1,1,95,0,0.0801958,"\int \sec ^2(c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(10*A + 7*C)*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b^2*d)","A",4,4,33,0.1212,1,"{16, 4046, 3772, 2643}"
2,1,92,0,0.0801193,"\int \sec (c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(7*A + 4*C)*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*b*d)","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
3,1,88,0,0.0641493,"\int \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\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}}","\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*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d)","A",3,3,25,0.1200,1,"{4046, 3772, 2643}"
4,1,89,0,0.0874223,"\int \cos (c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\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}}","\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*b^2*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
5,1,93,0,0.1044833,"\int \cos ^2(c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\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}}","\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*b*(2*A + 5*C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^2*Tan[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3))","A",4,4,33,0.1212,1,"{16, 4045, 3772, 2643}"
6,1,95,0,0.0857016,"\int \sec ^2(c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(3*(13*A + 10*C)*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(91*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(10/3)*Tan[c + d*x])/(13*b^2*d)","A",4,4,33,0.1212,1,"{16, 4046, 3772, 2643}"
7,1,92,0,0.0804184,"\int \sec (c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(10*A + 7*C)*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b*d)","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
8,1,90,0,0.0838983,"\int (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\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}","\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*b*(7*A + 4*C)*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d)","A",3,3,25,0.1200,1,"{4046, 3772, 2643}"
9,1,91,0,0.0944594,"\int \cos (c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\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}}","\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^2*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d)","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
10,1,91,0,0.1151959,"\int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\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}}","\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*b^3*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) + (3*b^2*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))","A",4,4,33,0.1212,1,"{16, 4046, 3772, 2643}"
11,1,95,0,0.0868797,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","\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}","\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*(8*A + 5*C)*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(16*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b^2*d)","A",4,4,33,0.1212,1,"{16, 4046, 3772, 2643}"
12,1,92,0,0.0835011,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","\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)}}","\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*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b*d)","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
13,1,90,0,0.0739019,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(1/3),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}}","\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*b*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(2*d*(b*Sec[c + d*x])^(1/3))","A",3,3,25,0.1200,1,"{4046, 3772, 2643}"
14,1,88,0,0.0968436,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),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)}}","\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*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))","A",4,4,31,0.1290,1,"{16, 4045, 3772, 2643}"
15,1,93,0,0.1093106,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),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}}","\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*b*(4*A + 7*C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^2*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3))","A",4,4,33,0.1212,1,"{16, 4045, 3772, 2643}"
16,1,95,0,0.079406,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\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)}}","\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*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d)","A",4,4,33,0.1212,1,"{16, 4046, 3772, 2643}"
17,1,92,0,0.0750487,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),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}}","\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*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(2*b*d*(b*Sec[c + d*x])^(1/3))","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
18,1,90,0,0.0707745,"\int \frac{A+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(4/3),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)}}","\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*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))","A",3,3,25,0.1200,1,"{4045, 3772, 2643}"
19,1,90,0,0.0901464,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),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}}","\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*(4*A + 7*C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3))","A",4,4,31,0.1290,1,"{16, 4045, 3772, 2643}"
20,1,93,0,0.1091261,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","\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}}","\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*b*(7*A + 10*C)*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(70*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^2*Tan[c + d*x])/(10*d*(b*Sec[c + d*x])^(10/3))","A",4,4,33,0.1212,1,"{16, 4045, 3772, 2643}"
21,1,146,0,0.1163826,"\int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\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)}","\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*b*C*Sec[c + d*x]^(2 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) + (3*b*(C*(4 + 3*m) + A*(7 + 3*m))*Hypergeometric2F1[1/2, (-1 - 3*m)/6, (5 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
22,1,146,0,0.1178091,"\int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2),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)}-\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 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*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 + 3*m)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Hypergeometric2F1[1/2, (1 - 3*m)/6, (7 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 - 3*m)*(5 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
23,1,144,0,0.1170125,"\int \sec ^m(c+d x) \sqrt[3]{b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),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)}-\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 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*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)) - (3*(C + 3*C*m + A*(4 + 3*m))*Hypergeometric2F1[1/2, (2 - 3*m)/6, (8 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(2 - 3*m)*(4 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
24,1,147,0,0.1198444,"\int \frac{\sec ^m(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt[3]{b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(1/3),x]","\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)}}","\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*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)) + (3*(C*(1 - 3*m) - A*(2 + 3*m))*Hypergeometric2F1[1/2, (4 - 3*m)/6, (10 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(4 - 3*m)*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
25,1,145,0,0.1267566,"\int \frac{\sec ^m(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{2/3}} \, dx","Int[(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(2/3),x]","\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}}","\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*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)) - (3*(A - C*(2 - 3*m) + 3*A*m)*Hypergeometric2F1[1/2, (5 - 3*m)/6, (11 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 - 3*m)*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
26,1,148,0,0.1344806,"\int \frac{\sec ^m(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(Sec[c + d*x]^m*(A + C*Sec[c + d*x]^2))/(b*Sec[c + d*x])^(4/3),x]","-\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)}}","-\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*C*Sec[c + d*x]^m*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3)) - (3*(A + C*(4 - 3*m) - 3*A*m)*Hypergeometric2F1[1/2, (7 - 3*m)/6, (13 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(1 - 3*m)*(7 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
27,1,145,0,0.1105745,"\int \sec ^m(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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)}}","\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,"(C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) - ((C*(m + n) + A*(1 + m + n))*Hypergeometric2F1[1/2, (1 - m - n)/2, (3 - m - n)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m - n)*(1 + m + n)*Sqrt[Sin[c + d*x]^2])","A",4,4,31,0.1290,1,"{20, 4046, 3772, 2643}"
28,1,120,0,0.1116134,"\int \sec ^2(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"((C*(2 + n) + A*(3 + n))*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*(3 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(2 + n)*Tan[c + d*x])/(b^2*d*(3 + n))","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
29,1,109,0,0.1020331,"\int \sec (c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"((C*(1 + n) + A*(2 + n))*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(2 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(1 + n)*Tan[c + d*x])/(b*d*(2 + n))","A",4,4,29,0.1379,1,"{16, 4046, 3772, 2643}"
30,1,113,0,0.0831302,"\int (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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)}}","\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,"-((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*(1 + n)*Sqrt[Sin[c + d*x]^2])) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n))","A",3,3,23,0.1304,1,"{4046, 3772, 2643}"
31,1,117,0,0.1183403,"\int \cos (c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(b^2*(C*(1 - n) - A*n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*n*Sqrt[Sin[c + d*x]^2]) + (b*C*(b*Sec[c + d*x])^(-1 + n)*Tan[c + d*x])/(d*n)","A",4,4,29,0.1379,1,"{16, 4046, 3772, 2643}"
32,1,132,0,0.145624,"\int \cos ^2(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\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)}","-\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,"-((b^3*(A*(1 - n) + C*(2 - n))*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(1 - n)*(3 - n)*Sqrt[Sin[c + d*x]^2])) - (b^2*C*(b*Sec[c + d*x])^(-2 + n)*Tan[c + d*x])/(d*(1 - n))","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
33,1,132,0,0.1434754,"\int \cos ^3(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\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)}","-\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^4*(A*(2 - n) + C*(3 - n))*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d*(2 - n)*(4 - n)*Sqrt[Sin[c + d*x]^2])) - (b^3*C*(b*Sec[c + d*x])^(-3 + n)*Tan[c + d*x])/(d*(2 - n))","A",4,4,31,0.1290,1,"{16, 4046, 3772, 2643}"
34,1,142,0,0.1237137,"\int \sec ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(2*C*Sec[c + d*x]^(7/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) + (2*(C*(5 + 2*n) + A*(7 + 2*n))*Hypergeometric2F1[1/2, (-3 - 2*n)/4, (1 - 2*n)/4, Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*(7 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
35,1,142,0,0.1269352,"\int \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(2*C*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) + (2*(C*(3 + 2*n) + A*(5 + 2*n))*Hypergeometric2F1[1/2, (-1 - 2*n)/4, (3 - 2*n)/4, Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
36,1,140,0,0.119767,"\int \sqrt{\sec (c+d x)} (b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),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)}-\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 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,"(2*C*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Hypergeometric2F1[1/2, (1 - 2*n)/4, (5 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(3 + 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
37,1,141,0,0.1138772,"\int \frac{(b \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\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)}","\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,"(2*C*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)) - (2*(A - C*(1 - 2*n) + 2*A*n)*Hypergeometric2F1[1/2, (3 - 2*n)/4, (7 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(1 + 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
38,1,140,0,0.125937,"\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","Int[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","-\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)}}","-\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,"(-2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hypergeometric2F1[1/2, (5 - 2*n)/4, (9 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
39,1,142,0,0.128632,"\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","Int[((b*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","-\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)}","-\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,"(-2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)) - (2*(A*(3 - 2*n) + C*(5 - 2*n))*Hypergeometric2F1[1/2, (7 - 2*n)/4, (11 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(7 - 2*n)*Sec[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2])","A",4,4,33,0.1212,1,"{20, 4046, 3772, 2643}"
40,1,167,0,0.1252124,"\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","Int[Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),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 \, _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)}}","\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,"(B*Hypergeometric2F1[1/2, (-m - n)/2, (2 - m - n)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(m + n)*Sqrt[Sin[c + d*x]^2]) + (C*Hypergeometric2F1[1/2, (-1 - m - n)/2, (1 - m - n)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)*Sqrt[Sin[c + d*x]^2])","A",7,5,38,0.1316,1,"{20, 4047, 3772, 2643, 12}"
41,1,154,0,0.1592256,"\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","Int[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 (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}","\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*(11*A + 8*C)*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(55*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-4/3, 1/2, -1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(8/3)*Sin[c + d*x])/(8*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(8/3)*Tan[c + d*x])/(11*b^2*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
42,1,151,0,0.1602071,"\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","Int[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 (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}","\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*(8*A + 5*C)*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(16*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b*d)","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
43,1,146,0,0.1396431,"\int (b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(-3*b*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d)","A",6,4,33,0.1212,1,"{4047, 3772, 2643, 4046}"
44,1,148,0,0.1634012,"\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","Int[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 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)}}","-\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*b^2*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*b*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*C*Tan[c + d*x])/(2*d*(b*Sec[c + d*x])^(1/3))","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
45,1,150,0,0.1952907,"\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","Int[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 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}}","\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*b^2*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*b*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^2*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4045}"
46,1,154,0,0.1920317,"\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","Int[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^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 A b^3 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\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}}","-\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 A b^3 \tan (c+d x)}{7 d (b \sec (c+d x))^{7/3}}-\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^3*B*Hypergeometric2F1[1/2, 7/6, 13/6, Cos[c + d*x]^2]*Sin[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3)*Sqrt[Sin[c + d*x]^2]) - (3*b^2*(4*A + 7*C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(28*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^3*Tan[c + d*x])/(7*d*(b*Sec[c + d*x])^(7/3))","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4045}"
47,1,154,0,0.1533465,"\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","Int[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 (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}","\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*(13*A + 10*C)*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(91*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-5/3, 1/2, -2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(10/3)*Sin[c + d*x])/(10*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(10/3)*Tan[c + d*x])/(13*b^2*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
48,1,151,0,0.1540351,"\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","Int[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{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}","\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*(10*A + 7*C)*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b*d)","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
49,1,146,0,0.1373471,"\int (b \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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*b*(7*A + 4*C)*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d)","A",6,4,33,0.1212,1,"{4047, 3772, 2643, 4046}"
50,1,146,0,0.1588267,"\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","Int[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^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}","-\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^2*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*Sqrt[Sin[c + d*x]^2]) + (3*b*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d)","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
51,1,150,0,0.1858511,"\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","Int[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 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}}","-\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*b^3*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) - (3*b^2*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*b^2*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
52,1,154,0,0.1912972,"\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","Int[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^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 A b^3 \tan (c+d x)}{5 d (b \sec (c+d x))^{5/3}}-\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}}","-\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 A b^3 \tan (c+d x)}{5 d (b \sec (c+d x))^{5/3}}-\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^3*B*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) - (3*b^2*(2*A + 5*C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(10*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*b^3*Tan[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3))","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4045}"
53,1,154,0,0.1518077,"\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","Int[(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 (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}","\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*(7*A + 4*C)*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*b^2*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
54,1,147,0,0.1463481,"\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","Int[(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 (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}","-\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*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*b*d)","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
55,1,142,0,0.1351532,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{2/3}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(2/3),x]","-\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}}","-\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*(A - 2*C)*Hypergeometric2F1[1/2, 5/6, 11/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*d*(b*Sec[c + d*x])^(5/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(2*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(d*(b*Sec[c + d*x])^(2/3))","A",6,4,33,0.1212,1,"{4047, 3772, 2643, 4046}"
56,1,147,0,0.1401604,"\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","Int[(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 (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}","-\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*(4*A + C)*Hypergeometric2F1[1/3, 1/2, 4/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(b*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*b*d)","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
57,1,154,0,0.1475161,"\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","Int[(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 (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}","\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*(7*A + 4*C)*Hypergeometric2F1[-1/6, 1/2, 5/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(7*b*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(4*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*b^2*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
58,1,154,0,0.1573987,"\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","Int[(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{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}","\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,"(3*(10*A + 7*C)*Hypergeometric2F1[-2/3, 1/2, 1/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(4/3)*Sin[c + d*x])/(40*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-7/6, 1/2, -1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(7/3)*Sin[c + d*x])/(7*b^3*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(7/3)*Tan[c + d*x])/(10*b^3*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
59,1,154,0,0.1506228,"\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","Int[(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{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}","-\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,"(-3*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
60,1,149,0,0.1440284,"\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","Int[(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 (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)}}","-\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*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(2*b*d*(b*Sec[c + d*x])^(1/3))","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
61,1,146,0,0.1381169,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(b \sec (c+d x))^{4/3}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(b*Sec[c + d*x])^(4/3),x]","-\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}}","-\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,"(-3*B*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*(A + 4*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(4*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*A*Tan[c + d*x])/(4*d*(b*Sec[c + d*x])^(4/3))","A",6,4,33,0.1212,1,"{4047, 3772, 2643, 4045}"
62,1,149,0,0.1411359,"\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","Int[(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 (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)}}","-\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*(2*A - C)*Hypergeometric2F1[1/2, 2/3, 5/3, Cos[c + d*x]^2]*Sin[c + d*x])/(8*d*(b*Sec[c + d*x])^(4/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*C*Tan[c + d*x])/(2*b*d*(b*Sec[c + d*x])^(1/3))","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
63,1,154,0,0.1475557,"\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","Int[(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{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}","-\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,"(-3*(5*A + 2*C)*Hypergeometric2F1[1/6, 1/2, 7/6, Cos[c + d*x]^2]*Sin[c + d*x])/(5*b*d*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(2*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*b^2*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
64,1,154,0,0.1507223,"\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","Int[(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{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}","\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,"(3*(8*A + 5*C)*Hypergeometric2F1[-1/3, 1/2, 2/3, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(16*b^2*d*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[-5/6, 1/2, 1/6, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(5/3)*Sin[c + d*x])/(5*b^3*d*Sqrt[Sin[c + d*x]^2]) + (3*C*(b*Sec[c + d*x])^(5/3)*Tan[c + d*x])/(8*b^3*d)","A",7,5,41,0.1220,1,"{16, 4047, 3772, 2643, 4046}"
65,1,230,0,0.1883537,"\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","Int[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 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)}","\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*b*C*Sec[c + d*x]^(2 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(7 + 3*m)) + (3*b*(C*(4 + 3*m) + A*(7 + 3*m))*Hypergeometric2F1[1/2, (-1 - 3*m)/6, (5 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*(7 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*b*B*Hypergeometric2F1[1/2, (-4 - 3*m)/6, (2 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
66,1,227,0,0.1905688,"\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","Int[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 (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)}","-\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*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(5 + 3*m)) - (3*(C*(2 + 3*m) + A*(5 + 3*m))*Hypergeometric2F1[1/2, (1 - 3*m)/6, (7 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(1 - 3*m)*(5 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (-2 - 3*m)/6, (4 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*Sin[c + d*x])/(d*(2 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
67,1,225,0,0.188582,"\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","Int[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 (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)}","-\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*C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(4 + 3*m)) - (3*(C + 3*C*m + A*(4 + 3*m))*Hypergeometric2F1[1/2, (2 - 3*m)/6, (8 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(2 - 3*m)*(4 + 3*m)*Sqrt[Sin[c + d*x]^2]) + (3*B*Hypergeometric2F1[1/2, (-1 - 3*m)/6, (5 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sin[c + d*x])/(d*(1 + 3*m)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
68,1,228,0,0.1908186,"\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","Int[(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 (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)}}","\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*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)) + (3*(C*(1 - 3*m) - A*(2 + 3*m))*Hypergeometric2F1[1/2, (4 - 3*m)/6, (10 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(4 - 3*m)*(2 + 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (1 - 3*m)/6, (7 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
69,1,226,0,0.1916458,"\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","Int[(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 (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}}","-\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*C*Sec[c + d*x]^(1 + m)*Sin[c + d*x])/(d*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)) - (3*(A - C*(2 - 3*m) + 3*A*m)*Hypergeometric2F1[1/2, (5 - 3*m)/6, (11 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(d*(5 - 3*m)*(1 + 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (2 - 3*m)/6, (8 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^m*Sin[c + d*x])/(d*(2 - 3*m)*(b*Sec[c + d*x])^(2/3)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
