^4 (4 A+3 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{64 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(3 a C+8 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{4 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Result too large to show","C",0
1348,1,4170,565,25.3361688,"\int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(3 a^2 (9 A+11 C)+44 a b B+5 A b^2\right) \sqrt{a+b \sec (c+d x)}}{231 d}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(539 a^3 B+5 a^2 b (229 A+297 C)+825 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{3465 a d}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-75 a^4 (9 A+11 C)-1793 a^3 b B-5 a^2 b^2 (205 A+297 C)-55 a b^3 B+20 A b^4\right) \sqrt{a+b \sec (c+d x)}}{3465 a^2 d}+\frac{2 \left(a^2-b^2\right) \left(75 a^4 (9 A+11 C)+1254 a^3 b B+15 a^2 b^2 (19 A+33 C)-110 a b^3 B+40 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left(1617 a^5 B+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+15 a^2 b^3 (17 A+33 C)-110 a b^4 B+40 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3465 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (11 a B+5 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{99 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{11 d}",1,"(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((6525*a^4*A + 9330*a^2*A*b^2 - 160*A*b^4 + 16434*a^3*b*B + 440*a*b^3*B + 7590*a^4*C + 11880*a^2*b^2*C)*Sin[c + d*x])/(6930*a^2) + ((3095*a^2*A*b + 30*A*b^3 + 1463*a^3*B + 1650*a*b^2*B + 2970*a^2*b*C)*Sin[2*(c + d*x)])/(3465*a) + ((513*a^2*A + 452*A*b^2 + 836*a*b*B + 396*a^2*C)*Sin[3*(c + d*x)])/2772 + (a*(23*A*b + 11*a*B)*Sin[4*(c + d*x)])/198 + (a^2*A*Sin[5*(c + d*x)])/44))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((494*a^2*A*b*Sqrt[Cos[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (34*A*b^3*Sqrt[Cos[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^5*Sqrt[Cos[c + d*x]])/(693*a^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (14*a^3*B*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (62*a*b^2*B*Sqrt[Cos[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^4*B*Sqrt[Cos[c + d*x]])/(63*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (58*a^2*b*C*Sqrt[Cos[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^3*C*Sqrt[Cos[c + d*x]])/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (30*a^3*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) + (442*a*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^4*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(693*a*Sqrt[b + a*Cos[c + d*x]]) + (58*a^2*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (62*b^3*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(63*Sqrt[b + a*Cos[c + d*x]]) + (10*a^3*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (18*a*b^2*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(7*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3465*a^3*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3465*a^2*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(1155*a^3*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(3465*a^3*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B + 33*C) + 3*a^4*(225*A + 539*B + 275*C) + 6*a^3*b*(505*A + 209*B + 660*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B + 15*a^2*b^3*(17*A + 33*C) + 15*a^4*b*(247*A + 319*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(1155*a^3*Sqrt[b + a*Cos[c + d*x]])))","C",0
1349,1,3785,452,24.568594,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(7 a^2 (7 A+9 C)+90 a b B+15 A b^2\right) \sqrt{a+b \sec (c+d x)}}{315 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(75 a^3 B+a^2 b (163 A+231 C)+135 a b^2 B+5 A b^3\right) \sqrt{a+b \sec (c+d x)}}{315 a d}-\frac{2 \left(a^2-b^2\right) \left(-75 a^3 B-6 a^2 b (19 A+28 C)-45 a b^2 B+10 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(-21 a^4 (7 A+9 C)-435 a^3 b B-3 a^2 b^2 (93 A+161 C)-45 a b^3 B+10 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{315 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (9 a B+5 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{63 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{9 d}",1,"(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(((747*a^2*A*b + 20*A*b^3 + 345*a^3*B + 540*a*b^2*B + 924*a^2*b*C)*Sin[c + d*x])/(315*a) + ((133*a^2*A + 150*A*b^2 + 270*a*b*B + 126*a^2*C)*Sin[2*(c + d*x)])/315 + (a*(19*A*b + 9*a*B)*Sin[3*(c + d*x)])/63 + (a^2*A*Sin[4*(c + d*x)])/18))/(d*(b + a*Cos[c + d*x])^2*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])) - (4*Cos[c + d*x]^(3/2)*((14*a^3*A*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (62*a*A*b^2*Sqrt[Cos[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*A*b^4*Sqrt[Cos[c + d*x]])/(63*a*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (58*a^2*b*B*Sqrt[Cos[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]])/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (6*a^3*C*Sqrt[Cos[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (46*a*b^2*C*Sqrt[Cos[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (58*a^2*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (62*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(63*Sqrt[b + a*Cos[c + d*x]]) + (10*a^3*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (18*a*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(7*Sqrt[b + a*Cos[c + d*x]]) + (34*a^2*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (2*b^3*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/Sqrt[b + a*Cos[c + d*x]])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^2*d*(b + a*Cos[c + d*x])^3*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sec[c + d*x]^(9/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^2*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(((10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2))/2 - I*(a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - a*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] + (3*(10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(315*a^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B + 21*a^4*(7*A + 9*C) + 3*a^2*b^2*(93*A + 161*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-10*A*b^3 + 6*a^2*b*(19*A + 60*B + 28*C) + 3*a^3*(49*A + 25*B + 63*C) + 15*a*b^2*(11*A + 3*(B + 7*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (10*A*b^4 - 435*a^3*b*B - 45*a*b^3*B - 21*a^4*(7*A + 9*C) - 3*a^2*b^2*(93*A + 161*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*a^2*Sqrt[b + a*Cos[c + d*x]])))","C",0
1350,1,64878,441,35.238213,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 (5 A+7 C)+56 a b B+15 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 d}+\frac{2 \sqrt{\cos (c+d x)} \left(63 a^3 B+5 a^2 b (29 A+49 C)+161 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \left(-5 a^4 (5 A+7 C)-56 a^3 b B+10 a^2 b^2 (A-7 C)+56 a b^3 B+15 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 (7 a B+5 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}+\frac{2 b^3 C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1351,1,86542,419,34.4119616,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{\cos (c+d x)} \left(6 a^2 (3 A+5 C)+70 a b B+b^2 (46 A-15 C)\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(10 a^3 B+4 a^2 b (4 A+15 C)+20 a b^2 B-b^3 (16 A-15 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{b \sin (c+d x) (10 a B+16 A b-15 b C) \sqrt{a+b \sec (c+d x)}}{15 d \sqrt{\cos (c+d x)}}+\frac{2 (a B+A b) \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}{3 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}{5 d}+\frac{b^2 (5 a C+2 b B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1352,1,129353,427,35.1150417,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sqrt{\cos (c+d x)} \left(24 a^2 B+a b (56 A-27 C)-12 b^2 B\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{12 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b \left(15 a^2 C+20 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(8 a^3 (A+3 C)+48 a^2 b B+a b^2 (16 A+33 C)+12 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{12 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{b \sin (c+d x) (8 a A-21 a C-12 b B) \sqrt{a+b \sec (c+d x)}}{12 d \sqrt{\cos (c+d x)}}-\frac{b (4 A-3 C) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{6 d \sqrt{\cos (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2}}{3 d}",1,"Result too large to show","C",0
1353,1,157926,453,35.5977662,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left(-\left(a^2 (48 A-33 C)\right)+54 a b B+8 b^2 (3 A+2 C)\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{8 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(5 a C+6 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d \sqrt{\cos (c+d x)}}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt{\cos (c+d x)}}",1,"Result too large to show","C",0
1354,1,180789,550,35.6275534,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(5 a^2 C+24 a b B+16 A b^2+12 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{32 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\left(a^3 (384 A+133 C)+472 a^2 b B+4 a b^2 (132 A+89 C)+128 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left(15 a^3 C+264 a^2 b B+4 a b^2 (108 A+71 C)+128 b^3 B\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{192 b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(-5 a^4 C+40 a^3 b B+120 a^2 b^2 (2 A+C)+160 a b^3 B+16 b^4 (4 A+3 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{64 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(5 a C+8 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{24 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{4 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