70,1,234,0,0.2047981,"\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","Int[(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]","-\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)}}","-\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,"(-3*C*Sec[c + d*x]^m*Sin[c + d*x])/(b*d*(1 - 3*m)*(b*Sec[c + d*x])^(1/3)) - (3*(A + C*(4 - 3*m) - 3*A*m)*Hypergeometric2F1[1/2, (7 - 3*m)/6, (13 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-2 + m)*Sin[c + d*x])/(b*d*(1 - 3*m)*(7 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2]) - (3*B*Hypergeometric2F1[1/2, (4 - 3*m)/6, (10 - 3*m)/6, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*Sin[c + d*x])/(b*d*(4 - 3*m)*(b*Sec[c + d*x])^(1/3)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
71,1,226,0,0.1807821,"\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","Int[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{(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)}","-\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,"(C*Sec[c + d*x]^(1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) - ((C*(m + n) + A*(1 + m + n))*Hypergeometric2F1[1/2, (1 - m - n)/2, (3 - m - n)/2, Cos[c + d*x]^2]*Sec[c + d*x]^(-1 + m)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - m - n)*(1 + m + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (-m - n)/2, (2 - m - n)/2, Cos[c + d*x]^2]*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(m + n)*Sqrt[Sin[c + d*x]^2])","A",7,5,39,0.1282,1,"{20, 4047, 3772, 2643, 4046}"
72,1,189,0,0.1951782,"\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","Int[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{(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)}","\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,"((C*(2 + n) + A*(3 + n))*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*(3 + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (-2 - n)/2, -n/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(2 + n)*Sin[c + d*x])/(b^2*d*(2 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(2 + n)*Tan[c + d*x])/(b^2*d*(3 + n))","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
73,1,182,0,0.185408,"\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","Int[Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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)}","\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,"((C*(1 + n) + A*(2 + n))*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(2 + n)*Sqrt[Sin[c + d*x]^2]) + (B*Hypergeometric2F1[1/2, (-1 - n)/2, (1 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(1 + n)*Sin[c + d*x])/(b*d*(1 + n)*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^(1 + n)*Tan[c + d*x])/(b*d*(2 + n))","A",7,5,37,0.1351,1,"{16, 4047, 3772, 2643, 4046}"
74,1,175,0,0.1427664,"\int (b \sec (c+d x))^n \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\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)}","-\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,"-((b*(A + A*n + C*n)*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*(1 + n)*Sqrt[Sin[c + d*x]^2])) + (B*Hypergeometric2F1[1/2, -n/2, (2 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*Sqrt[Sin[c + d*x]^2]) + (C*(b*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + n))","A",6,4,31,0.1290,1,"{4047, 3772, 2643, 4046}"
75,1,191,0,0.1935489,"\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","Int[Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(b^2*(C*(1 - n) - A*n)*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*n*Sqrt[Sin[c + d*x]^2]) - (b*B*Hypergeometric2F1[1/2, (1 - n)/2, (3 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-1 + n)*Sin[c + d*x])/(d*(1 - n)*Sqrt[Sin[c + d*x]^2]) + (b*C*(b*Sec[c + d*x])^(-1 + n)*Tan[c + d*x])/(d*n)","A",7,5,37,0.1351,1,"{16, 4047, 3772, 2643, 4046}"
76,1,208,0,0.2168607,"\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","Int[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{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)}","-\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,"-((b^3*(A*(1 - n) + C*(2 - n))*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(1 - n)*(3 - n)*Sqrt[Sin[c + d*x]^2])) - (b^2*B*Hypergeometric2F1[1/2, (2 - n)/2, (4 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-2 + n)*Sin[c + d*x])/(d*(2 - n)*Sqrt[Sin[c + d*x]^2]) - (b^2*C*(b*Sec[c + d*x])^(-2 + n)*Tan[c + d*x])/(d*(1 - n))","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
77,1,208,0,0.2104111,"\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","Int[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^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)}","-\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^4*(A*(2 - n) + C*(3 - n))*Hypergeometric2F1[1/2, (4 - n)/2, (6 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-4 + n)*Sin[c + d*x])/(d*(2 - n)*(4 - n)*Sqrt[Sin[c + d*x]^2])) - (b^3*B*Hypergeometric2F1[1/2, (3 - n)/2, (5 - n)/2, Cos[c + d*x]^2]*(b*Sec[c + d*x])^(-3 + n)*Sin[c + d*x])/(d*(3 - n)*Sqrt[Sin[c + d*x]^2]) - (b^3*C*(b*Sec[c + d*x])^(-3 + n)*Tan[c + d*x])/(d*(2 - n))","A",7,5,39,0.1282,1,"{16, 4047, 3772, 2643, 4046}"
78,1,223,0,0.1959543,"\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","Int[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]","\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)}","\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,"(2*C*Sec[c + d*x]^(7/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(7 + 2*n)) + (2*(C*(5 + 2*n) + A*(7 + 2*n))*Hypergeometric2F1[1/2, (-3 - 2*n)/4, (1 - 2*n)/4, Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*(7 + 2*n)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (-5 - 2*n)/4, (-1 - 2*n)/4, Cos[c + d*x]^2]*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
79,1,223,0,0.1878709,"\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","Int[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{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)}","\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,"(2*C*Sec[c + d*x]^(5/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 + 2*n)) + (2*(C*(3 + 2*n) + A*(5 + 2*n))*Hypergeometric2F1[1/2, (-1 - 2*n)/4, (3 - 2*n)/4, Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*(5 + 2*n)*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (-3 - 2*n)/4, (1 - 2*n)/4, Cos[c + d*x]^2]*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
80,1,221,0,0.1789897,"\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","Int[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{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)}","-\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,"(2*C*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 + 2*n)) - (2*(C + 2*C*n + A*(3 + 2*n))*Hypergeometric2F1[1/2, (1 - 2*n)/4, (5 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(3 + 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2]) + (2*B*Hypergeometric2F1[1/2, (-1 - 2*n)/4, (3 - 2*n)/4, Cos[c + d*x]^2]*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
81,1,222,0,0.1806549,"\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","Int[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","-\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)}","-\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,"(2*C*Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + 2*n)) - (2*(A - C*(1 - 2*n) + 2*A*n)*Hypergeometric2F1[1/2, (3 - 2*n)/4, (7 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(1 + 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (1 - 2*n)/4, (5 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
82,1,221,0,0.1991442,"\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","Int[((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{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)}}","-\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,"(-2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*Sqrt[Sec[c + d*x]]) - (2*(A + C*(3 - 2*n) - 2*A*n)*Hypergeometric2F1[1/2, (5 - 2*n)/4, (9 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 - 2*n)*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (3 - 2*n)/4, (7 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
83,1,223,0,0.2010438,"\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","Int[((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]","-\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)}","-\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,"(-2*C*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*Sec[c + d*x]^(3/2)) - (2*(A*(3 - 2*n) + C*(5 - 2*n))*Hypergeometric2F1[1/2, (7 - 2*n)/4, (11 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(3 - 2*n)*(7 - 2*n)*Sec[c + d*x]^(7/2)*Sqrt[Sin[c + d*x]^2]) - (2*B*Hypergeometric2F1[1/2, (5 - 2*n)/4, (9 - 2*n)/4, Cos[c + d*x]^2]*(b*Sec[c + d*x])^n*Sin[c + d*x])/(d*(5 - 2*n)*Sec[c + d*x]^(5/2)*Sqrt[Sin[c + d*x]^2])","A",7,5,41,0.1220,1,"{20, 4047, 3772, 2643, 4046}"
84,1,140,0,0.1854961,"\int \sec ^3(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(5*A + 4*C)*Tan[c + d*x])/(5*d) + (a*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(5*A + 4*C)*Tan[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4077, 4047, 3767, 4046, 3768, 3770}"
85,1,117,0,0.1700437,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (a*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,31,0.2258,1,"{4077, 4047, 3768, 3770, 4046, 3767, 8}"
86,1,86,0,0.1076089,"\int \sec (c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,29,0.2069,1,"{4077, 4047, 3767, 8, 4046, 3770}"
87,1,58,0,0.0543391,"\int (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(2*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",5,4,23,0.1739,1,"{4049, 3770, 3767, 8}"
88,1,42,0,0.1005618,"\int \cos (c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*Sin[c + d*x])/d + (a*C*Tan[c + d*x])/d","A",5,5,29,0.1724,1,"{4077, 4047, 8, 4045, 3770}"
89,1,58,0,0.1267989,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(A + 2*C)*x)/2 + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",5,5,31,0.1613,1,"{4075, 4047, 8, 4045, 3770}"
90,1,77,0,0.1450365,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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)","\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*(A + 2*C)*x)/2 + (a*(2*A + 3*C)*Sin[c + d*x])/(3*d) + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{4075, 4047, 2637, 4045, 8}"
91,1,95,0,0.1773009,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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)","\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*(3*A + 4*C)*x)/8 + (a*(A + C)*Sin[c + d*x])/d + (a*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*A*Sin[c + d*x]^3)/(3*d)","A",7,6,31,0.1935,1,"{4075, 4047, 2635, 8, 4044, 3013}"
92,1,131,0,0.1866071,"\int \cos ^5(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","-\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)","-\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*(3*A + 4*C)*x)/8 + (a*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (a*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*C)*Sin[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4075, 4047, 2633, 4045, 2635, 8}"
93,1,172,0,0.3853513,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a^2*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (a^2*(4*A + 3*C)*Tan[c + d*x])/(3*d) + (a^2*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(12*d) + ((10*A + 3*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(30*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(5*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(10*a*d)","A",8,8,33,0.2424,1,"{4089, 4010, 4001, 3788, 3767, 8, 4046, 3770}"
94,1,132,0,0.2122919,"\int \sec (c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a^2*(12*A + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(12*A + 7*C)*Tan[c + d*x])/(6*d) + (a^2*(12*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)","A",7,7,31,0.2258,1,"{4083, 4001, 3788, 3767, 8, 4046, 3770}"
95,1,96,0,0.1400934,"\int (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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^2*A*x + (a^2*(2*A + C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Tan[c + d*x])/d + (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + (C*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(3*d)","A",6,6,25,0.2400,1,"{4055, 3917, 3914, 3767, 8, 3770}"
96,1,112,0,0.1899764,"\int \cos (c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"2*a^2*A*x + (a^2*(2*A + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/d - (a^2*(2*A - 3*C)*Tan[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",6,6,31,0.1935,1,"{4087, 3917, 3914, 3767, 8, 3770}"
97,1,119,0,0.2896749,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a^2*(3*A + 2*C)*x)/2 + (2*a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*A - 2*C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((A - 2*C)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",5,4,33,0.1212,1,"{4087, 4018, 3996, 3770}"
98,1,110,0,0.2678765,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(A + 2*C)*x + (a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(A + C)*Sin[c + d*x])/d + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",5,4,33,0.1212,1,"{4087, 4017, 3996, 3770}"
99,1,136,0,0.3073534,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(7*A + 12*C)*x)/8 + (a^2*(7*A + 12*C)*Sin[c + d*x])/(6*d) + (a^2*(7*A + 12*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)","A",6,6,33,0.1818,1,"{4087, 4013, 3788, 2637, 4045, 8}"
100,1,169,0,0.3873581,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(3*A + 4*C)*x)/4 + (a^2*(18*A + 25*C)*Sin[c + d*x])/(15*d) + (a^2*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^2*(9*A + 10*C)*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (A*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(10*d)","A",7,7,33,0.2121,1,"{4087, 4017, 3996, 3787, 2635, 8, 2637}"
101,1,194,0,0.4124872,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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*(11*A + 14*C)*x)/16 + (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (a^2*(11*A + 14*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(9*A + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (2*a^2*(4*A + 5*C)*Sin[c + d*x]^3)/(15*d)","A",8,7,33,0.2121,1,"{4087, 4017, 3996, 3787, 2633, 2635, 8}"
102,1,197,0,0.4166587,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a^3*(30*A + 23*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(30*A + 23*C)*Tan[c + d*x])/(10*d) + (3*a^3*(30*A + 23*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + ((30*A + 7*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(120*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(6*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(10*a*d) + (a^3*(30*A + 23*C)*Tan[c + d*x]^3)/(120*d)","A",12,8,33,0.2424,1,"{4089, 4010, 4001, 3791, 3770, 3767, 8, 3768}"
103,1,157,0,0.2557875,"\int \sec (c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a^3*(20*A + 13*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(20*A + 13*C)*Tan[c + d*x])/(5*d) + (3*a^3*(20*A + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(40*d) - (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(20*A + 13*C)*Tan[c + d*x]^3)/(60*d)","A",11,7,31,0.2258,1,"{4083, 4001, 3791, 3770, 3767, 8, 3768}"
104,1,147,0,0.2189767,"\int (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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*A*x + (a^3*(28*A + 15*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(4*A + 3*C)*Tan[c + d*x])/(8*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (C*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(4*a*d) + ((4*A + 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(8*d)","A",7,6,25,0.2400,1,"{4055, 3917, 3914, 3767, 8, 3770}"
105,1,145,0,0.250234,"\int \cos (c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}-\frac{(3 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 a d}+3 a^3 A x+\frac{5 a^3 C \tan (c+d x)}{2 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}","\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}-\frac{(3 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 a d}+3 a^3 A x+\frac{5 a^3 C \tan (c+d x)}{2 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}",1,"3*a^3*A*x + (a^3*(6*A + 5*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/d + (5*a^3*C*Tan[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(3*a*d) - ((6*A - 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)","A",7,6,31,0.1935,1,"{4087, 3917, 3914, 3767, 8, 3770}"
106,1,162,0,0.4139539,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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{(A-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a d}+\frac{1}{2} a^3 x (7 A+2 C)+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{2 d}","\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{(A-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a d}+\frac{1}{2} a^3 x (7 A+2 C)+\frac{A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{2 d}",1,"(a^3*(7*A + 2*C)*x)/2 + (a^3*(2*A + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A - C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - ((A - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((A - 4*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",6,4,33,0.1212,1,"{4087, 4018, 3996, 3770}"
107,1,156,0,0.3965847,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\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{A \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a 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 ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}","-\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{A \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 a 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 ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"(a^3*(5*A + 6*C)*x)/2 + (3*a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*A*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((5*A - 6*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",6,5,33,0.1515,1,"{4087, 4017, 4018, 3996, 3770}"
108,1,169,0,0.4101384,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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{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{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 ^3(c+d x) (a \sec (c+d x)+a)^3}{4 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{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{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 ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(15*A + 28*C)*x)/8 + (a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(3*A + 4*C)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(4*a*d) + ((5*A + 4*C)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(8*d)","A",6,4,33,0.1212,1,"{4087, 4017, 3996, 3770}"
109,1,161,0,0.3298118,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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*(13*A + 20*C)*x)/8 + (a^3*(13*A + 20*C)*Sin[c + d*x])/(5*d) + (3*a^3*(13*A + 20*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (3*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) - (a^3*(13*A + 20*C)*Sin[c + d*x]^3)/(60*d)","A",9,7,33,0.2121,1,"{4087, 4013, 3791, 2637, 2635, 8, 2633}"
110,1,216,0,0.5507661,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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{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{1}{16} a^3 x (23 A+30 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 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{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{1}{16} a^3 x (23 A+30 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}",1,"(a^3*(23*A + 30*C)*x)/16 + (a^3*(34*A + 45*C)*Sin[c + d*x])/(15*d) + (a^3*(23*A + 30*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(73*A + 90*C)*Cos[c + d*x]^2*Sin[c + d*x])/(120*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(10*a*d) + ((31*A + 30*C)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(120*d)","A",8,7,33,0.2121,1,"{4087, 4017, 3996, 3787, 2635, 8, 2637}"
111,1,228,0,0.4812392,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a^4*(14*A + 11*C)*ArcTanh[Sin[c + d*x]])/(4*d) + (16*a^4*(14*A + 11*C)*Tan[c + d*x])/(35*d) + (27*a^4*(14*A + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(140*d) + (a^4*(14*A + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(70*d) + ((21*A + 4*C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(105*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(7*d) + (2*C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(21*a*d) + (8*a^4*(14*A + 11*C)*Tan[c + d*x]^3)/(105*d)","A",15,8,33,0.2424,1,"{4089, 4010, 4001, 3791, 3770, 3767, 8, 3768}"
112,1,188,0,0.3027747,"\int \sec (c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(7*a^4*(10*A + 7*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(10*A + 7*C)*Tan[c + d*x])/(5*d) + (27*a^4*(10*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + (a^4*(10*A + 7*C)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) - (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(30*d) + (C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (2*a^4*(10*A + 7*C)*Tan[c + d*x]^3)/(15*d)","A",14,7,31,0.2258,1,"{4083, 4001, 3791, 3770, 3767, 8, 3768}"
113,1,177,0,0.2943049,"\int (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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{(5 A+7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{15 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{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}","\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{(5 A+7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{15 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{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*A*x + (a^4*(12*A + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^4*(10*A + 7*C)*Tan[c + d*x])/(2*d) + (a*C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + ((5*A + 7*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + ((8*A + 7*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(6*d)","A",8,6,25,0.2400,1,"{4055, 3917, 3914, 3767, 8, 3770}"
114,1,181,0,0.3417415,"\int \cos (c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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-7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 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{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}","\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-7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 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{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,"4*a^4*A*x + (a^4*(52*A + 35*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/d + (5*a^4*(4*A + 7*C)*Tan[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) - ((12*A - 7*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - ((12*A - 35*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)","A",8,6,31,0.1935,1,"{4087, 3917, 3914, 3767, 8, 3770}"
115,1,192,0,0.5276478,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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{(A-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 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 (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}","\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{(A-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 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 (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*(13*A + 2*C)*x)/2 + (2*a^4*(2*A + 3*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(A - 2*C)*Sin[c + d*x])/(2*d) - (a*(3*A - 2*C)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - ((A - 2*C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((3*A + 22*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",7,4,33,0.1212,1,"{4087, 4018, 3996, 3770}"
116,1,198,0,0.5461374,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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{(2 A-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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{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}","\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{(2 A-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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{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,"2*a^4*(3*A + 2*C)*x + (a^4*(2*A + 13*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(2*A - C)*Sin[c + d*x])/(2*d) + (2*a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - ((2*A - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((4*A - 9*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",7,5,33,0.1515,1,"{4087, 4017, 4018, 3996, 3770}"
117,1,200,0,0.5738739,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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{(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{1}{8} a^4 x (35 A+52 C)+\frac{4 a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \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}","\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{(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{1}{8} a^4 x (35 A+52 C)+\frac{4 a^4 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \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 + 52*C)*x)/8 + (4*a^4*C*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(7*A + 4*C)*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) + ((7*A + 4*C)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) - ((35*A - 12*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)","A",7,5,33,0.1515,1,"{4087, 4017, 4018, 3996, 3770}"
118,1,207,0,0.5516856,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{a^4 (7 A+10 C) \sin (c+d x)}{2 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{(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{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{5 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}","\frac{a^4 (7 A+10 C) \sin (c+d x)}{2 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{(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{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{5 d}+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}",1,"(a^4*(7*A + 12*C)*x)/2 + (a^4*C*ArcTanh[Sin[c + d*x]])/d + (a^4*(7*A + 10*C)*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) + ((7*A + 5*C)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + ((7*A + 8*C)*Cos[c + d*x]*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",7,4,33,0.1212,1,"{4087, 4017, 3996, 3770}"
119,1,192,0,0.3664567,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(7*a^4*(7*A + 10*C)*x)/16 + (4*a^4*(7*A + 10*C)*Sin[c + d*x])/(5*d) + (27*a^4*(7*A + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(7*A + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (2*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(6*d) - (2*a^4*(7*A + 10*C)*Sin[c + d*x]^3)/(15*d)","A",12,7,33,0.2121,1,"{4087, 4013, 3791, 2637, 2635, 8, 2633}"
120,1,254,0,0.7186469,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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{(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{(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{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}","\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{(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{(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{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*(11*A + 14*C)*x)/4 + (a^4*(454*A + 581*C)*Sin[c + d*x])/(105*d) + (a^4*(11*A + 14*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a^4*(247*A + 308*C)*Cos[c + d*x]^2*Sin[c + d*x])/(210*d) + (2*a*A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(21*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + ((8*A + 7*C)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*d) + ((109*A + 126*C)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(210*d)","A",9,7,33,0.2121,1,"{4087, 4017, 3996, 3787, 2635, 8, 2637}"
121,1,165,0,0.2019549,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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)*ArcTanh[Sin[c + d*x]])/(8*a*d) - ((3*A + 4*C)*Tan[c + d*x])/(a*d) + (3*(4*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + ((4*A + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*A + 4*C)*Tan[c + d*x]^3)/(3*a*d)","A",7,5,33,0.1515,1,"{4085, 3787, 3767, 3768, 3770}"
122,1,133,0,0.1788738,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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*A + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) + ((3*A + 4*C)*Tan[c + d*x])/(a*d) - ((2*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((3*A + 4*C)*Tan[c + d*x]^3)/(3*a*d)","A",6,5,33,0.1515,1,"{4085, 3787, 3768, 3770, 3767}"
123,1,107,0,0.1667284,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"((2*A + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((A + 2*C)*Tan[c + d*x])/(a*d) + ((2*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,6,33,0.1818,1,"{4085, 3787, 3767, 8, 3768, 3770}"
124,1,57,0,0.1519811,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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,"-((C*ArcTanh[Sin[c + d*x]])/(a*d)) + (C*Tan[c + d*x])/(a*d) + ((A + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",4,4,31,0.1290,1,"{4083, 3998, 3770, 3794}"
125,1,49,0,0.107575,"\int \frac{A+C \sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(a*d) - ((A + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",4,4,25,0.1600,1,"{4051, 3770, 3919, 3794}"
126,1,52,0,0.1089174,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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,"-((A*x)/a) + ((2*A + C)*Sin[c + d*x])/(a*d) - ((A + C)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",4,4,31,0.1290,1,"{4085, 3787, 2637, 8}"
127,1,96,0,0.1525735,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"((3*A + 2*C)*x)/(2*a) - ((2*A + C)*Sin[c + d*x])/(a*d) + ((3*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",5,5,33,0.1515,1,"{4085, 3787, 2635, 8, 2637}"
128,1,124,0,0.1687569,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"-((3*A + 2*C)*x)/(2*a) + ((4*A + 3*C)*Sin[c + d*x])/(a*d) - ((3*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*A + 3*C)*Sin[c + d*x]^3)/(3*a*d)","A",6,5,33,0.1515,1,"{4085, 3787, 2633, 2635, 8}"
129,1,156,0,0.1867139,"\int \frac{\cos ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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,"(3*(5*A + 4*C)*x)/(8*a) - ((4*A + 3*C)*Sin[c + d*x])/(a*d) + (3*(5*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((5*A + 4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((4*A + 3*C)*Sin[c + d*x]^3)/(3*a*d)","A",7,5,33,0.1515,1,"{4085, 3787, 2635, 8, 2633}"
130,1,172,0,0.3291261,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"-(((2*A + 5*C)*ArcTanh[Sin[c + d*x]])/(a^2*d)) + ((5*A + 12*C)*Tan[c + d*x])/(a^2*d) - ((2*A + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(a^2*d) - (2*(2*A + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) + ((5*A + 12*C)*Tan[c + d*x]^3)/(3*a^2*d)","A",7,6,33,0.1818,1,"{4085, 4019, 3787, 3768, 3770, 3767}"
131,1,150,0,0.3050025,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"((2*A + 7*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (4*(A + 4*C)*Tan[c + d*x])/(3*a^2*d) + ((2*A + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - (2*(A + 4*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,7,33,0.2121,1,"{4085, 4019, 3787, 3767, 8, 3768, 3770}"
132,1,99,0,0.2524959,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"(-2*C*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A + 4*C)*Tan[c + d*x])/(3*a^2*d) + (2*C*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,6,33,0.1818,1,"{4085, 4008, 3787, 3770, 3767, 8}"
133,1,81,0,0.1622272,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\frac{2 (A-2 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) \sec (c+d x)}{3 d (a \sec (c+d x)+a)^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,"(C*ArcTanh[Sin[c + d*x]])/(a^2*d) + (2*(A - 2*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",4,4,31,0.1290,1,"{4079, 3998, 3770, 3794}"
134,1,68,0,0.1206531,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"(A*x)/a^2 - (2*(2*A - C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",3,3,25,0.1200,1,"{4053, 3919, 3794}"
135,1,82,0,0.2178718,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"(-2*A*x)/a^2 + ((10*A + C)*Sin[c + d*x])/(3*a^2*d) - (2*A*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",5,5,31,0.1613,1,"{4085, 4020, 3787, 2637, 8}"
136,1,137,0,0.3104327,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"((7*A + 2*C)*x)/(2*a^2) - (4*(4*A + C)*Sin[c + d*x])/(3*a^2*d) + ((7*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - (2*(4*A + C)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,6,33,0.1818,1,"{4085, 4020, 3787, 2635, 8, 2637}"
137,1,163,0,0.3246365,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"-(((5*A + 2*C)*x)/a^2) + ((12*A + 5*C)*Sin[c + d*x])/(a^2*d) - ((5*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(a^2*d) - (2*(5*A + 2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - ((12*A + 5*C)*Sin[c + d*x]^3)/(3*a^2*d)","A",7,6,33,0.1818,1,"{4085, 4020, 3787, 2633, 2635, 8}"
138,1,198,0,0.4898208,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"((2*A + 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(11*A + 76*C)*Tan[c + d*x])/(15*a^3*d) + ((2*A + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((11*A + 76*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",8,7,33,0.2121,1,"{4085, 4019, 3787, 3767, 8, 3768, 3770}"
139,1,145,0,0.4274675,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"(-3*C*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((2*A + 27*C)*Tan[c + d*x])/(15*a^3*d) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (3*C*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",7,7,33,0.2121,1,"{4085, 4019, 4008, 3787, 3770, 3767, 8}"
140,1,123,0,0.330017,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((3*A - 7*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((6*A - 29*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",5,5,33,0.1515,1,"{4085, 4008, 3998, 3770, 3794}"
141,1,104,0,0.1870346,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((A + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - C)*Tan[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) + ((2*A + 7*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",3,3,31,0.09677,1,"{4079, 4000, 3794}"
142,1,106,0,0.1804717,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"(A*x)/a^3 - ((A + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*A - 3*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((22*A - 3*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",4,4,25,0.1600,1,"{4053, 3922, 3919, 3794}"
143,1,120,0,0.354935,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"(-3*A*x)/a^3 + (2*(36*A + C)*Sin[c + d*x])/(15*a^3*d) - ((A + C)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*A - C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (3*A*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",6,5,31,0.1613,1,"{4085, 4020, 3787, 2637, 8}"
144,1,183,0,0.4660274,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"((13*A + 2*C)*x)/(2*a^3) - (2*(76*A + 11*C)*Sin[c + d*x])/(15*a^3*d) + ((13*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*A + C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((76*A + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",7,6,33,0.1818,1,"{4085, 4020, 3787, 2635, 8, 2637}"
145,1,216,0,0.4972609,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"-((23*A + 6*C)*x)/(2*a^3) + (4*(34*A + 9*C)*Sin[c + d*x])/(5*a^3*d) - ((23*A + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((13*A + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((23*A + 6*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a^3 + a^3*Sec[c + d*x])) - (4*(34*A + 9*C)*Sin[c + d*x]^3)/(15*a^3*d)","A",8,6,33,0.1818,1,"{4085, 4020, 3787, 2633, 2635, 8}"
146,1,232,0,0.6489497,"\int \frac{\sec ^5(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^5*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"((2*A + 21*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (32*(5*A + 54*C)*Tan[c + d*x])/(105*a^4*d) + ((2*A + 21*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((10*A + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (16*(5*A + 54*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",9,7,33,0.2121,1,"{4085, 4019, 3787, 3767, 8, 3768, 3770}"
147,1,183,0,0.5890162,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"(-4*C*ArcTanh[Sin[c + d*x]])/(a^4*d) + (2*(3*A + 122*C)*Tan[c + d*x])/(105*a^4*d) + ((3*A - 88*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*C*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (2*(A - 6*C)*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",8,7,33,0.2121,1,"{4085, 4019, 4008, 3787, 3770, 3767, 8}"
148,1,161,0,0.4841863,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((8*A - 55*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((16*A - 215*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (2*(2*A - 5*C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",6,6,33,0.1818,1,"{4085, 4019, 4008, 3998, 3770, 3794}"
149,1,138,0,0.3743425,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"((23*A - 54*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + (4*(2*A + 9*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(3*A - 4*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",4,4,33,0.1212,1,"{4085, 4008, 4000, 3794}"
150,1,142,0,0.2469615,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"-((A + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + (2*(4*A - 3*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + ((6*A + 13*C)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((6*A + 13*C)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))","A",4,4,31,0.1290,1,"{4079, 4000, 3796, 3794}"
151,1,136,0,0.2623562,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"(A*x)/a^4 - ((55*A - 8*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (8*(20*A - C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(5*A - 2*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",5,4,25,0.1600,1,"{4053, 3922, 3919, 3794}"
152,1,152,0,0.4994501,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","\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}","\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,"(-4*A*x)/a^4 + (2*(332*A + 3*C)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 3*C)*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*A*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(6*A - C)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",7,5,31,0.1613,1,"{4085, 4020, 3787, 2637, 8}"
153,1,215,0,0.6271046,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"((21*A + 2*C)*x)/(2*a^4) - (32*(54*A + 5*C)*Sin[c + d*x])/(105*a^4*d) + ((21*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((129*A + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (16*(54*A + 5*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*A*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",8,6,33,0.1818,1,"{4085, 4020, 3787, 2635, 8, 2637}"
154,1,248,0,0.6874387,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"(-2*(11*A + 2*C)*x)/a^4 + (4*(454*A + 83*C)*Sin[c + d*x])/(35*a^4*d) - (2*(11*A + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(a^4*d) - ((178*A + 31*C)*Cos[c + d*x]^2*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(11*A + 2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^4*d*(1 + Sec[c + d*x])) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - (2*(8*A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) - (4*(454*A + 83*C)*Sin[c + d*x]^3)/(105*a^4*d)","A",9,6,33,0.1818,1,"{4085, 4020, 3787, 2633, 2635, 8}"
155,1,223,0,0.5157759,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(4*a*(99*A + 80*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(99*A + 80*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(99*A + 80*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*C*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (4*(99*A + 80*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*a*d)","A",6,6,35,0.1714,1,"{4089, 4016, 3803, 3800, 4001, 3792}"
156,1,180,0,0.4454808,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(2*a*(21*A + 16*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(21*A + 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(21*A + 16*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)","A",5,5,35,0.1429,1,"{4089, 4016, 3800, 4001, 3792}"
157,1,137,0,0.3905002,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*a*(35*A + 27*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A + 18*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(7*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)","A",4,4,35,0.1143,1,"{4089, 4010, 4001, 3792}"
158,1,95,0,0.198207,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*a*(15*A + 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (4*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)","A",3,3,33,0.09091,1,"{4083, 4001, 3792}"
159,1,96,0,0.1457209,"\int \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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*Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*C*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",5,5,27,0.1852,1,"{4055, 3915, 3774, 203, 3792}"
160,1,94,0,0.2007834,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d - (a*(A - 2*C)*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,33,0.1515,1,"{4087, 3915, 3774, 203, 3792}"
161,1,110,0,0.2509979,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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[a]*(3*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*A*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,35,0.1143,1,"{4087, 4015, 3774, 203}"
162,1,153,0,0.3514548,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),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}}","\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,"(Sqrt[a]*(5*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*A + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{4087, 4015, 3805, 3774, 203}"
163,1,196,0,0.419558,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(Sqrt[a]*(35*A + 48*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(35*A + 48*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(35*A + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{4087, 4015, 3805, 3774, 203}"
164,1,225,0,0.6549565,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*a^2*(143*A + 112*C)*Tan[c + d*x])/(165*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(33*A + 28*C)*Sec[c + d*x]^3*Tan[c + d*x])/(231*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(143*A + 112*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(1155*d) + (2*a*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(33*d) + (2*(143*A + 112*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(385*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)","A",6,6,35,0.1714,1,"{4089, 4018, 4016, 3800, 4001, 3792}"
165,1,174,0,0.4766655,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(8*a^2*(63*A + 47*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(63*A + 47*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*(63*A + 22*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(9*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(21*a*d)","A",5,5,35,0.1429,1,"{4089, 4010, 4001, 3793, 3792}"
166,1,132,0,0.2631475,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(8*a^2*(35*A + 19*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(35*A + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) - (4*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)","A",4,4,33,0.1212,1,"{4083, 4001, 3793, 3792}"
167,1,133,0,0.2226691,"\int (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{2 a^2 (5 A+4 C) \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\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 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}","\frac{2 a^2 (5 A+4 C) \tan (c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}+\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 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,"(2*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(5*A + 4*C)*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",6,6,27,0.2222,1,"{4055, 3917, 3915, 3774, 203, 3792}"
168,1,136,0,0.2888964,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","-\frac{a^2 (3 A-8 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","-\frac{a^2 (3 A-8 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\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 (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,"(3*a^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (a^2*(3*A - 8*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{4087, 3917, 3915, 3774, 203, 3792}"
169,1,151,0,0.4278228,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (5 A-8 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(7*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*A - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(A - 4*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",5,5,35,0.1429,1,"{4087, 4018, 4015, 3774, 203}"
170,1,155,0,0.4622733,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 (19 A+24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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 \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}","\frac{a^2 (19 A+24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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 \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^(3/2)*(11*A + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(19*A + 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{4087, 4017, 4015, 3774, 203}"
171,1,200,0,0.5714302,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 (75 A+112 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (75 A+112 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(75*A + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*A + 112*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(13*A + 16*C)*Cos[c + d*x]*Sin[c + d*x])/(32*d*Sqrt[a + a*Sec[c + d*x]]) + (a*A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,6,35,0.1714,1,"{4087, 4017, 4015, 3805, 3774, 203}"
172,1,245,0,0.6392456,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (133 A+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(133*A + 176*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(133*A + 176*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(133*A + 176*C)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(67*A + 80*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (3*a*A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,6,35,0.1714,1,"{4087, 4017, 4015, 3805, 3774, 203}"
173,1,273,0,0.8645255,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^2 (143 A+136 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}-\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}","\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^2 (143 A+136 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{1287 d}+\frac{2 a^3 (10439 A+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}-\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,"(2*a^3*(10439*A + 8368*C)*Tan[c + d*x])/(6435*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2717*A + 2224*C)*Sec[c + d*x]^3*Tan[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(10439*A + 8368*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(45045*d) + (2*a^2*(143*A + 136*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(1287*d) + (2*a*(10439*A + 8368*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(15015*d) + (10*a*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)","A",7,6,35,0.1714,1,"{4089, 4018, 4016, 3800, 4001, 3792}"
174,1,211,0,0.532672,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{16 a^2 (33 A+25 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{693 d}+\frac{64 a^3 (33 A+25 C) \tan (c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\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}","\frac{16 a^2 (33 A+25 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{693 d}+\frac{64 a^3 (33 A+25 C) \tan (c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\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,"(64*a^3*(33*A + 25*C)*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(33*A + 25*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(693*d) + (2*a*(33*A + 25*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(231*d) + (2*(99*A + 26*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*d) + (10*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*a*d)","A",6,5,35,0.1429,1,"{4089, 4010, 4001, 3793, 3792}"
175,1,169,0,0.3163478,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{16 a^2 (21 A+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\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}","\frac{16 a^2 (21 A+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (21 A+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\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,"(64*a^3*(21*A + 13*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(21*A + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(21*A + 13*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) - (4*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)","A",5,4,33,0.1212,1,"{4083, 4001, 3793, 3792}"
176,1,170,0,0.3083616,"\int (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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 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}","\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^{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 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,"(2*a^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(49*A + 32*C)*Tan[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(7*A + 8*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",7,6,27,0.2222,1,"{4055, 3917, 3915, 3774, 203, 3792}"
177,1,173,0,0.3694429,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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{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 (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}","\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{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 (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,"(5*a^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d + (a^3*(15*A + 64*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(15*A - 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",7,6,33,0.1818,1,"{4087, 3917, 3915, 3774, 203, 3792}"
178,1,188,0,0.5982088,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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 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}","\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^{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 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^(5/2)*(19*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(27*A - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(A - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) - (a*(3*A - 4*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d)","A",6,5,35,0.1429,1,"{4087, 4018, 4015, 3774, 203}"
179,1,192,0,0.6191276,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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{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 \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}","\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{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 \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,"(5*a^(5/2)*(5*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*A - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(3*A - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (5*a*A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",6,6,35,0.1714,1,"{4087, 4017, 4018, 4015, 3774, 203}"
180,1,200,0,0.6547425,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 (299 A+432 C) \sin (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\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^2 (17 A+16 C) \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d}+\frac{5 a A \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}","\frac{a^3 (299 A+432 C) \sin (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\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^2 (17 A+16 C) \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{32 d}+\frac{5 a A \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^(5/2)*(163*A + 304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(299*A + 432*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(17*A + 16*C)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",6,5,35,0.1429,1,"{4087, 4017, 4015, 3774, 203}"
181,1,245,0,0.7612136,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^3 (787 A+1040 C) \sin (c+d x) \cos (c+d x)}{960 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)^{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}","\frac{a^3 (283 A+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^3 (787 A+1040 C) \sin (c+d x) \cos (c+d x)}{960 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)^{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^(5/2)*(283*A + 400*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 400*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(787*A + 1040*C)*Cos[c + d*x]*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(79*A + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,35,0.1714,1,"{4087, 4017, 4015, 3805, 3774, 203}"
182,1,290,0,0.8584703,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{a^3 (1015 A+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 \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}","\frac{a^3 (1015 A+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 \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^(5/2)*(1015*A + 1304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1015*A + 1304*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1015*A + 1304*C)*Cos[c + d*x]*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(109*A + 136*C)*Cos[c + d*x]^2*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(23*A + 24*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)","A",8,6,35,0.1714,1,"{4087, 4017, 4015, 3805, 3774, 203}"
183,1,236,0,0.8026111,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"-((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A + 143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(21*A + 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(21*A + 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*a*d)","A",7,6,35,0.1714,1,"{4089, 4021, 4010, 4001, 3795, 203}"
184,1,193,0,0.5936521,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"(Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A + 37*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)","A",6,6,35,0.1714,1,"{4089, 4021, 4010, 4001, 3795, 203}"
185,1,152,0,0.4167576,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}","-\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,"-((Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A + 14*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) - (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)","A",5,5,35,0.1429,1,"{4089, 4010, 4001, 3795, 203}"
186,1,109,0,0.2058208,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"(Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*C*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)","A",4,4,33,0.1212,1,"{4083, 4001, 3795, 203}"
187,1,115,0,0.1621897,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,27,0.1852,1,"{4055, 3920, 3774, 203, 3795}"
188,1,113,0,0.2243492,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}","\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*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,33,0.1515,1,"{4087, 3920, 3774, 203, 3795}"
189,1,159,0,0.3693505,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"((7*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (A*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,35,0.1714,1,"{4087, 4022, 3920, 3774, 203, 3795}"
190,1,200,0,0.5629301,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"-((9*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*A + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - (A*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,35,0.1714,1,"{4087, 4022, 3920, 3774, 203, 3795}"
191,1,243,0,0.7285251,"\int \frac{\cos ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"((107*A + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((21*A + 16*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + ((43*A + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,35,0.1714,1,"{4087, 4022, 3920, 3774, 203, 3795}"
192,1,259,0,0.8450675,"\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","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((11*A + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((455*A + 799*C)*Tan[c + d*x])/(105*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((35*A + 67*C)*Sec[c + d*x]^2*Tan[c + d*x])/(70*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((7*A + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(14*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((245*A + 397*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(210*a^2*d)","A",7,6,35,0.1714,1,"{4085, 4021, 4010, 4001, 3795, 203}"
193,1,214,0,0.6271999,"\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","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((7*A + 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((15*A + 31*C)*Tan[c + d*x])/(5*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A + 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((5*A + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(10*a^2*d)","A",6,6,35,0.1714,1,"{4085, 4021, 4010, 4001, 3795, 203}"
194,1,169,0,0.4484175,"\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","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((3*A + 11*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((3*A + 13*C)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((3*A + 7*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)","A",5,5,35,0.1429,1,"{4085, 4010, 4001, 3795, 203}"