1355,1,211844,674,36.3720543,"\int \frac{(a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\sin (c+d x) \left(15 a^2 C+110 a b B+80 A b^2+64 b^2 C\right) \sqrt{a+b \sec (c+d x)}}{240 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{\sin (c+d x) \left(15 a^3 C+590 a^2 b B+4 a b^2 (260 A+193 C)+360 b^3 B\right) \sqrt{a+b \sec (c+d x)}}{960 b d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\sin (c+d x) \left(-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right) \sqrt{a+b \sec (c+d x)}}{1920 b^2 d \sqrt{\cos (c+d x)}}+\frac{\left(-15 a^4 C+1330 a^3 b B+4 a^2 b^2 (1180 A+809 C)+3560 a b^3 B+256 b^4 (5 A+4 C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{1920 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left(-45 a^4 C+150 a^3 b B+12 a^2 b^2 (220 A+141 C)+2840 a b^3 B+256 b^4 (5 A+4 C)\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{1920 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{\left(-3 a^5 C+10 a^4 b B-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B-80 a b^4 (4 A+3 C)-96 b^5 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{128 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a C+2 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{8 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Result too large to show","C",0
1356,1,492,380,19.041548,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(a \cos (c+d x)+b) \left(\frac{(7 a B-6 A b) \sin (2 (c+d x))}{35 a^2}+\frac{\sin (c+d x) \left(115 a^2 A+140 a^2 C-112 a b B+96 A b^2\right)}{210 a^3}+\frac{A \sin (3 (c+d x))}{14 a}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \left(-63 a^3 B+a^2 (44 A b+70 b C)-56 a b^2 B+48 A b^3\right) (a \cos (c+d x)+b)+i a \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(a^3 (25 A+63 B+35 C)-2 a^2 b (22 A-7 B+35 C)+4 a b^2 (14 B-3 A)-48 A b^3\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(63 a^3 B-2 a^2 b (22 A+35 C)+56 a b^2 B-48 A b^3\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{105 a^4 d \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 (6 A b-7 a B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 a^2 d}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^2 (5 A+7 C)-28 a b B+24 A b^2\right) \sqrt{a+b \sec (c+d x)}}{105 a^3 d}-\frac{2 \sqrt{\cos (c+d x)} \left(-63 a^3 B+a^2 (44 A b+70 b C)-56 a b^2 B+48 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(5 a^4 (5 A+7 C)-49 a^3 b B+2 a^2 b^2 (16 A+35 C)-56 a b^3 B+48 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{105 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 a d}",1,"((b + a*Cos[c + d*x])*(((115*a^2*A + 96*A*b^2 - 112*a*b*B + 140*a^2*C)*Sin[c + d*x])/(210*a^3) + ((-6*A*b + 7*a*B)*Sin[2*(c + d*x)])/(35*a^2) + (A*Sin[3*(c + d*x)])/(14*a)))/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(-48*A*b^3 + 63*a^3*B + 56*a*b^2*B - 2*a^2*b*(22*A + 35*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(-48*A*b^3 + 4*a*b^2*(-3*A + 14*B) - 2*a^2*b*(22*A - 7*B + 35*C) + a^3*(25*A + 63*B + 35*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^4*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","C",0
1357,1,379,291,17.0650907,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) (a \cos (c+d x)+b) (3 a A \cos (c+d x)+5 a B-4 A b)+\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) (a \cos (c+d x)+b)-i a \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(a^2 (9 A+5 (B+3 C))+2 a b (A-5 B)+8 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 (4 A b-5 a B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^2 d}+\frac{2 \sqrt{\cos (c+d x)} \left(3 a^2 (3 A+5 C)-10 a b B+8 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \left(-5 a^3 B+a^2 b (7 A+15 C)-10 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}",1,"(2*a*(b + a*Cos[c + d*x])*(-4*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x] + (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(I*(a + b)*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2))/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","C",0
1358,1,359,216,10.3232129,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{4 \cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(a A \sin (c+d x) (a \cos (c+d x)+b)-\frac{\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(i a \sec ^2\left(\frac{1}{2} (c+d x)\right) (a (A+3 (B+C))-2 A b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+(2 A b-3 a B) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) (3 a B-2 A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)} (A \cos (2 (c+d x))+A+2 B \cos (c+d x)+2 C)}","\frac{2 \left(a^2 (A+3 C)-3 a b B+2 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 (2 A b-3 a B) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a d}",1,"(4*Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(a*A*(b + a*Cos[c + d*x])*Sin[c + d*x] - ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(-2*A*b + 3*a*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(-2*A*b + a*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*A*b - 3*a*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2)))/(3*a^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sqrt[a + b*Sec[c + d*x]])","C",0