195,1,126,0,0.2301684,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((A - 7*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A + 5*C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,33,0.1212,1,"{4079, 4001, 3795, 203}"
196,1,125,0,0.1902615,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,5,27,0.1852,1,"{4053, 3920, 3774, 203, 3795}"
197,1,158,0,0.386058,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(-3*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((9*A + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A + C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,33,0.1818,1,"{4085, 4022, 3920, 3774, 203, 3795}"
198,1,217,0,0.5879373,"\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","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((19*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*A + 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*A + 2*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*A + C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,35,0.1714,1,"{4085, 4022, 3920, 3774, 203, 3795}"
199,1,266,0,0.7757007,"\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","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((47*A + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*a^(3/2)*d) + ((17*A + 9*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (3*(7*A + 4*C)*Sin[c + d*x])/(8*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((13*A + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,35,0.1714,1,"{4085, 4022, 3920, 3774, 203, 3795}"
200,1,259,0,0.837182,"\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","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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{(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{(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}}","\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{(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{(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,"-((75*A + 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A + 21*C)*Sec[c + d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((465*A + 1729*C)*Tan[c + d*x])/(120*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((45*A + 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((195*A + 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)","A",7,7,35,0.2000,1,"{4085, 4019, 4021, 4010, 4001, 3795, 203}"
201,1,212,0,0.6593666,"\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","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"((19*A + 163*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((A + 17*C)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((21*A + 197*C)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + (5*(3*A + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)","A",6,6,35,0.1714,1,"{4085, 4019, 4010, 4001, 3795, 203}"
202,1,165,0,0.4563726,"\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","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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*(A - 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((3*A - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((A + 9*C)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,35,0.1429,1,"{4085, 4008, 4001, 3795, 203}"
203,1,130,0,0.260597,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((7*A - 9*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,33,0.1212,1,"{4079, 4000, 3795, 203}"
204,1,162,0,0.2608478,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*A - 5*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,6,27,0.2222,1,"{4053, 3922, 3920, 3774, 203, 3795}"
205,1,199,0,0.5576192,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{(35 A+3 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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}}","\frac{(35 A+3 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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,"(-5*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((115*A + 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*A - C)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*A + 3*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,33,0.2121,1,"{4085, 4020, 4022, 3920, 3774, 203, 3795}"
206,1,262,0,0.8029547,"\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","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{(63 A+11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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}}","-\frac{(63 A+11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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,"((39*A + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(5/2)*d) - ((219*A + 43*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((19*A + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((63*A + 11*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((31*A + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",9,7,35,0.2000,1,"{4085, 4020, 4022, 3920, 3774, 203, 3795}"
207,1,205,0,0.2222094,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(5*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,7,33,0.2121,1,"{4077, 4047, 3768, 3771, 2641, 4046, 2639}"
208,1,172,0,0.2015299,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,7,33,0.2121,1,"{4077, 4047, 3768, 3771, 2639, 4046, 2641}"
209,1,135,0,0.1831112,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\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}","\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,"(2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,6,33,0.1818,1,"{4077, 4047, 3771, 2641, 4046, 2639}"
210,1,135,0,0.184622,"\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","Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\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}","\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,"(2*a*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,33,0.1818,1,"{4075, 4047, 3771, 2641, 4046, 2639}"
211,1,141,0,0.187561,"\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","Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\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)}}","\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,"(2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,33,0.1818,1,"{4075, 4047, 3771, 2639, 4045, 2641}"
212,1,174,0,0.204178,"\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","Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),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 (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)}","\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,"(2*a*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,33,0.2121,1,"{4075, 4047, 3769, 3771, 2641, 4045, 2639}"
213,1,205,0,0.2315716,"\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","Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\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)}","\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,"(2*a*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(7*A + 9*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,7,33,0.2121,1,"{4075, 4047, 3769, 3771, 2639, 4045, 2641}"
214,1,270,0,0.4389744,"\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","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),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}","\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,"(-16*a^2*(3*A + 2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (16*a^2*(3*A + 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(7*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 19*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d) + (8*C*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d)","A",10,8,35,0.2286,1,"{4089, 4018, 3997, 3787, 3768, 3771, 2639, 2641}"
215,1,237,0,0.4139341,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(-4*a^2*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(7*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(5*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(35*A + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (8*C*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d)","A",9,8,35,0.2286,1,"{4089, 4018, 3997, 3787, 3771, 2641, 3768, 2639}"
216,1,196,0,0.389384,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\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}","\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,"(-16*a^2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(15*A + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (8*C*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",8,7,35,0.2000,1,"{4089, 4018, 3997, 3787, 3771, 2639, 2641}"
217,1,198,0,0.402225,"\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","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","-\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)}}","-\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,"(4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (8*a^2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(A - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(A - C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",8,7,35,0.2000,1,"{4087, 4018, 3997, 3787, 3771, 2639, 2641}"
218,1,196,0,0.3922904,"\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","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","-\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)}","-\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,"(16*a^2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(7*A - 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",8,7,35,0.2000,1,"{4087, 4017, 3997, 3787, 3771, 2639, 2641}"
219,1,204,0,0.4214242,"\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","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\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)}","\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,"(4*a^2*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (8*a^2*(3*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(33*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))","A",8,7,35,0.2000,1,"{4087, 4017, 3996, 3787, 3771, 2639, 2641}"
220,1,237,0,0.4360347,"\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","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\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)}","\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,"(16*a^2*(2*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(19*A + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^2*(5*A + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))","A",9,8,35,0.2286,1,"{4087, 4017, 3996, 3787, 3769, 3771, 2641, 2639}"
221,1,270,0,0.4728698,"\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","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\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)}","\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,"(4*a^2*(7*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (8*a^2*(25*A + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(89*A + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (4*a^2*(7*A + 9*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (8*a^2*(25*A + 33*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (8*A*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2))","A",10,8,35,0.2286,1,"{4087, 4017, 3996, 3787, 3769, 3771, 2639, 2641}"
222,1,319,0,0.6181463,"\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","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),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}","\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,"(-4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(143*A + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(7*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (4*a^3*(143*A + 105*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (8*a^3*(44*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(385*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d) + (4*C*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(33*a*d) + (2*(33*A + 35*C)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(231*d)","A",11,8,35,0.2286,1,"{4089, 4018, 3997, 3787, 3768, 3771, 2639, 2641}"
223,1,286,0,0.5896918,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),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}","\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,"(-4*a^3*(27*A + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(27*A + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (8*a^3*(21*A + 16*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + (4*C*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 73*C)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d)","A",10,8,35,0.2286,1,"{4089, 4018, 3997, 3787, 3771, 2641, 3768, 2639}"
224,1,253,0,0.5627824,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\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}","\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,"(-4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(70*A + 53*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + (12*C*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d) + (2*(5*A + 7*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",9,7,35,0.2000,1,"{4089, 4018, 3997, 3787, 3771, 2639, 2641}"
225,1,259,0,0.564843,"\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","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\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{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{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 A \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d \sqrt{\sec (c+d x)}}","\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{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{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 A \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d \sqrt{\sec (c+d x)}}",1,"(4*a^3*(5*A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(5*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - 3*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*a*d) - (2*(5*A - 9*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",9,7,35,0.2000,1,"{4087, 4018, 3997, 3787, 3771, 2639, 2641}"
226,1,253,0,0.5695391,"\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","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","-\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)}","-\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,"(4*a^3*(9*A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (8*a^3*(3*A - 10*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (4*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(5*a*d*Sqrt[Sec[c + d*x]]) - (2*(9*A - 5*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",9,8,35,0.2286,1,"{4087, 4017, 4018, 3997, 3787, 3771, 2639, 2641}"
227,1,253,0,0.5728711,"\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","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","-\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)}","-\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,"(4*a^3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(41*A - 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (12*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(7*A + 5*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",9,7,35,0.2000,1,"{4087, 4017, 3997, 3787, 3771, 2639, 2641}"
228,1,253,0,0.5799808,"\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","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),x]","\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)}","\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,"(4*a^3*(17*A + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (8*a^3*(16*A + 21*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (4*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sec[c + d*x]^(5/2)) + (2*(73*A + 63*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2))","A",9,7,35,0.2000,1,"{4087, 4017, 3996, 3787, 3771, 2639, 2641}"
229,1,286,0,0.6051795,"\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","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),x]","\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)}","\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,"(4*a^3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (8*a^3*(35*A + 44*C)*Sin[c + d*x])/(385*d*Sec[c + d*x]^(3/2)) + (4*a^3*(105*A + 143*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (4*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(33*a*d*Sec[c + d*x]^(7/2)) + (2*(35*A + 33*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2))","A",10,8,35,0.2286,1,"{4087, 4017, 3996, 3787, 3769, 3771, 2641, 2639}"
230,1,319,0,0.6550252,"\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","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),x]","\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)}","\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,"(4*a^3*(175*A + 221*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(95*A + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (40*a^3*(118*A + 143*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (4*a^3*(175*A + 221*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(95*A + 121*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2)) + (12*A*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(9/2)) + (2*(145*A + 143*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2))","A",11,8,35,0.2286,1,"{4087, 4017, 3996, 3787, 3769, 3771, 2639, 2641}"
231,1,232,0,0.236459,"\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","Int[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(-3*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(5*A + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((3*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((5*A + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",9,6,35,0.1714,1,"{4085, 3787, 3768, 3771, 2641, 2639}"
232,1,190,0,0.2167247,"\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","Int[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((3*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",8,6,35,0.1714,1,"{4085, 3787, 3768, 3771, 2639, 2641}"
233,1,152,0,0.1893587,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"-(((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,35,0.1714,1,"{4085, 3787, 3771, 2641, 3768, 2639}"
234,1,124,0,0.1709531,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\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}","-\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,"((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,5,35,0.1429,1,"{4085, 3787, 3771, 2639, 2641}"
235,1,162,0,0.2002034,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\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}","\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,"-(((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","A",7,6,35,0.1714,1,"{4085, 3787, 3769, 3771, 2641, 2639}"
236,1,199,0,0.2136403,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","-\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}","-\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,"(3*(7*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((7*A + 5*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((5*A + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))","A",8,6,35,0.1714,1,"{4085, 3787, 3769, 3771, 2639, 2641}"
237,1,229,0,0.3794982,"\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","Int[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"((A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + (2*(A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((A + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",9,7,35,0.2000,1,"{4085, 4019, 3787, 3768, 3771, 2639, 2641}"
238,1,191,0,0.3364715,"\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","Int[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"(-4*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((A - 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,7,35,0.2000,1,"{4085, 4019, 3787, 3771, 2641, 3768, 2639}"
239,1,165,0,0.3156458,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"-(((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,6,35,0.1714,1,"{4085, 4019, 3787, 3771, 2639, 2641}"
240,1,170,0,0.3173107,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\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}","-\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,"(4*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((5*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,6,35,0.1714,1,"{4085, 4020, 3787, 3771, 2639, 2641}"
241,1,201,0,0.3624941,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\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}","\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,"-(((7*A + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + (2*(5*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (2*(5*A + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((7*A + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)","A",8,7,35,0.2000,1,"{4085, 4020, 3787, 3769, 3771, 2641, 2639}"
242,1,236,0,0.3760413,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","-\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}","-\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,"(4*(14*A + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + (4*(14*A + 5*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(3*A + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((3*A + C)*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","A",9,7,35,0.2000,1,"{4085, 4020, 3787, 3769, 3771, 2639, 2641}"
243,1,282,0,0.5369544,"\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","Int[(Sec[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"((9*A + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) - ((9*A + 119*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((A + 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - ((9*A + 119*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))","A",10,7,35,0.2000,1,"{4085, 4019, 3787, 3768, 3771, 2639, 2641}"
244,1,249,0,0.5237582,"\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","Int[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"((A - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*(A - 4*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A - 13*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,7,35,0.2000,1,"{4085, 4019, 3787, 3771, 2641, 3768, 2639}"
245,1,220,0,0.5102153,"\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","Int[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((A - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*(2*A - 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A - 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,35,0.1714,1,"{4085, 4019, 3787, 3771, 2639, 2641}"
246,1,222,0,0.4921526,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((9*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + (2*(3*A - 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",8,7,35,0.2000,1,"{4085, 4019, 4020, 3787, 3771, 2639, 2641}"
247,1,226,0,0.5062037,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\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}","-\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,"((49*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - (2*(4*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,35,0.1714,1,"{4085, 4020, 3787, 3771, 2639, 2641}"
248,1,249,0,0.5334039,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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,"-((119*A + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((11*A + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(2*a^3*d) + ((11*A + C)*Sin[c + d*x])/(2*a^3*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - (2*A*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - ((119*A + 9*C)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,7,35,0.2000,1,"{4085, 4020, 3787, 3769, 3771, 2641, 2639}"
249,1,290,0,0.5703373,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","-\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}","-\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,"(7*(33*A + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((63*A + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (7*(33*A + 7*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((63*A + 13*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - (2*(6*A + C)*Sin[c + d*x])/(15*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - ((63*A + 13*C)*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))","A",10,7,35,0.2000,1,"{4085, 4020, 3787, 3769, 3771, 2639, 2641}"
250,1,214,0,0.4596202,"\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","Int[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(Sqrt[a]*(48*A + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(48*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,37,0.1351,1,"{4089, 4016, 3803, 3801, 215}"
251,1,169,0,0.3877146,"\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","Int[Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(Sqrt[a]*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(8*A + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,37,0.1351,1,"{4089, 4016, 3803, 3801, 215}"
252,1,124,0,0.3058162,"\int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(Sqrt[a]*(8*A + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,37,0.1081,1,"{4089, 4016, 3801, 215}"
253,1,115,0,0.3039445,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\sec (c+d x)}} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\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}","\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,"(Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*(2*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d","A",4,4,37,0.1081,1,"{4089, 4015, 3801, 215}"
254,1,116,0,0.2909419,"\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","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\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}","\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,"(2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",4,4,37,0.1081,1,"{4087, 4015, 3801, 215}"
255,1,122,0,0.3145916,"\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","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),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)}}","\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,"(2*a*(7*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",3,3,37,0.08108,1,"{4087, 4013, 3804}"
256,1,168,0,0.3878085,"\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","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),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}}","\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,"(2*a*A*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(24*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",4,4,37,0.1081,1,"{4087, 4015, 3805, 3804}"
257,1,213,0,0.4589365,"\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","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),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}}","\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,"(2*a*A*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(16*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",5,4,37,0.1081,1,"{4087, 4015, 3805, 3804}"
258,1,265,0,0.6721744,"\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","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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{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{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}","\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{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{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,"(a^(3/2)*(176*A + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(176*A + 133*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 67*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,6,37,0.1622,1,"{4089, 4018, 4016, 3803, 3801, 215}"
259,1,218,0,0.6013219,"\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","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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{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{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}","\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{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{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,"(a^(3/2)*(112*A + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(112*A + 75*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 13*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(32*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,6,37,0.1622,1,"{4089, 4018, 4016, 3803, 3801, 215}"
260,1,171,0,0.5049448,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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 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}","\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^{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 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^(3/2)*(24*A + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(24*A + 19*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,37,0.1351,1,"{4089, 4018, 4016, 3801, 215}"
261,1,171,0,0.4898339,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\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{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{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}","\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{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{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^(3/2)*(8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(8*A - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",5,5,37,0.1351,1,"{4089, 4018, 4015, 3801, 215}"
262,1,169,0,0.4673083,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\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{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 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)}}","\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{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 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,"(3*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(8*A - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",5,5,37,0.1351,1,"{4087, 4018, 4015, 3801, 215}"
263,1,163,0,0.4644012,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\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^{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 \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)}}","\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^{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 \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,"(2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(4*A + 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",5,5,37,0.1351,1,"{4087, 4017, 4015, 3801, 215}"
264,1,169,0,0.4136741,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),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)}","\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,"(8*a^2*(19*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 35*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (6*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))","A",4,4,37,0.1081,1,"{4087, 4013, 3809, 3804}"
265,1,219,0,0.5862971,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),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)}","\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,"(2*a^2*(52*A + 63*C)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(136*A + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",5,5,37,0.1351,1,"{4087, 4017, 4015, 3805, 3804}"
266,1,266,0,0.6773109,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),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)}","\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,"(2*a^2*(28*A + 33*C)*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(112*A + 143*C)*Sin[c + d*x])/(385*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(112*A + 143*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(33*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",6,5,37,0.1351,1,"{4087, 4017, 4015, 3805, 3804}"
267,1,312,0,0.897397,"\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","Int[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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{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^{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{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}","\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{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^{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{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,"(a^(5/2)*(1304*A + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1304*A + 1015*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1015*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(136*A + 109*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 23*C)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)","A",8,6,37,0.1622,1,"{4089, 4018, 4016, 3803, 3801, 215}"
268,1,265,0,0.783079,"\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","Int[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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 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}","\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^{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 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^(5/2)*(400*A + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(400*A + 283*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1040*A + 787*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 79*C)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,37,0.1622,1,"{4089, 4018, 4016, 3803, 3801, 215}"
269,1,218,0,0.6774484,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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{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{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}","\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{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{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,"(a^(5/2)*(304*A + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(432*A + 299*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 17*C)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (5*a*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",6,5,37,0.1351,1,"{4089, 4018, 4016, 3801, 215}"
270,1,218,0,0.6581383,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Sec[c + d*x]],x]","\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^{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{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}","\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^{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{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*a^(5/2)*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(24*A - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 31*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (5*a*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",6,5,37,0.1351,1,"{4089, 4018, 4015, 3801, 215}"
271,1,224,0,0.6742659,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(3/2),x]","\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^{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 (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)}}","\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^{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 (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^(5/2)*(8*A + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(56*A - 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 21*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d) - (a*(4*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,5,37,0.1351,1,"{4087, 4018, 4015, 3801, 215}"
272,1,210,0,0.6568142,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(5/2),x]","\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{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{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)}}","\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{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{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,"(5*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*(64*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A - 15*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",6,6,37,0.1622,1,"{4087, 4017, 4018, 4015, 3801, 215}"
273,1,210,0,0.6404944,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(7/2),x]","\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^{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 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)}","\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^{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 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,"(2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(32*A + 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",6,5,37,0.1351,1,"{4087, 4017, 4015, 3801, 215}"
274,1,216,0,0.5004939,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(9/2),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)}","\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,"(64*a^3*(13*A + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) + (2*a*(13*A + 21*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (10*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))","A",5,4,37,0.1081,1,"{4087, 4013, 3809, 3804}"
275,1,266,0,0.7970176,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(11/2),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{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{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{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)}","\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{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{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{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,"(2*a^3*(232*A + 297*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^3*(568*A + 759*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 33*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (10*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",6,5,37,0.1351,1,"{4087, 4017, 4015, 3805, 3804}"
276,1,313,0,0.8650554,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sec[c + d*x]^(13/2),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{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{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{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)}","\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{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{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{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,"(2*a^3*(2224*A + 2717*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(15015*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^3*(8368*A + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 143*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (10*a*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2))","A",7,5,37,0.1351,1,"{4087, 4017, 4015, 3805, 3804}"
277,1,226,0,0.7177705,"\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","Int[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"-((8*A + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((8*A + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - (C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,37,0.1892,1,"{4089, 4021, 4023, 3808, 206, 3801, 215}"
278,1,183,0,0.5459883,"\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","Int[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"((8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4089, 4021, 4023, 3808, 206, 3801, 215}"
279,1,133,0,0.3676086,"\int \frac{\sqrt{\sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}","\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,"-((C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4089, 4023, 3808, 206, 3801, 215}"
280,1,135,0,0.3639273,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","-\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}","-\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,"(2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4087, 4023, 3808, 206, 3801, 215}"
281,1,136,0,0.3371446,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\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}}","\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[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,37,0.1081,1,"{4087, 4013, 3808, 206}"
282,1,181,0,0.4861486,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\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}}","\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,"-((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(13*A + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,37,0.1351,1,"{4087, 4022, 4013, 3808, 206}"
283,1,224,0,0.6746196,"\int \frac{A+C \sec ^2(c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\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}}","-\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,"(Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(43*A + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,37,0.1351,1,"{4087, 4022, 4013, 3808, 206}"
284,1,188,0,0.562369,"\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","Int[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(-3*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((A + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4085, 4021, 4023, 3808, 206, 3801, 215}"
285,1,145,0,0.3917857,"\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","Int[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,6,37,0.1622,1,"{4085, 4023, 3808, 206, 3801, 215}"
286,1,152,0,0.3508368,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","-\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}}","-\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,"-((7*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((5*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,37,0.1081,1,"{4085, 4013, 3808, 206}"
287,1,201,0,0.5149509,"\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","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\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}}","\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,"((11*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((7*A + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((19*A + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,37,0.1351,1,"{4085, 4022, 4013, 3808, 206}"
288,1,248,0,0.7022562,"\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","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\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}}","-\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,"-((15*A + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((9*A + 5*C)*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((13*A + 5*C)*Sin[c + d*x])/(10*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((49*A + 25*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,37,0.1351,1,"{4085, 4022, 4013, 3808, 206}"
289,1,237,0,0.7675416,"\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","Int[(Sec[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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{(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{(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}}","\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{(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{(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,"(-5*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((3*A + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((A - 15*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,8,37,0.2162,1,"{4085, 4019, 4021, 4023, 3808, 206, 3801, 215}"
290,1,192,0,0.5605807,"\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","Int[(Sec[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"(2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A - 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,7,37,0.1892,1,"{4085, 4019, 4023, 3808, 206, 3801, 215}"
291,1,154,0,0.3693461,"\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","Int[(Sqrt[Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"((19*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,37,0.1081,1,"{4085, 4012, 3808, 206}"
292,1,199,0,0.5456135,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\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{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{(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}}","\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{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{(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,"(-5*(15*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,37,0.1351,1,"{4085, 4020, 4013, 3808, 206}"
293,1,246,0,0.7033222,"\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","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\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{(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{(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}}","-\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{(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{(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,"((163*A + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((17*A + C)*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + (5*(19*A + 3*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((299*A + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4085, 4020, 4022, 4013, 3808, 206}"
294,1,295,0,0.9089511,"\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","Int[(A + C*Sec[c + d*x]^2)/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\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{(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{(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}}","\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{(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{(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,"-((283*A + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((21*A + 5*C)*Sin[c + d*x])/(16*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((157*A + 45*C)*Sin[c + d*x])/(80*a^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((787*A + 195*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((2671*A + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,37,0.1622,1,"{4085, 4020, 4022, 4013, 3808, 206}"
295,1,434,0,0.7043327,"\int (a+a \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(2/3)*(A + C*Sec[c + d*x]^2),x]","\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}}}","\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,"(3*C*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (3*Sqrt[2]*A*AppellF1[7/6, 1/2, 1, 13/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]]) + (3*C*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d*(1 + Sec[c + d*x])) - (3^(3/4)*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",10,10,27,0.3704,1,"{4055, 3924, 3779, 3778, 136, 3828, 3827, 50, 63, 225}"
296,1,384,0,0.4339701,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(1/3),x]","\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}}","\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,"(3*C*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(1/3)) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",9,9,27,0.3333,1,"{4055, 3924, 3779, 3778, 136, 3828, 3827, 63, 225}"
297,1,396,0,0.4577202,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{4/3}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(4/3),x]","\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}}","\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,"(-3*(A + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^(4/3)) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(A - 4*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",9,9,27,0.3333,1,"{4053, 3924, 3779, 3778, 136, 3828, 3827, 63, 225}"
298,1,457,0,0.5042445,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{7/3}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(7/3),x]","-\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}}","-\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,"(-3*(A + C)*Tan[c + d*x])/(11*d*(a + a*Sec[c + d*x])^(7/3)) - (3*(4*A - 7*C)*Tan[c + d*x])/(55*a^2*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3*Sqrt[2]*A*AppellF1[-5/6, 1/2, 1, 1/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(5*a^2*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(4*A - 7*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(55*2^(1/3)*a^2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",10,10,27,0.3704,1,"{4053, 3924, 3779, 3778, 136, 3828, 3827, 51, 63, 225}"
299,1,815,0,1.0161554,"\int (a+a \sec (c+d x))^{4/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(4/3)*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(3*a*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*d) + (3*Sqrt[2]*a*A*AppellF1[11/6, 1/2, 1, 17/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(11*d*Sqrt[1 - Sec[c + d*x]]) + (3*C*(a + a*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d) - (15*(1 + Sqrt[3])*a*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (15*2^(1/3)*3^(1/4)*a*C*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (5*3^(3/4)*(1 - Sqrt[3])*a*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",12,12,27,0.4444,1,"{4055, 3924, 3779, 3778, 136, 3828, 3827, 50, 63, 308, 225, 1881}"
300,1,774,0,0.8267505,"\int \sqrt[3]{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(1/3)*(A + C*Sec[c + d*x]^2),x]","\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}}}","\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,"(3*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*3^(1/4)*C*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*C*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",11,11,27,0.4074,1,"{4055, 3924, 3779, 3778, 136, 3828, 3827, 63, 308, 225, 1881}"
301,1,791,0,0.8785525,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(2/3),x]","\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}}}","\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,"(-3*(A + C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3)) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*a*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*(A + 2*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(A + 2*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(A + 2*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",11,11,27,0.4074,1,"{4053, 3924, 3779, 3778, 136, 3828, 3827, 63, 308, 225, 1881}"
302,1,841,0,0.926865,"\int \frac{A+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/3),x]","\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}}","\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,"(-3*(A + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3)) - (3*(2*A - 5*C)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) - (3*Sqrt[2]*A*AppellF1[-1/6, 1/2, 1, 5/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)) - (3*(1 + Sqrt[3])*(2*A - 5*C)*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(2*A - 5*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(2*A - 5*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",12,12,27,0.4444,1,"{4053, 3924, 3779, 3778, 136, 3828, 3827, 51, 63, 308, 225, 1881}"
303,1,244,0,0.5333919,"\int \sec ^m(c+d x) (a+a \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^m*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(C*Sec[c + d*x]^(1 + m)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) + (2^(3/2 + n)*C*n*AppellF1[1/2, 1 - m, -1/2 - n, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n)) + (2^(1/2 + n)*(C*(m - n) + A*(1 + m + n))*AppellF1[1/2, 1 - m, 1/2 - n, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n))","A",8,5,33,0.1515,1,"{4089, 4023, 3828, 3825, 133}"
304,1,253,0,0.5215558,"\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n) - ((C - A*n + C*n)*Hypergeometric2F1[1/2 - n, -n, 1 - n, (-2*Sec[c + d*x])/(1 - Sec[c + d*x])]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + n)*(1 + Sec[c + d*x])) + (2^(3/2 + n)*C*AppellF1[1/2, 1 + n, -1/2 - n, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/d","A",8,6,37,0.1622,1,"{4087, 4023, 3828, 3825, 132, 133}"
305,1,38,0,0.9278606,"\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","Int[((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)+a)^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 + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n)","A",16,6,88,0.06818,1,"{4023, 3828, 3825, 132, 133, 4087}"
306,1,106,0,0.1756297,"\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","Int[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 (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}","\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,"(a*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(B + C)*Tan[c + d*x])/d + (a*(4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*(B + C)*Tan[c + d*x]^3)/(3*d)","A",7,6,38,0.1579,1,"{4072, 3997, 3787, 3768, 3770, 3767}"
307,1,86,0,0.1447685,"\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","Int[Sec[c + d*x]*(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(a*(B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*B + 2*C)*Tan[c + d*x])/(3*d) + (a*(B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,7,36,0.1944,1,"{4072, 3997, 3787, 3767, 8, 3768, 3770}"
308,1,56,0,0.0586209,"\int (a+a \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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*(2*B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,4,30,0.1333,1,"{4048, 3770, 3767, 8}"
309,1,32,0,0.0717028,"\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","Int[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+C) \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\frac{a C \tan (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 + C)*ArcTanh[Sin[c + d*x]])/d + (a*C*Tan[c + d*x])/d","A",5,5,36,0.1389,1,"{4072, 3914, 3767, 8, 3770}"
310,1,32,0,0.0969974,"\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","Int[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+d x)}{d}+a x (B+C)+\frac{a C \tanh ^{-1}(\sin (c+d x))}{d}","\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 + C)*x + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Sin[c + d*x])/d","A",4,3,38,0.07895,1,"{4072, 3996, 3770}"
311,1,47,0,0.1345819,"\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","Int[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 (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)","\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*(B + 2*C)*x)/2 + (a*(B + C)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",5,5,38,0.1316,1,"{4072, 3996, 3787, 2637, 8}"
312,1,77,0,0.1568802,"\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","Int[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 (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)","\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*(B + C)*x)/2 + (a*(2*B + 3*C)*Sin[c + d*x])/(3*d) + (a*(B + C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",6,6,38,0.1579,1,"{4072, 3996, 3787, 2635, 8, 2637}"
313,1,97,0,0.1693429,"\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","Int[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 (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)","-\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*(3*B + 4*C)*x)/8 + (a*(B + C)*Sin[c + d*x])/d + (a*(3*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(B + C)*Sin[c + d*x]^3)/(3*d)","A",7,6,38,0.1579,1,"{4072, 3996, 3787, 2633, 2635, 8}"
314,1,169,0,0.323242,"\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","Int[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 (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}","\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,"(a^2*(7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(10*B + 9*C)*Tan[c + d*x])/(5*d) + (a^2*(7*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a^2*(5*B + 6*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (C*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(5*d) + (a^2*(10*B + 9*C)*Tan[c + d*x]^3)/(15*d)","A",8,7,40,0.1750,1,"{4072, 4018, 3997, 3787, 3768, 3770, 3767}"
315,1,138,0,0.267637,"\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","Int[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 (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}","\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,"(a^2*(8*B + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(8*B + 7*C)*Tan[c + d*x])/(6*d) + (a^2*(8*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*B - C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)","A",8,8,38,0.2105,1,"{4072, 4010, 4001, 3788, 3767, 8, 4046, 3770}"
316,1,103,0,0.1068571,"\int (a+a \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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*(3*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*a^2*(3*B + 2*C)*Tan[c + d*x])/(3*d) + (a^2*(3*B + 2*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",7,7,32,0.2188,1,"{4054, 12, 3788, 3767, 8, 4046, 3770}"
317,1,82,0,0.146151,"\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","Int[Cos[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 (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}","\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*B*x + (a^2*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*B + 3*C)*Tan[c + d*x])/(2*d) + (C*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",6,6,38,0.1579,1,"{4072, 3917, 3914, 3767, 8, 3770}"
318,1,73,0,0.2062693,"\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","Int[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 (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}","\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)*x + (a^2*(B + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(B - C)*Sin[c + d*x])/d + (C*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/d","A",5,4,40,0.1000,1,"{4072, 4018, 3996, 3770}"