1359,1,36160,219,33.951021,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","-\frac{2 (A b-a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1360,1,52620,260,32.4690115,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{(2 A+C) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(2 b B-a C) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b d \sqrt{\cos (c+d x)}}-\frac{C \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
1361,1,98830,350,33.259221,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{\left(3 a^2 C-4 a b B+8 A b^2+4 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(4 b B-3 a C) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{(4 b B-3 a C) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(4 b B-a C) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{C \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
1362,1,25325,208,29.6210035,"\int \frac{\sqrt{\cos (c+d x)} \left(a A+(A b+a B) \sec (c+d x)+b B \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(a*A + (A*b + a*B)*Sec[c + d*x] + b*B*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{2 A \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1363,1,3870,461,24.4052294,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-\left(a^2 (A-5 C)\right)-5 a b B+6 A b^2\right) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(5 a^3 B-a^2 (9 A b-15 b C)-20 a b^2 B+24 A b^3\right) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)}-\frac{2 \left(-5 a^3 B+6 a^2 b (2 A+5 C)-40 a b^2 B+48 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(-3 a^4 (3 A+5 C)+25 a^3 b B-6 a^2 b^2 (4 A-5 C)-40 a b^3 B+48 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^4 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(-9*A*b + 5*a*B)*Sin[c + d*x])/(15*a^3) + (4*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (2*A*Sin[2*(c + d*x)])/(5*a^2)))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) - (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*((6*a*A*Sqrt[Cos[c + d*x]])/(5*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^2*Sqrt[Cos[c + d*x]])/(5*a*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (32*A*b^4*Sqrt[Cos[c + d*x]])/(5*a^3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (10*b*B*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*b^3*B*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^2*C*Sqrt[Cos[c + d*x]])/(a*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(5*a^2*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*a*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (4*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^4*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(5*a^4*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(((48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2))/2 - I*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - a*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] + (3*(48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(15*a^4*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B + 6*a^2*b^2*(4*A - 5*C) + 3*a^4*(3*A + 5*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-48*A*b^3 + 4*a*b^2*(9*A + 10*B) - 6*a^2*b*(2*A + 5*(B + C)) + a^3*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (48*A*b^4 + 25*a^3*b*B - 40*a*b^3*B - 6*a^2*b^2*(4*A - 5*C) - 3*a^4*(3*A + 5*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(5*a^4*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]])))","C",0
1364,1,3283,350,22.8463382,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-\left(a^2 (A-3 C)\right)-3 a b B+4 A b^2\right) \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2 (A+3 C)-6 a b B+8 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left(3 a^3 B-a^2 (5 A b-3 b C)-6 a b^2 B+8 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*A*Sin[c + d*x])/(3*a^2) - (4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) + (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*((-10*A*b*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^3*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*B*Sqrt[Cos[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^2*B*Sqrt[Cos[c + d*x]])/(a*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*b*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) - (2*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*a*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]))*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*((2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*(I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^(3/2)) - (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*(I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(((8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2))/2 + I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - a*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] + (3*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 + ((I/2)*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - ((I/2)*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) - ((a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(3*a^3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(I*(a + b)*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(8*A*b^2 - 6*a*b*(A + B) + a^2*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (8*A*b^3 + 3*a^3*B - 6*a*b^2*B + a^2*(-5*A*b + 3*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(a^3*(a^2 - b^2)*Sqrt[b + a*Cos[c + d*x]])))","C",0
1365,1,517,249,17.239656,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{4 \sqrt{\cos (c+d x)} (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)+A b^2 \sin (c+d x)\right)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}-\frac{4 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \left(a^2 (C-A)-a b B+2 A b^2\right) (a \cos (c+d x)+b)-i (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(a^2 (A-C)+a b B-2 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i a (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a (A+B-C)-2 A b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(-\left(a^2 (A-C)\right)-a b B+2 A b^2\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 (2 A b-a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"(4*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(a*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(3/2)) - (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-2*A*b^2 + a*b*B + a^2*(A - C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-2*A*b + a*(A + B - C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*A*b^2 - a*b*B + a^2*(-A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^2*(a^2 - b^2)*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2))","C",0
1366,1,63246,311,34.8423455,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a b d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 A \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{a d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1367,1,111509,393,34.5044306,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}+\frac{\sin (c+d x) \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\sqrt{\cos (c+d x)} \left(3 a^2 C-2 a b B+2 A b^2-b^2 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 b B-3 a C) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1368,1,4917,663,27.144701,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)^2}-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right) \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left(a^2-b^2\right)^2}+\frac{2 \left(5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left(3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(-14*A*b + 5*a*B)*Sin[c + d*x])/(15*a^4) - (4*(A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x] + a^2*b^3*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (4*(-15*a^2*A*b^4*Sin[c + d*x] + 11*A*b^6*Sin[c + d*x] + 12*a^3*b^3*B*Sin[c + d*x] - 8*a*b^5*B*Sin[c + d*x] - 9*a^4*b^2*C*Sin[c + d*x] + 5*a^2*b^4*C*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + (2*A*Sin[2*(c + d*x)])/(5*a^3)))/(d*Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*((6*a^2*A*Sqrt[Cos[c + d*x]])/(5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (22*A*b^2*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (424*A*b^4*Sqrt[Cos[c + d*x]])/(15*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (256*A*b^6*Sqrt[Cos[c + d*x]])/(15*a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a*b*B*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (56*b^3*B*Sqrt[Cos[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (32*b^5*B*Sqrt[Cos[c + d*x]])/(3*a^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (10*b^2*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*b^4*C*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*a*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (88*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (64*A*b^5*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*a^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (14*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*b^4*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*a*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*b^3*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + 2*a^3*b^2*(36*A + 40*B - 15*C) + 4*a^2*b^3*(-29*A + 15*B + 10*C) - a^4*b*(17*A + 45*(B + C)) + a^5*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^5*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + 2*a^3*b^2*(36*A + 40*B - 15*C) + 4*a^2*b^3*(-29*A + 15*B + 10*C) - a^4*b*(17*A + 45*(B + C)) + a^5*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + 2*a^3*b^2*(36*A + 40*B - 15*C) + 4*a^2*b^3*(-29*A + 15*B + 10*C) - a^4*b*(17*A + 45*(B + C)) + a^5*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(5*a^5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + 2*a^3*b^2*(36*A + 40*B - 15*C) + 4*a^2*b^3*(-29*A + 15*B + 10*C) - a^4*b*(17*A + 45*(B + C)) + a^5*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + 2*a^3*b^2*(36*A + 40*B - 15*C) + 4*a^2*b^3*(-29*A + 15*B + 10*C) - a^4*b*(17*A + 45*(B + C)) + a^5*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + 2*a^3*b^2*(36*A + 40*B - 15*C) + 4*a^2*b^3*(-29*A + 15*B + 10*C) - a^4*b*(17*A + 45*(B + C)) + a^5*(9*A + 5*(B + 3*C)))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(15*a^5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + 2*a^3*b^2*(36*A + 40*B - 15*C) + 4*a^2*b^3*(-29*A + 15*B + 10*C) - a^4*b*(17*A + 45*(B + C)) + a^5*(9*A + 5*(B + 3*C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B + 5*a^4*b^2*(11*A - 15*C) + 3*a^6*(3*A + 5*C) + 4*a^2*b^4*(-53*A + 10*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(5*a^5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]])))","C",0
1369,1,4327,521,25.5031223,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^4 (A-5 C)+8 a^3 b B-a^2 b^2 (13 A-C)-4 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(4 a^4 C-7 a^3 b B+10 a^2 A b^2+3 a b^3 B-6 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-\left(a^4 (A+3 C)\right)+9 a^3 b B-2 a^2 b^2 (8 A-C)-8 a b^3 B+16 A b^4\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(-3 a^5 B+a^4 (8 A b-6 b C)+15 a^3 b^2 B-2 a^2 b^3 (14 A-C)-8 a b^4 B+16 A b^5\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*A*Sin[c + d*x])/(3*a^3) + (4*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(-12*a^2*A*b^3*Sin[c + d*x] + 8*A*b^5*Sin[c + d*x] + 9*a^3*b^2*B*Sin[c + d*x] - 5*a*b^4*B*Sin[c + d*x] - 6*a^4*b*C*Sin[c + d*x] + 2*a^2*b^3*C*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*((-16*a*A*b*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (56*A*b^3*Sqrt[Cos[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (32*A*b^5*Sqrt[Cos[c + d*x]])/(3*a^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*B*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (10*b^2*B*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*b^4*B*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a*b*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^3*C*Sqrt[Cos[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (14*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*A*b^4*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*a*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*b^3*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B + 2*a^2*b^3*(-14*A + C) + a^4*(8*A*b - 6*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a^4*(a^2 - b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B + 2*a^2*b^3*(-14*A + C) + a^4*(8*A*b - 6*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B + 2*a^2*b^3*(-14*A + C) + a^4*(8*A*b - 6*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(((16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B + 2*a^2*b^3*(-14*A + C) + a^4*(8*A*b - 6*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2))/2 - I*(a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] - a*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B + 2*a^2*b^3*(-14*A + C) + a^4*(8*A*b - 6*b*C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] + (3*(16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B + 2*a^2*b^3*(-14*A + C) + a^4*(8*A*b - 6*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(3*a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B + 2*a^2*b^3*(14*A - C) + a^4*(-8*A*b + 6*b*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 4*a*b^3*(3*A + 2*B) + 2*a^2*b^2*(8*A - 3*B - C) + 3*a^3*b*(-3*A - 3*B + C) + a^4*(A + 3*(B + C)))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (16*A*b^5 - 3*a^5*B + 15*a^3*b^2*B - 8*a*b^4*B + 2*a^2*b^3*(-14*A + C) + a^4*(8*A*b - 6*b*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]])))","C",0