319,1,88,0,0.2196438,"\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","Int[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 (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}","\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*(3*B + 4*C)*x)/2 + (a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*B + 2*C)*Sin[c + d*x])/(2*d) + (B*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",5,4,40,0.1000,1,"{4072, 4017, 3996, 3770}"
320,1,102,0,0.2298995,"\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","Int[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{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}","\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*(2*B + 3*C)*x)/2 + (2*a^2*(2*B + 3*C)*Sin[c + d*x])/(3*d) + (a^2*(2*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (B*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)","A",6,6,40,0.1500,1,"{4072, 4013, 3788, 2637, 4045, 8}"
321,1,135,0,0.3068122,"\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","Int[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 (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)","\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*(7*B + 8*C)*x)/8 + (a^2*(4*B + 5*C)*Sin[c + d*x])/(3*d) + (a^2*(7*B + 8*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(5*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(12*d) + (B*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(4*d)","A",7,7,40,0.1750,1,"{4072, 4017, 3996, 3787, 2635, 8, 2637}"
322,1,160,0,0.3291315,"\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","Int[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 (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)","-\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*(6*B + 7*C)*x)/8 + (a^2*(9*B + 10*C)*Sin[c + d*x])/(5*d) + (a^2*(6*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(6*B + 5*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (B*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(5*d) - (a^2*(9*B + 10*C)*Sin[c + d*x]^3)/(15*d)","A",8,7,40,0.1750,1,"{4072, 4017, 3996, 3787, 2633, 2635, 8}"
323,1,163,0,0.312872,"\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","Int[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 (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}","\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,"(a^3*(15*B + 13*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(15*B + 13*C)*Tan[c + d*x])/(5*d) + (3*a^3*(15*B + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + ((5*B - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(15*B + 13*C)*Tan[c + d*x]^3)/(60*d)","A",12,8,38,0.2105,1,"{4072, 4010, 4001, 3791, 3770, 3767, 8, 3768}"
324,1,125,0,0.1394773,"\int (a+a \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(5*a^3*(4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(4*B + 3*C)*Tan[c + d*x])/d + (3*a^3*(4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + (a^3*(4*B + 3*C)*Tan[c + d*x]^3)/(12*d)","A",11,7,32,0.2188,1,"{4054, 12, 3791, 3770, 3767, 8, 3768}"
325,1,111,0,0.2048055,"\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","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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 + (a^3*(7*B + 5*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(B + C)*Tan[c + d*x])/(2*d) + (a*C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((3*B + 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)","A",7,6,38,0.1579,1,"{4072, 3917, 3914, 3767, 8, 3770}"
326,1,108,0,0.3117844,"\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","Int[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 (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}","\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*(3*B + C)*x + (a^3*(6*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) - (5*a^3*C*Sin[c + d*x])/(2*d) + (a*C*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((B + 2*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/d","A",6,4,40,0.1000,1,"{4072, 4018, 3996, 3770}"
327,1,117,0,0.3345133,"\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","Int[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{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}","\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*(7*B + 6*C)*x)/2 + (a^3*(B + 3*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*B*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((B - 2*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",6,5,40,0.1250,1,"{4072, 4017, 4018, 3996, 3770}"
328,1,125,0,0.3392679,"\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","Int[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{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}","\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*(5*B + 7*C)*x)/2 + (a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(B + C)*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + ((5*B + 3*C)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",6,4,40,0.1000,1,"{4072, 4017, 3996, 3770}"
329,1,124,0,0.251688,"\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","Int[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 (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}","-\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,"(5*a^3*(3*B + 4*C)*x)/8 + (a^3*(3*B + 4*C)*Sin[c + d*x])/d + (3*a^3*(3*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (B*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) - (a^3*(3*B + 4*C)*Sin[c + d*x]^3)/(12*d)","A",9,7,40,0.1750,1,"{4072, 4013, 3791, 2637, 2635, 8, 2633}"
330,1,176,0,0.4466181,"\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","Int[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 (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}","\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*(13*B + 15*C)*x)/8 + (a^3*(38*B + 45*C)*Sin[c + d*x])/(15*d) + (a^3*(13*B + 15*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^3*(43*B + 45*C)*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + (a*B*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((7*B + 5*C)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(20*d)","A",8,7,40,0.1750,1,"{4072, 4017, 3996, 3787, 2635, 8, 2637}"
331,1,201,0,0.4792425,"\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","Int[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 (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}","-\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*(23*B + 26*C)*x)/16 + (a^3*(17*B + 19*C)*Sin[c + d*x])/(5*d) + (a^3*(23*B + 26*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(21*B + 22*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((4*B + 3*C)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(17*B + 19*C)*Sin[c + d*x]^3)/(15*d)","A",9,7,40,0.1750,1,"{4072, 4017, 3996, 3787, 2633, 2635, 8}"
332,1,131,0,0.2533344,"\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","Int[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(3*(B - C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((3*B - 4*C)*Tan[c + d*x])/(a*d) + (3*(B - C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*B - 4*C)*Tan[c + d*x]^3)/(3*a*d)","A",7,6,40,0.1500,1,"{4072, 4019, 3787, 3768, 3770, 3767}"
333,1,108,0,0.2398667,"\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","Int[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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,"-((2*B - 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) + (2*(B - C)*Tan[c + d*x])/(a*d) - ((2*B - 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,7,40,0.1750,1,"{4072, 4019, 3787, 3767, 8, 3768, 3770}"
334,1,62,0,0.1658126,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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,"((B - C)*ArcTanh[Sin[c + d*x]])/(a*d) + (C*Tan[c + d*x])/(a*d) - ((B - C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,6,38,0.1579,1,"{4072, 4008, 3787, 3770, 3767, 8}"
335,1,44,0,0.0728113,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]),x]","\frac{(B-C) \tan (c+d x)}{a d (\sec (c+d x)+1)}+\frac{C \tanh ^{-1}(\sin (c+d x))}{a d}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a*d) + ((B - C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",4,4,32,0.1250,1,"{4050, 3770, 12, 3794}"
336,1,35,0,0.1316121,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\frac{B x}{a}-\frac{(B-C) \tan (c+d x)}{d (a \sec (c+d x)+a)}","\frac{B x}{a}-\frac{(B-C) \tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"(B*x)/a - ((B - C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",3,3,38,0.07895,1,"{4072, 3919, 3794}"
337,1,60,0,0.1958948,"\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","Int[(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 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}","\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,"-(((B - C)*x)/a) + ((2*B - C)*Sin[c + d*x])/(a*d) - ((B - C)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",5,5,40,0.1250,1,"{4072, 4020, 3787, 2637, 8}"
338,1,98,0,0.2317414,"\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","Int[(Cos[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"((3*B - 2*C)*x)/(2*a) - (2*(B - C)*Sin[c + d*x])/(a*d) + ((3*B - 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,6,40,0.1500,1,"{4072, 4020, 3787, 2635, 8, 2637}"
339,1,122,0,0.2412749,"\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","Int[(Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(-3*(B - C)*x)/(2*a) + ((4*B - 3*C)*Sin[c + d*x])/(a*d) - (3*(B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*B - 3*C)*Sin[c + d*x]^3)/(3*a*d)","A",7,6,40,0.1500,1,"{4072, 4020, 3787, 2633, 2635, 8}"
340,1,156,0,0.3806795,"\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","Int[(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{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}","\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,"-((4*B - 7*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) + (2*(5*B - 8*C)*Tan[c + d*x])/(3*a^2*d) - ((4*B - 7*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) + ((5*B - 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,7,40,0.1750,1,"{4072, 4019, 3787, 3767, 8, 3768, 3770}"
341,1,108,0,0.3356953,"\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","Int[(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{(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}","-\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,"((B - 2*C)*ArcTanh[Sin[c + d*x]])/(a^2*d) - ((B - 4*C)*Tan[c + d*x])/(3*a^2*d) - ((B - 2*C)*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,7,40,0.1750,1,"{4072, 4019, 4008, 3787, 3770, 3767, 8}"
342,1,79,0,0.2331213,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((2*B - 5*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",5,5,38,0.1316,1,"{4072, 4008, 3998, 3770, 3794}"
343,1,62,0,0.073853,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2,x]","\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}","\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,"((B + 2*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) + ((B - C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",3,3,32,0.09375,1,"{4052, 12, 3794}"
344,1,70,0,0.1776131,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"(B*x)/a^2 - ((4*B - C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",4,4,38,0.1053,1,"{4072, 3922, 3919, 3794}"
345,1,98,0,0.3151748,"\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","Int[(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{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}","\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,"-(((2*B - C)*x)/a^2) + (2*(5*B - 2*C)*Sin[c + d*x])/(3*a^2*d) - ((2*B - C)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,5,40,0.1250,1,"{4072, 4020, 3787, 2637, 8}"
346,1,143,0,0.3779559,"\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","Int[(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{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}","-\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,"((7*B - 4*C)*x)/(2*a^2) - (2*(8*B - 5*C)*Sin[c + d*x])/(3*a^2*d) + ((7*B - 4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((8*B - 5*C)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,6,40,0.1500,1,"{4072, 4020, 3787, 2635, 8, 2637}"
347,1,170,0,0.4016847,"\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","Int[(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{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}","-\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,"-((10*B - 7*C)*x)/(2*a^2) + (4*(3*B - 2*C)*Sin[c + d*x])/(a^2*d) - ((10*B - 7*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((10*B - 7*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((B - C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - (4*(3*B - 2*C)*Sin[c + d*x]^3)/(3*a^2*d)","A",8,6,40,0.1500,1,"{4072, 4020, 3787, 2633, 2635, 8}"
348,1,202,0,0.552848,"\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","Int[(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{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}","\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,"-((6*B - 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) + (8*(9*B - 19*C)*Tan[c + d*x])/(15*a^3*d) - ((6*B - 13*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) + ((B - C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((6*B - 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + (4*(9*B - 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",9,7,40,0.1750,1,"{4072, 4019, 3787, 3767, 8, 3768, 3770}"
349,1,156,0,0.4983979,"\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","Int[(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{(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}","-\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,"((B - 3*C)*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((7*B - 27*C)*Tan[c + d*x])/(15*a^3*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((4*B - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((B - 3*C)*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",8,7,40,0.1750,1,"{4072, 4019, 4008, 3787, 3770, 3767, 8}"
350,1,125,0,0.403038,"\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","Int[(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{(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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((2*B - 7*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((4*B - 29*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",6,6,40,0.1500,1,"{4072, 4019, 4008, 3998, 3770, 3794}"
351,1,102,0,0.2509115,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"-((B - C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((3*B - 8*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*B + 7*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",4,4,38,0.1053,1,"{4072, 4008, 4000, 3794}"
352,1,102,0,0.1095761,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3,x]","\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}","\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,"((B - C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*B + 3*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((2*B + 3*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",4,4,32,0.1250,1,"{4052, 12, 3796, 3794}"
353,1,108,0,0.2522864,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"(B*x)/a^3 - ((B - C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*B - 2*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (2*(11*B - C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",5,4,38,0.1053,1,"{4072, 3922, 3919, 3794}"
354,1,136,0,0.4435101,"\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","Int[(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{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}","\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,"-(((3*B - C)*x)/a^3) + (2*(36*B - 11*C)*Sin[c + d*x])/(15*a^3*d) - ((B - C)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*B - 4*C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*B - C)*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",7,5,40,0.1250,1,"{4072, 4020, 3787, 2637, 8}"
355,1,187,0,0.5452471,"\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","Int[(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{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}","-\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,"((13*B - 6*C)*x)/(2*a^3) - (8*(19*B - 9*C)*Sin[c + d*x])/(15*a^3*d) + ((13*B - 6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*B - 6*C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - (4*(19*B - 9*C)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,40,0.1500,1,"{4072, 4020, 3787, 2635, 8, 2637}"
356,1,230,0,0.4852663,"\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","Int[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 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}}","\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,"(32*a*(11*B + 10*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(11*B + 10*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(11*B + 10*C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^5*Tan[c + d*x])/(11*d*Sqrt[a + a*Sec[c + d*x]]) - (64*(11*B + 10*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (32*(11*B + 10*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*a*d)","A",7,6,42,0.1429,1,"{4072, 4016, 3803, 3800, 4001, 3792}"
357,1,187,0,0.4188512,"\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","Int[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 (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}}","\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,"(4*a*(9*B + 8*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(9*B + 8*C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(9*B + 8*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (4*(9*B + 8*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)","A",6,6,42,0.1429,1,"{4072, 4016, 3803, 3800, 4001, 3792}"
358,1,144,0,0.3593722,"\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","Int[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 (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}}","\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*(7*B + 6*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(7*B + 6*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*B + 6*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)","A",5,5,42,0.1190,1,"{4072, 4016, 3800, 4001, 3792}"
359,1,101,0,0.2769745,"\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","Int[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 (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}","\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*a*(5*B + 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)","A",4,4,40,0.1000,1,"{4072, 4010, 4001, 3792}"
360,1,62,0,0.0842717,"\int \sqrt{a+a \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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 + C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",3,3,34,0.08824,1,"{4054, 12, 3792}"
361,1,66,0,0.1710621,"\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","Int[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}}","\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,"(2*Sqrt[a]*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,40,0.1250,1,"{4072, 3915, 3774, 203, 3792}"
362,1,68,0,0.2097681,"\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","Int[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{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}}","\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[a]*(B + 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*B*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,42,0.09524,1,"{4072, 4015, 3774, 203}"
363,1,117,0,0.2813409,"\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","Int[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{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}}","\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,"(Sqrt[a]*(3*B + 4*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(3*B + 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,42,0.1190,1,"{4072, 4015, 3805, 3774, 203}"
364,1,160,0,0.3361695,"\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","Int[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{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}}","\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,"(Sqrt[a]*(5*B + 6*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*B + 6*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(5*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,42,0.1190,1,"{4072, 4015, 3805, 3774, 203}"
365,1,234,0,0.637845,"\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","Int[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 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}","\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,"(4*a^2*(187*B + 168*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(187*B + 168*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(11*B + 12*C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (8*a*(187*B + 168*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a*C*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (4*(187*B + 168*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d)","A",7,7,42,0.1667,1,"{4072, 4018, 4016, 3803, 3800, 4001, 3792}"
366,1,189,0,0.5643492,"\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","Int[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 (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}","\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*(39*B + 34*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(9*B + 10*C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(39*B + 34*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(39*B + 34*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d)","A",6,6,42,0.1429,1,"{4072, 4018, 4016, 3800, 4001, 3792}"
367,1,138,0,0.3533511,"\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","Int[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{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}","\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,"(8*a^2*(21*B + 19*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(21*B + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)","A",5,5,40,0.1250,1,"{4072, 4010, 4001, 3793, 3792}"
368,1,101,0,0.1290319,"\int (a+a \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(8*a^2*(5*B + 3*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(5*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",4,4,34,0.1176,1,"{4054, 12, 3793, 3792}"
369,1,105,0,0.2427678,"\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","Int[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{2 a^2 (3 B+4 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\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 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 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^{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 C \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a^(3/2)*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(3*B + 4*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,40,0.1500,1,"{4072, 3917, 3915, 3774, 203, 3792}"
370,1,103,0,0.3566274,"\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","Int[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^2 (B-2 C) \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\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{2 a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}","\frac{a^2 (B-2 C) \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\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{2 a C \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(a^(3/2)*(3*B + 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(B - 2*C)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d","A",5,5,42,0.1190,1,"{4072, 4018, 4015, 3774, 203}"
371,1,119,0,0.3776369,"\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","Int[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^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 B \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}","\frac{a^2 (5 B+4 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 B \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(a^(3/2)*(7*B + 12*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*B + 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",5,5,42,0.1190,1,"{4072, 4017, 4015, 3774, 203}"
372,1,164,0,0.4703706,"\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","Int[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]","\frac{a^2 (11 B+14 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (11 B+14 C) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(11*B + 14*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(11*B + 14*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(7*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,42,0.1429,1,"{4072, 4017, 4015, 3805, 3774, 203}"
373,1,209,0,0.5575012,"\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","Int[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]","\frac{a^2 (75 B+88 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (75 B+88 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(75*B + 88*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*B + 88*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(75*B + 88*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(9*B + 8*C)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*B*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",7,6,42,0.1429,1,"{4072, 4017, 4015, 3805, 3774, 203}"
374,1,282,0,0.8395593,"\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","Int[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^2 (13 B+16 C) \tan (c+d x) \sec ^4(c+d x) \sqrt{a \sec (c+d x)+a}}{143 d}+\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{8 a^2 (4615 B+4184 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{45045 d}+\frac{4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}+\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}","\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{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{8 a^2 (4615 B+4184 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{45045 d}+\frac{4 a^3 (4615 B+4184 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}+\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,"(4*a^3*(4615*B + 4184*C)*Tan[c + d*x])/(6435*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(4615*B + 4184*C)*Sec[c + d*x]^3*Tan[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(299*B + 280*C)*Sec[c + d*x]^4*Tan[c + d*x])/(1287*d*Sqrt[a + a*Sec[c + d*x]]) - (8*a^2*(4615*B + 4184*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(45045*d) + (2*a^2*(13*B + 16*C)*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(143*d) + (4*a*(4615*B + 4184*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(15015*d) + (2*a*C*Sec[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(13*d)","A",8,7,42,0.1667,1,"{4072, 4018, 4016, 3803, 3800, 4001, 3792}"
375,1,237,0,0.7586717,"\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","Int[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 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^2 (11 B+14 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 a^3 (803 B+710 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}-\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}","\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^2 (11 B+14 C) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}+\frac{2 a^3 (803 B+710 C) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}-\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,"(2*a^3*(803*B + 710*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(209*B + 194*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(803*B + 710*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a^2*(11*B + 14*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*a*(803*B + 710*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*a*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)","A",7,6,42,0.1429,1,"{4072, 4018, 4016, 3800, 4001, 3792}"
376,1,175,0,0.4097843,"\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","Int[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{16 a^2 (15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (15 B+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\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}","\frac{16 a^2 (15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{64 a^3 (15 B+13 C) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\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,"(64*a^3*(15*B + 13*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(15*B + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(15*B + 13*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)","A",6,5,40,0.1250,1,"{4072, 4010, 4001, 3793, 3792}"
377,1,138,0,0.1745036,"\int (a+a \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(64*a^3*(7*B + 5*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(7*B + 5*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*a*(7*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",5,4,34,0.1176,1,"{4054, 12, 3793, 3792}"
378,1,142,0,0.3172877,"\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","Int[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{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^{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 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 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^{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 C \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(2*a^(5/2)*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(35*B + 32*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(5*B + 8*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*a*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",7,6,40,0.1500,1,"{4072, 3917, 3915, 3774, 203, 3792}"
379,1,143,0,0.5153744,"\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","Int[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^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{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{2 a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 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{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{2 a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^(5/2)*(5*B + 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d - (a^3*(3*B + 14*C)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(B + 2*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",6,5,42,0.1190,1,"{4072, 4018, 4015, 3774, 203}"
380,1,154,0,0.5285588,"\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","Int[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^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^{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 B \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 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^{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 B \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a^(5/2)*(19*B + 20*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(9*B - 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(B - 4*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",6,6,42,0.1429,1,"{4072, 4017, 4018, 4015, 3774, 203}"
381,1,164,0,0.5665146,"\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","Int[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^3 (49 B+54 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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^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}","\frac{a^3 (49 B+54 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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^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^(5/2)*(25*B + 38*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*B + 54*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(3*B + 2*C)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",6,5,42,0.1190,1,"{4072, 4017, 4015, 3774, 203}"
382,1,209,0,0.6669672,"\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","Int[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^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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^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^3 (95 B+104 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) (a \sec (c+d x)+a)^{3/2}}{4 d}","\frac{a^3 (163 B+200 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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^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^3 (95 B+104 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) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a^(5/2)*(163*B + 200*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(163*B + 200*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(95*B + 104*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(11*B + 8*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",7,6,42,0.1429,1,"{4072, 4017, 4015, 3805, 3774, 203}"
383,1,254,0,0.7494177,"\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","Int[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^3 (283 B+326 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}","\frac{a^3 (283 B+326 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^(5/2)*(283*B + 326*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*B + 326*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(283*B + 326*C)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(157*B + 170*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(13*B + 10*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (a*B*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",8,6,42,0.1429,1,"{4072, 4017, 4015, 3805, 3774, 203}"
384,1,243,0,0.8761972,"\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","Int[(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{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}}","\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,"(Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(111*B - 143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(3*B - 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(9*B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(93*B - 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*a*d)","A",8,6,42,0.1429,1,"{4072, 4021, 4010, 4001, 3795, 203}"
385,1,202,0,0.6939536,"\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","Int[(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{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}}","\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,"-((Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(49*B - 37*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(7*B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(7*B - 31*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)","A",7,6,42,0.1429,1,"{4072, 4021, 4010, 4001, 3795, 203}"
386,1,159,0,0.5143661,"\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","Int[(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{\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}}","\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,"(Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(5*B - 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)","A",6,6,42,0.1429,1,"{4072, 4021, 4010, 4001, 3795, 203}"
387,1,118,0,0.3037582,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}","-\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,"-((Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(3*B - 2*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)","A",5,5,40,0.1250,1,"{4072, 4010, 4001, 3795, 203}"
388,1,78,0,0.0949332,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,34,0.1176,1,"{4054, 12, 3795, 203}"
389,1,91,0,0.186105,"\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","Int[(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 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}","\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*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)","A",6,5,40,0.1250,1,"{4072, 3920, 3774, 203, 3795}"
390,1,119,0,0.3200733,"\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","Int[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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,"-(((B - 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (B*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,42,0.1429,1,"{4072, 4022, 3920, 3774, 203, 3795}"
391,1,165,0,0.4708531,"\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","Int[(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{(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}}","-\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)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((B - 4*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,42,0.1429,1,"{4072, 4022, 3920, 3774, 203, 3795}"
392,1,206,0,0.6454785,"\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","Int[(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{(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}}","\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,"-((9*B - 14*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*B - 2*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - ((B - 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,42,0.1429,1,"{4072, 4022, 3920, 3774, 203, 3795}"
393,1,261,0,0.9191547,"\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","Int[(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{(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}}","-\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,"-((15*B - 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Sec[c + d*x]^4*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((651*B - 799*C)*Tan[c + d*x])/(105*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((63*B - 67*C)*Sec[c + d*x]^2*Tan[c + d*x])/(70*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((7*B - 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(14*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((273*B - 397*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(210*a^2*d)","A",8,7,42,0.1667,1,"{4072, 4019, 4021, 4010, 4001, 3795, 203}"
394,1,216,0,0.7251426,"\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","Int[(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{(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}}","\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,"((11*B - 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((65*B - 93*C)*Tan[c + d*x])/(15*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((5*B - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((35*B - 39*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(30*a^2*d)","A",7,7,42,0.1667,1,"{4072, 4019, 4021, 4010, 4001, 3795, 203}"
395,1,171,0,0.5559128,"\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","Int[(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{(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}}","-\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,"-((7*B - 11*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((9*B - 13*C)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((3*B - 7*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)","A",6,6,42,0.1429,1,"{4072, 4019, 4010, 4001, 3795, 203}"
396,1,118,0,0.3193766,"\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","Int[(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{(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}}","\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,"((3*B - 7*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + (2*C*Tan[c + d*x])/(a*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,40,0.1250,1,"{4072, 4008, 4001, 3795, 203}"
397,1,87,0,0.1057511,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((B + 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) + ((B - C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,34,0.1176,1,"{4052, 12, 3795, 203}"
398,1,127,0,0.2696562,"\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","Int[(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{(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}}","-\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,"(2*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*B - C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",7,6,40,0.1500,1,"{4072, 3922, 3920, 3774, 203, 3795}"
399,1,170,0,0.4939353,"\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","Int[(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]","-\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}}","-\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,"-(((3*B - 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + ((9*B - 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*B - C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,42,0.1667,1,"{4072, 4020, 4022, 3920, 3774, 203, 3795}"
400,1,221,0,0.6852496,"\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","Int[(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{(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}}","\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,"((19*B - 12*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*B - 9*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*B - 6*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*B - C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",9,7,42,0.1667,1,"{4072, 4020, 4022, 3920, 3774, 203, 3795}"
401,1,261,0,0.9281382,"\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","Int[(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{(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{(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{(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}}","-\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{(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{(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,"((163*B - 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((13*B - 21*C)*Sec[c + d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((985*B - 1729*C)*Tan[c + d*x])/(120*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((85*B - 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((475*B - 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)","A",8,7,42,0.1667,1,"{4072, 4019, 4021, 4010, 4001, 3795, 203}"
402,1,216,0,0.7479352,"\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","Int[(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{(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}}","-\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,"-((75*B - 163*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((9*B - 17*C)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((93*B - 197*C)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((39*B - 95*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)","A",7,6,42,0.1429,1,"{4072, 4019, 4010, 4001, 3795, 203}"
403,1,169,0,0.5643633,"\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","Int[(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{(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}}","\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,"((19*B - 75*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*B - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((B - 9*C)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,42,0.1429,1,"{4072, 4019, 4008, 4001, 3795, 203}"
404,1,126,0,0.3344199,"\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","Int[(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{(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}}","\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,"((5*B + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*B - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",5,5,40,0.1250,1,"{4072, 4008, 4000, 3795, 203}"
405,1,126,0,0.1516744,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2),x]","\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}}","\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,"((3*B + 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) + ((B - C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((3*B + 5*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",5,5,34,0.1471,1,"{4052, 12, 3796, 3795, 203}"
406,1,164,0,0.3460063,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"(2*B*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*B - 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*B - 3*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",8,6,40,0.1500,1,"{4072, 3922, 3920, 3774, 203, 3795}"
407,1,207,0,0.6720904,"\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","Int[(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]","\frac{(35 B-11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\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{(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}}","\frac{(35 B-11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\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{(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,"-(((5*B - 2*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + ((115*B - 43*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((B - C)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*B - 7*C)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*B - 11*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",9,7,42,0.1667,1,"{4072, 4020, 4022, 3920, 3774, 203, 3795}"
408,1,152,0,0.2130463,"\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","Int[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 (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}","\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*(4*A + 3*(B + C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(5*A + 5*B + 4*C)*Tan[c + d*x])/(5*d) + (a*(4*A + 3*(B + C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (a*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (a*(5*A + 5*B + 4*C)*Tan[c + d*x]^3)/(15*d)","A",7,6,39,0.1538,1,"{4076, 4047, 3767, 4046, 3768, 3770}"
409,1,127,0,0.190692,"\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","Int[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 (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}","\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*(4*A + 4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(3*A + 2*(B + C))*Tan[c + d*x])/(3*d) + (a*(4*A + 4*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*(B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,39,0.1795,1,"{4076, 4047, 3768, 3770, 4046, 3767, 8}"
410,1,92,0,0.1202092,"\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","Int[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 (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}","\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*(2*A + B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(3*A + 3*B + 2*C)*Tan[c + d*x])/(3*d) + (a*(B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,37,0.1622,1,"{4076, 4047, 3767, 8, 4046, 3770}"
411,1,63,0,0.0642943,"\int (a+a \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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*A*x + (a*(2*A + 2*B + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*(B + C)*Tan[c + d*x])/d + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,4,31,0.1290,1,"{4048, 3770, 3767, 8}"
412,1,46,0,0.1162039,"\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","Int[Cos[c + d*x]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","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}","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 + B)*x + (a*(B + C)*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (a*C*Tan[c + d*x])/d","A",5,5,37,0.1351,1,"{4076, 4047, 8, 4045, 3770}"
413,1,62,0,0.1497711,"\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","Int[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 (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}","\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*(A + 2*(B + C))*x)/2 + (a*C*ArcTanh[Sin[c + d*x]])/d + (a*(A + B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",5,5,39,0.1282,1,"{4074, 4047, 8, 4045, 3770}"
414,1,82,0,0.17611,"\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","Int[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 (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}","\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*(A + B + 2*C)*x)/2 + (a*(2*A + 3*(B + C))*Sin[c + d*x])/(3*d) + (a*(A + B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",5,5,39,0.1282,1,"{4074, 4047, 2637, 4045, 8}"
415,1,102,0,0.2108593,"\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","Int[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 (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}","\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*(3*A + 4*(B + C))*x)/8 + (a*(A + B + C)*Sin[c + d*x])/d + (a*(3*A + 4*(B + C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(A + B)*Sin[c + d*x]^3)/(3*d)","A",7,6,39,0.1538,1,"{4074, 4047, 2635, 8, 4044, 3013}"
416,1,141,0,0.225113,"\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","Int[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 (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}","-\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*(3*(A + B) + 4*C)*x)/8 + (a*(4*A + 5*(B + C))*Sin[c + d*x])/(5*d) + (a*(3*(A + B) + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*(B + C))*Sin[c + d*x]^3)/(15*d)","A",7,6,39,0.1538,1,"{4074, 4047, 2633, 4045, 2635, 8}"
417,1,222,0,0.4297617,"\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","Int[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 (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}","\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,"(a^2*(14*A + 12*B + 11*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^2*(10*A + 9*B + 8*C)*Tan[c + d*x])/(5*d) + (a^2*(14*A + 12*B + 11*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^2*(10*A + 12*B + 9*C)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + (C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(6*d) + ((3*B + C)*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(15*d) + (a^2*(10*A + 9*B + 8*C)*Tan[c + d*x]^3)/(15*d)","A",8,7,41,0.1707,1,"{4088, 4018, 3997, 3787, 3768, 3770, 3767}"
418,1,190,0,0.4101203,"\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","Int[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 (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}","\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,"(a^2*(8*A + 7*B + 6*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(8*A + 7*B + 6*C)*Tan[c + d*x])/(6*d) + (a^2*(8*A + 7*B + 6*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((20*A - 5*B + 6*C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(5*d) + ((5*B + 2*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*a*d)","A",8,8,41,0.1951,1,"{4088, 4010, 4001, 3788, 3767, 8, 4046, 3770}"
419,1,147,0,0.2348713,"\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","Int[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 (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}","\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,"(a^2*(12*A + 8*B + 7*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^2*(12*A + 8*B + 7*C)*Tan[c + d*x])/(6*d) + (a^2*(12*A + 8*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*B - C)*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*a*d)","A",7,7,39,0.1795,1,"{4082, 4001, 3788, 3767, 8, 4046, 3770}"
420,1,120,0,0.1572219,"\int (a+a \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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*A*x + (a^2*(4*A + 3*B + 2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a^2*(2*A + 3*B + 2*C)*Tan[c + d*x])/(2*d) + (C*(a + a*Sec[c + d*x])^2*Tan[c + d*x])/(3*d) + ((3*B + 2*C)*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(6*d)","A",6,6,33,0.1818,1,"{4054, 3917, 3914, 3767, 8, 3770}"
421,1,121,0,0.216887,"\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","Int[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 (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}","-\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*(2*A + B)*x + (a^2*(2*A + 4*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/d - (a^2*(2*A - 2*B - 3*C)*Tan[c + d*x])/(2*d) - ((2*A - C)*(a^2 + a^2*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",6,6,39,0.1538,1,"{4086, 3917, 3914, 3767, 8, 3770}"
422,1,128,0,0.287309,"\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","Int[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 (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}","\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*(3*A + 4*B + 2*C)*x)/2 + (a^2*(B + 2*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(3*A + 2*B - 2*C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((A - 2*C)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",5,4,41,0.09756,1,"{4086, 4018, 3996, 3770}"
423,1,134,0,0.2917589,"\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","Int[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 (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}","\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*(2*A + 3*B + 4*C)*x)/2 + (a^2*C*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*A + 3*B + 2*C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + ((2*A + 3*B)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",5,4,41,0.09756,1,"{4086, 4017, 3996, 3770}"
424,1,149,0,0.3278044,"\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","Int[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 (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}","\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*(7*A + 8*B + 12*C)*x)/8 + (a^2*(7*A + 8*B + 12*C)*Sin[c + d*x])/(6*d) + (a^2*(7*A + 8*B + 12*C)*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((A + 2*B)*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)","A",6,6,41,0.1463,1,"{4086, 4013, 3788, 2637, 4045, 8}"
425,1,187,0,0.414596,"\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","Int[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 (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}","\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*(6*A + 7*B + 8*C)*x)/8 + (a^2*(18*A + 20*B + 25*C)*Sin[c + d*x])/(15*d) + (a^2*(6*A + 7*B + 8*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(18*A + 25*B + 20*C)*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ((2*A + 5*B)*Cos[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(20*d)","A",7,7,41,0.1707,1,"{4086, 4017, 3996, 3787, 2635, 8, 2637}"
426,1,213,0,0.4410334,"\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","Int[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 (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}","-\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*(11*A + 12*B + 14*C)*x)/16 + (a^2*(8*A + 9*B + 10*C)*Sin[c + d*x])/(5*d) + (a^2*(11*A + 12*B + 14*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^2*(9*A + 12*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((A + 3*B)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^2*(8*A + 9*B + 10*C)*Sin[c + d*x]^3)/(15*d)","A",8,7,41,0.1707,1,"{4086, 4017, 3996, 3787, 2633, 2635, 8}"
427,1,274,0,0.5980194,"\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","Int[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 (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}","\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,"(a^3*(26*A + 23*B + 21*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(133*A + 119*B + 108*C)*Tan[c + d*x])/(35*d) + (a^3*(26*A + 23*B + 21*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (a^3*(154*A + 147*B + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(280*d) + (C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(7*d) + ((7*B + 3*C)*Sec[c + d*x]^3*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(42*a*d) + ((3*A + 4*B + 3*C)*Sec[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(15*d) + (a^3*(133*A + 119*B + 108*C)*Tan[c + d*x]^3)/(105*d)","A",9,7,41,0.1707,1,"{4088, 4018, 3997, 3787, 3768, 3770, 3767}"
428,1,216,0,0.4551225,"\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","Int[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 (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}","\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,"(a^3*(30*A + 26*B + 23*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (a^3*(30*A + 26*B + 23*C)*Tan[c + d*x])/(10*d) + (3*a^3*(30*A + 26*B + 23*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + ((30*A - 6*B + 7*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(120*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(6*d) + ((2*B + C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(10*a*d) + (a^3*(30*A + 26*B + 23*C)*Tan[c + d*x]^3)/(120*d)","A",12,8,41,0.1951,1,"{4088, 4010, 4001, 3791, 3770, 3767, 8, 3768}"
429,1,175,0,0.2765642,"\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","Int[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 (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}","\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,"(a^3*(20*A + 15*B + 13*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^3*(20*A + 15*B + 13*C)*Tan[c + d*x])/(5*d) + (3*a^3*(20*A + 15*B + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(40*d) + ((5*B - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*a*d) + (a^3*(20*A + 15*B + 13*C)*Tan[c + d*x]^3)/(60*d)","A",11,7,39,0.1795,1,"{4082, 4001, 3791, 3770, 3767, 8, 3768}"
430,1,162,0,0.2372571,"\int (a+a \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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*A*x + (a^3*(28*A + 20*B + 15*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (5*a^3*(4*A + 4*B + 3*C)*Tan[c + d*x])/(8*d) + (C*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) + ((4*B + 3*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*a*d) + ((12*A + 20*B + 15*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(24*d)","A",7,6,33,0.1818,1,"{4054, 3917, 3914, 3767, 8, 3770}"
431,1,156,0,0.2843269,"\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","Int[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{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{(3 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 a d}+\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}","\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{(3 A-C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{3 a d}+\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d}",1,"a^3*(3*A + B)*x + (a^3*(6*A + 7*B + 5*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/d + (5*a^3*(B + C)*Tan[c + d*x])/(2*d) - ((3*A - C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(3*a*d) - ((6*A - 3*B - 5*C)*(a^3 + a^3*Sec[c + d*x])*Tan[c + d*x])/(6*d)","A",7,6,39,0.1538,1,"{4086, 3917, 3914, 3767, 8, 3770}"