1370,1,3834,401,24.522785,"\int \frac{\sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^3 B-a^2 b (9 A+C)-2 a b^2 B+8 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sin (c+d x) \left(-2 a^4 C+5 a^3 b B-2 a^2 b^2 (4 A+C)-a b^3 B+4 A b^4\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\cos (c+d x)} \left(3 a^4 (A-C)+6 a^3 b B-a^2 b^2 (15 A+C)-2 a b^3 B+8 A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-4*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x] + a^2*b*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(9*a^2*A*b^2*Sin[c + d*x] - 5*A*b^4*Sin[c + d*x] - 6*a^3*b*B*Sin[c + d*x] + 2*a*b^3*B*Sin[c + d*x] + 3*a^4*C*Sin[c + d*x] + a^2*b^2*C*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) - (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*((2*a^2*A*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (10*A*b^2*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*A*b^4*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (4*a*b*B*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*b^3*B*Sqrt[Cos[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a^2*C*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*b^2*C*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (4*a*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (4*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a*b*C*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a*(a^3 - a*b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)*((-2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*(b + a*Cos[c + d*x])^(3/2)) + (2*Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a*(a^3 - a*b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(3*a*(a^3 - a*b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 2*a*b^2*(3*A + B) + 3*a^3*(A + B - C) - a^2*b*(9*A - 3*B + C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B + 3*a^4*(A - C) - a^2*b^2*(15*A + C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(a*(a^3 - a*b^2)^2*Sqrt[b + a*Cos[c + d*x]])))","C",0
1371,1,673,378,19.001124,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{(a \cos (c+d x)+b)^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(\frac{4 \left(a^2 C \sin (c+d x)-a b B \sin (c+d x)+A b^2 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{4 \left(3 a^3 B \sin (c+d x)-6 a^2 A b \sin (c+d x)-4 a^2 b C \sin (c+d x)+a b^2 B \sin (c+d x)+2 A b^3 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac{4 \cos ^{\frac{3}{2}}(c+d x) \sqrt{\sec (c+d x)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b)^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \left(-i a (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(a^2 (3 A-3 B+C)+a b (3 A-B+3 C)-2 A b^2\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} \left(3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right) (a \cos (c+d x)+b)-i (a+b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 d \left(a^3-a b^2\right)^2 (a+b \sec (c+d x))^{5/2} (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(-\left(a^2 (3 A+C)\right)+a b B+2 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(3 a^3 B-2 a^2 b (3 A+2 C)+a b^2 B+2 A b^3\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \sin (c+d x) \left(a^4 C+2 a^3 b B-5 a^2 b^2 (A+C)+2 a b^3 B+A b^4\right)}{3 a b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((4*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x] + a^2*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (4*(-6*a^2*A*b*Sin[c + d*x] + 2*A*b^3*Sin[c + d*x] + 3*a^3*B*Sin[c + d*x] + a*b^2*B*Sin[c + d*x] - 4*a^2*b*C*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Sqrt[Cos[c + d*x]]*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2)) + (4*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sqrt[Sec[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)*((-I)*(a + b)*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(-2*A*b^2 + a^2*(3*A - 3*B + C) + a*b*(3*A - B + 3*C))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*(a^3 - a*b^2)^2*d*(A + 2*C + 2*B*Cos[c + d*x] + A*Cos[2*c + 2*d*x])*(a + b*Sec[c + d*x])^(5/2))","C",0
1372,1,119861,447,36.6850172,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \left(A b^2-a (b B-a C)\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left(-3 a^4 C+a^2 b^2 (3 A+7 C)-4 a b^3 B+A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \sqrt{\cos (c+d x)} \left(-3 a^4 C+a^2 b^2 (3 A+7 C)-4 a b^3 B+A b^4\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 a b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 C \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
1373,1,215866,563,37.5434335,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","-\frac{2 \sin (c+d x) \left(A b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac{\left(5 a^2 C-2 a b B+2 A b^2-3 b^2 C\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \sin (c+d x) \left(-5 a^4 C+2 a^3 b B+a^2 b^2 (A+9 C)-6 a b^3 B+3 A b^4\right)}{3 b^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}-\frac{\sin (c+d x) \left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\sqrt{\cos (c+d x)} \left(-15 a^4 C+6 a^3 b B+26 a^2 b^2 C-14 a b^3 B+8 A b^4-3 b^4 C\right) \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 b B-5 a C) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0