432,1,171,0,0.4254455,"\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","Int[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 (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}","\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*(7*A + 6*B + 2*C)*x)/2 + (a^3*(2*A + 6*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^3*(A - C)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - ((A - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((A - 2*B - 4*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(2*d)","A",6,4,41,0.09756,1,"{4086, 4018, 3996, 3770}"
433,1,169,0,0.4415586,"\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","Int[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{(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{(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{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 \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}","-\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{(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{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 \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"(a^3*(5*A + 7*B + 6*C)*x)/2 + (a^3*(B + 3*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(A + B)*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + ((A + B)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*a*d) - ((5*A + 3*B - 6*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",6,5,41,0.1220,1,"{4086, 4017, 4018, 3996, 3770}"
434,1,183,0,0.4412209,"\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","Int[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{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{(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{1}{8} a^3 x (15 A+20 B+28 C)+\frac{a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 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{(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{1}{8} a^3 x (15 A+20 B+28 C)+\frac{a^3 C \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(15*A + 20*B + 28*C)*x)/8 + (a^3*C*ArcTanh[Sin[c + d*x]])/d + (5*a^3*(3*A + 4*(B + C))*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(4*d) + ((3*A + 4*B)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(12*a*d) + ((15*A + 20*B + 12*C)*Cos[c + d*x]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(24*d)","A",6,4,41,0.09756,1,"{4086, 4017, 3996, 3770}"
435,1,179,0,0.365264,"\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","Int[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 (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}","-\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*(13*A + 15*B + 20*C)*x)/8 + (a^3*(13*A + 15*B + 20*C)*Sin[c + d*x])/(5*d) + (3*a^3*(13*A + 15*B + 20*C)*Cos[c + d*x]*Sin[c + d*x])/(40*d) + ((3*A + 5*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) - (a^3*(13*A + 15*B + 20*C)*Sin[c + d*x]^3)/(60*d)","A",9,7,41,0.1707,1,"{4086, 4013, 3791, 2637, 2635, 8, 2633}"
436,1,235,0,0.5910146,"\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","Int[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 (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{(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{1}{16} a^3 x (23 A+26 B+30 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 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{(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{1}{16} a^3 x (23 A+26 B+30 C)+\frac{A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}",1,"(a^3*(23*A + 26*B + 30*C)*x)/16 + (a^3*(34*A + 38*B + 45*C)*Sin[c + d*x])/(15*d) + (a^3*(23*A + 26*B + 30*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(73*A + 86*B + 90*C)*Cos[c + d*x]^2*Sin[c + d*x])/(120*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((A + 2*B)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(10*a*d) + ((31*A + 42*B + 30*C)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(120*d)","A",8,7,41,0.1707,1,"{4086, 4017, 3996, 3787, 2635, 8, 2637}"
437,1,265,0,0.6106141,"\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","Int[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 (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{(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{1}{16} a^3 x (21 A+23 B+26 C)+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 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{(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{1}{16} a^3 x (21 A+23 B+26 C)+\frac{A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}",1,"(a^3*(21*A + 23*B + 26*C)*x)/16 + (a^3*(108*A + 119*B + 133*C)*Sin[c + d*x])/(35*d) + (a^3*(21*A + 23*B + 26*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^3*(129*A + 147*B + 154*C)*Cos[c + d*x]^3*Sin[c + d*x])/(280*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + ((3*A + 7*B)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(42*a*d) + ((3*A + 4*B + 3*C)*Cos[c + d*x]^4*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d) - (a^3*(108*A + 119*B + 133*C)*Sin[c + d*x]^3)/(105*d)","A",9,7,41,0.1707,1,"{4086, 4017, 3996, 3787, 2633, 2635, 8}"
438,1,252,0,0.520057,"\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","Int[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{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}","\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,"(a^4*(56*A + 49*B + 44*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(56*A + 49*B + 44*C)*Tan[c + d*x])/(35*d) + (27*a^4*(56*A + 49*B + 44*C)*Sec[c + d*x]*Tan[c + d*x])/(560*d) + (a^4*(56*A + 49*B + 44*C)*Sec[c + d*x]^3*Tan[c + d*x])/(280*d) + ((42*A - 7*B + 8*C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(210*d) + (C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(7*d) + ((7*B + 4*C)*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(42*a*d) + (2*a^4*(56*A + 49*B + 44*C)*Tan[c + d*x]^3)/(105*d)","A",15,8,41,0.1951,1,"{4088, 4010, 4001, 3791, 3770, 3767, 8, 3768}"
439,1,209,0,0.3283114,"\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","Int[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{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}","\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,"(7*a^4*(10*A + 8*B + 7*C)*ArcTanh[Sin[c + d*x]])/(16*d) + (4*a^4*(10*A + 8*B + 7*C)*Tan[c + d*x])/(5*d) + (27*a^4*(10*A + 8*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(80*d) + (a^4*(10*A + 8*B + 7*C)*Sec[c + d*x]^3*Tan[c + d*x])/(40*d) + ((6*B - C)*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(30*d) + (C*(a + a*Sec[c + d*x])^5*Tan[c + d*x])/(6*a*d) + (2*a^4*(10*A + 8*B + 7*C)*Tan[c + d*x]^3)/(15*d)","A",14,7,39,0.1795,1,"{4082, 4001, 3791, 3770, 3767, 8, 3768}"
440,1,195,0,0.3040174,"\int (a+a \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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{(20 A+35 B+28 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{60 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{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}","\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{(20 A+35 B+28 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{60 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{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*A*x + (a^4*(48*A + 35*B + 28*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a^4*(40*A + 35*B + 28*C)*Tan[c + d*x])/(8*d) + (a*(5*B + 4*C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + a*Sec[c + d*x])^4*Tan[c + d*x])/(5*d) + ((20*A + 35*B + 28*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((32*A + 35*B + 28*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)","A",8,6,33,0.1818,1,"{4054, 3917, 3914, 3767, 8, 3770}"
441,1,196,0,0.3820377,"\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","Int[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{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-4 B-7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 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{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}","\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-4 B-7 C) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 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{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*(4*A + B)*x + (a^4*(52*A + 48*B + 35*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/d + (5*a^4*(4*A + 8*B + 7*C)*Tan[c + d*x])/(8*d) - (a*(4*A - C)*(a + a*Sec[c + d*x])^3*Tan[c + d*x])/(4*d) - ((12*A - 4*B - 7*C)*(a^2 + a^2*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - ((12*A - 32*B - 35*C)*(a^4 + a^4*Sec[c + d*x])*Tan[c + d*x])/(24*d)","A",8,6,39,0.1538,1,"{4086, 3917, 3914, 3767, 8, 3770}"
442,1,209,0,0.5656568,"\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","Int[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{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{(A-B-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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 (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}","\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{(A-B-2 C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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 (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*(13*A + 8*B + 2*C)*x)/2 + (a^4*(8*A + 13*B + 12*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(A - B - 2*C)*Sin[c + d*x])/(2*d) - (a*(3*A - 2*C)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - ((A - B - 2*C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + ((3*A + 18*B + 22*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",7,4,41,0.09756,1,"{4086, 4018, 3996, 3770}"
443,1,217,0,0.6018408,"\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","Int[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{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{(2 A+B-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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{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}","\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{(2 A+B-C) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{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{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,"(a^4*(12*A + 13*B + 8*C)*x)/2 + (a^4*(2*A + 8*B + 13*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (5*a^4*(2*A + B - C)*Sin[c + d*x])/(2*d) + (a*(4*A + 3*B)*Cos[c + d*x]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - ((2*A + B - C)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - ((8*A - 3*B - 18*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(6*d)","A",7,5,41,0.1220,1,"{4086, 4017, 4018, 3996, 3770}"
444,1,217,0,0.6218986,"\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","Int[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]","\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{(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{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{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}","\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{(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{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{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)/8 + (a^4*(B + 4*C)*ArcTanh[Sin[c + d*x]])/d + (5*a^4*(7*A + 8*B + 4*C)*Sin[c + d*x])/(8*d) + (a*(A + B)*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) + ((7*A + 8*B + 4*C)*Cos[c + d*x]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) - ((35*A + 32*B - 12*C)*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)","A",7,5,41,0.1220,1,"{4086, 4017, 4018, 3996, 3770}"
445,1,225,0,0.582938,"\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","Int[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 (28 A+35 B+40 C) \sin (c+d x)}{8 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{(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{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}","\frac{a^4 (28 A+35 B+40 C) \sin (c+d x)}{8 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{(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{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*(28*A + 35*B + 48*C)*x)/8 + (a^4*C*ArcTanh[Sin[c + d*x]])/d + (a^4*(28*A + 35*B + 40*C)*Sin[c + d*x])/(8*d) + (a*(4*A + 5*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(5*d) + ((28*A + 35*B + 20*C)*Cos[c + d*x]^2*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(60*d) + ((28*A + 35*B + 32*C)*Cos[c + d*x]*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(24*d)","A",7,4,41,0.09756,1,"{4086, 4017, 3996, 3770}"
446,1,213,0,0.4092301,"\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","Int[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{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}","-\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,"(7*a^4*(7*A + 8*B + 10*C)*x)/16 + (4*a^4*(7*A + 8*B + 10*C)*Sin[c + d*x])/(5*d) + (27*a^4*(7*A + 8*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(80*d) + (a^4*(7*A + 8*B + 10*C)*Cos[c + d*x]^3*Sin[c + d*x])/(40*d) + ((2*A + 3*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(6*d) - (2*a^4*(7*A + 8*B + 10*C)*Sin[c + d*x]^3)/(15*d)","A",12,7,41,0.1707,1,"{4086, 4013, 3791, 2637, 2635, 8, 2633}"
447,1,278,0,0.7633545,"\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","Int[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 (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{(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{(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{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}","\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{(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{(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{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*(44*A + 49*B + 56*C)*x)/16 + (a^4*(454*A + 504*B + 581*C)*Sin[c + d*x])/(105*d) + (a^4*(44*A + 49*B + 56*C)*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a^4*(988*A + 1113*B + 1232*C)*Cos[c + d*x]^2*Sin[c + d*x])/(840*d) + (a*(4*A + 7*B)*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + ((16*A + 21*B + 14*C)*Cos[c + d*x]^4*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(70*d) + ((436*A + 511*B + 504*C)*Cos[c + d*x]^3*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(840*d)","A",9,7,41,0.1707,1,"{4086, 4017, 3996, 3787, 2635, 8, 2637}"
448,1,303,0,0.7941969,"\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","Int[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 (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{(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{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{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}","-\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{(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{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{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*(323*A + 352*B + 392*C)*x)/128 + (a^4*(208*A + 227*B + 252*C)*Sin[c + d*x])/(35*d) + (a^4*(323*A + 352*B + 392*C)*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (a^4*(2007*A + 2208*B + 2408*C)*Cos[c + d*x]^3*Sin[c + d*x])/(2240*d) + (a*(A + 2*B)*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(14*d) + (A*Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(8*d) + ((61*A + 80*B + 56*C)*Cos[c + d*x]^5*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(336*d) + (7*(7*A + 8*(B + C))*Cos[c + d*x]^4*(a^4 + a^4*Sec[c + d*x])*Sin[c + d*x])/(120*d) - (a^4*(208*A + 227*B + 252*C)*Sin[c + d*x]^3)/(105*d)","A",10,7,41,0.1707,1,"{4086, 4017, 3996, 3787, 2633, 2635, 8}"
449,1,183,0,0.2182199,"\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","Int[(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{(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}","-\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)*ArcTanh[Sin[c + d*x]])/(8*a*d) - ((3*A - 4*B + 4*C)*Tan[c + d*x])/(a*d) + (3*(4*A - 4*B + 5*C)*Sec[c + d*x]*Tan[c + d*x])/(8*a*d) + ((4*A - 4*B + 5*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*a*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((3*A - 4*B + 4*C)*Tan[c + d*x]^3)/(3*a*d)","A",7,5,41,0.1220,1,"{4084, 3787, 3767, 3768, 3770}"
450,1,148,0,0.1983015,"\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","Int[(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{(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}","\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*A - 3*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) + ((3*A - 3*B + 4*C)*Tan[c + d*x])/(a*d) - ((2*A - 3*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((3*A - 3*B + 4*C)*Tan[c + d*x]^3)/(3*a*d)","A",6,5,41,0.1220,1,"{4084, 3787, 3768, 3770, 3767}"
451,1,119,0,0.1919905,"\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","Int[(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{(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}","-\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,"((2*A - 2*B + 3*C)*ArcTanh[Sin[c + d*x]])/(2*a*d) - ((A - 2*B + 2*C)*Tan[c + d*x])/(a*d) + ((2*A - 2*B + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,6,41,0.1463,1,"{4084, 3787, 3767, 8, 3768, 3770}"
452,1,63,0,0.1682638,"\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","Int[(Sec[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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,"((B - C)*ArcTanh[Sin[c + d*x]])/(a*d) + (C*Tan[c + d*x])/(a*d) + ((A - B + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",4,4,39,0.1026,1,"{4082, 3998, 3770, 3794}"
453,1,52,0,0.1157258,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{a+a \sec (c+d x)} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(a*d) - ((A - B + C)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x]))","A",4,4,33,0.1212,1,"{4050, 3770, 3919, 3794}"
454,1,62,0,0.1303082,"\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","Int[(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 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}","\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,"-(((A - B)*x)/a) + ((2*A - B + C)*Sin[c + d*x])/(a*d) - ((A - B + C)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",4,4,39,0.1026,1,"{4084, 3787, 2637, 8}"
455,1,108,0,0.1816879,"\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","Int[(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{(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}","-\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,"((3*A - 2*B + 2*C)*x)/(2*a) - ((2*A - 2*B + C)*Sin[c + d*x])/(a*d) + ((3*A - 2*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",5,5,41,0.1220,1,"{4084, 3787, 2635, 8, 2637}"
456,1,139,0,0.1965478,"\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","Int[(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{(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}","-\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,"-((3*A - 3*B + 2*C)*x)/(2*a) + ((4*A - 3*B + 3*C)*Sin[c + d*x])/(a*d) - ((3*A - 3*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) - ((4*A - 3*B + 3*C)*Sin[c + d*x]^3)/(3*a*d)","A",6,5,41,0.1220,1,"{4084, 3787, 2633, 2635, 8}"
457,1,174,0,0.212429,"\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","Int[(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{(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}","\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,"(3*(5*A - 4*B + 4*C)*x)/(8*a) - ((4*A - 4*B + 3*C)*Sin[c + d*x])/(a*d) + (3*(5*A - 4*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*a*d) + ((5*A - 4*B + 4*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d) - ((A - B + C)*Cos[c + d*x]^3*Sin[c + d*x])/(d*(a + a*Sec[c + d*x])) + ((4*A - 4*B + 3*C)*Sin[c + d*x]^3)/(3*a*d)","A",7,5,41,0.1220,1,"{4084, 3787, 2635, 8, 2633}"
458,1,194,0,0.3630637,"\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","Int[(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{(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}","\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*A - 7*B + 10*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) + ((5*A - 8*B + 12*C)*Tan[c + d*x])/(a^2*d) - ((4*A - 7*B + 10*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((4*A - 7*B + 10*C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) + ((5*A - 8*B + 12*C)*Tan[c + d*x]^3)/(3*a^2*d)","A",7,6,41,0.1463,1,"{4084, 4019, 3787, 3768, 3770, 3767}"
459,1,169,0,0.3372339,"\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","Int[(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{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}","-\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,"((2*A - 4*B + 7*C)*ArcTanh[Sin[c + d*x]])/(2*a^2*d) - (2*(2*A - 5*B + 8*C)*Tan[c + d*x])/(3*a^2*d) + ((2*A - 4*B + 7*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^2*d) - ((2*A - 5*B + 8*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,7,41,0.1707,1,"{4084, 4019, 3787, 3767, 8, 3768, 3770}"
460,1,112,0,0.2826134,"\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","Int[(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{(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}","\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,"((B - 2*C)*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((A - B + 4*C)*Tan[c + d*x])/(3*a^2*d) - ((B - 2*C)*Tan[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,6,41,0.1463,1,"{4084, 4008, 3787, 3770, 3767, 8}"
461,1,87,0,0.1717574,"\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","Int[(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 A+B-4 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) \sec (c+d x)}{3 d (a \sec (c+d x)+a)^2}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a^2*d) + ((2*A + B - 4*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",4,4,39,0.1026,1,"{4078, 3998, 3770, 3794}"
462,1,74,0,0.1287297,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"(A*x)/a^2 - ((4*A - B - 2*C)*Tan[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",3,3,33,0.09091,1,"{4052, 3919, 3794}"
463,1,100,0,0.2586296,"\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","Int[(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{(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}","\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,"-(((2*A - B)*x)/a^2) + ((10*A - 4*B + C)*Sin[c + d*x])/(3*a^2*d) - ((2*A - B)*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",5,5,39,0.1282,1,"{4084, 4020, 3787, 2637, 8}"
464,1,156,0,0.3348716,"\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","Int[(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{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}","-\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,"((7*A - 4*B + 2*C)*x)/(2*a^2) - (2*(8*A - 5*B + 2*C)*Sin[c + d*x])/(3*a^2*d) + ((7*A - 4*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((8*A - 5*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",6,6,41,0.1463,1,"{4084, 4020, 3787, 2635, 8, 2637}"
465,1,185,0,0.3628682,"\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","Int[(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{(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}","-\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,"-((10*A - 7*B + 4*C)*x)/(2*a^2) + ((12*A - 8*B + 5*C)*Sin[c + d*x])/(a^2*d) - ((10*A - 7*B + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) - ((10*A - 7*B + 4*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2) - ((12*A - 8*B + 5*C)*Sin[c + d*x]^3)/(3*a^2*d)","A",7,6,41,0.1463,1,"{4084, 4020, 3787, 2633, 2635, 8}"
466,1,216,0,0.5162818,"\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","Int[(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{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}","-\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,"((2*A - 6*B + 13*C)*ArcTanh[Sin[c + d*x]])/(2*a^3*d) - (2*(11*A - 36*B + 76*C)*Tan[c + d*x])/(15*a^3*d) + ((2*A - 6*B + 13*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^3*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((A - 6*B + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((11*A - 36*B + 76*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",8,7,41,0.1707,1,"{4084, 4019, 3787, 3767, 8, 3768, 3770}"
467,1,161,0,0.4500602,"\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","Int[(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{(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}","\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,"((B - 3*C)*ArcTanh[Sin[c + d*x]])/(a^3*d) + ((2*A - 7*B + 27*C)*Tan[c + d*x])/(15*a^3*d) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A + 4*B - 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((B - 3*C)*Tan[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",7,7,41,0.1707,1,"{4084, 4019, 4008, 3787, 3770, 3767, 8}"
468,1,132,0,0.3508236,"\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","Int[(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{(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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a^3*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((3*A + 2*B - 7*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((6*A + 4*B - 29*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",5,5,41,0.1220,1,"{4084, 4008, 3998, 3770, 3794}"
469,1,110,0,0.2072776,"\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","Int[(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{(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}","\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,"-((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((A - C)*Tan[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) + ((2*A + 3*B + 7*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",3,3,39,0.07692,1,"{4078, 4000, 3794}"
470,1,115,0,0.1963759,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"(A*x)/a^3 - ((A - B + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((7*A - 2*B - 3*C)*Tan[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((22*A - 2*B - 3*C)*Tan[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",4,4,33,0.1212,1,"{4052, 3922, 3919, 3794}"
471,1,141,0,0.4036636,"\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","Int[(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{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}","\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,"-(((3*A - B)*x)/a^3) + (2*(36*A - 11*B + C)*Sin[c + d*x])/(15*a^3*d) - ((A - B + C)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((9*A - 4*B - C)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((3*A - B)*Sin[c + d*x])/(d*(a^3 + a^3*Sec[c + d*x]))","A",6,5,39,0.1282,1,"{4084, 4020, 3787, 2637, 8}"
472,1,201,0,0.5173985,"\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","Int[(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{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}","-\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,"((13*A - 6*B + 2*C)*x)/(2*a^3) - (2*(76*A - 36*B + 11*C)*Sin[c + d*x])/(15*a^3*d) + ((13*A - 6*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((11*A - 6*B + C)*Cos[c + d*x]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((76*A - 36*B + 11*C)*Cos[c + d*x]*Sin[c + d*x])/(15*d*(a^3 + a^3*Sec[c + d*x]))","A",7,6,41,0.1463,1,"{4084, 4020, 3787, 2635, 8, 2637}"
473,1,237,0,0.5448663,"\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","Int[(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{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}","-\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,"-((23*A - 13*B + 6*C)*x)/(2*a^3) + (4*(34*A - 19*B + 9*C)*Sin[c + d*x])/(5*a^3*d) - ((23*A - 13*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((13*A - 8*B + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((23*A - 13*B + 6*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*(a^3 + a^3*Sec[c + d*x])) - (4*(34*A - 19*B + 9*C)*Sin[c + d*x]^3)/(15*a^3*d)","A",8,6,41,0.1463,1,"{4084, 4020, 3787, 2633, 2635, 8}"
474,1,254,0,0.6899817,"\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","Int[(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{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}","-\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,"((2*A - 8*B + 21*C)*ArcTanh[Sin[c + d*x]])/(2*a^4*d) - (8*(20*A - 83*B + 216*C)*Tan[c + d*x])/(105*a^4*d) + ((2*A - 8*B + 21*C)*Sec[c + d*x]*Tan[c + d*x])/(2*a^4*d) - ((10*A - 52*B + 129*C)*Sec[c + d*x]^3*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(20*A - 83*B + 216*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^5*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((B - 2*C)*Sec[c + d*x]^4*Tan[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",9,7,41,0.1707,1,"{4084, 4019, 3787, 3767, 8, 3768, 3770}"
475,1,204,0,0.6290717,"\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","Int[(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{(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}","\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,"((B - 4*C)*ArcTanh[Sin[c + d*x]])/(a^4*d) + ((6*A - 55*B + 244*C)*Tan[c + d*x])/(105*a^4*d) + ((3*A + 25*B - 88*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((B - 4*C)*Tan[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((2*A + 5*B - 12*C)*Sec[c + d*x]^3*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",8,7,41,0.1707,1,"{4084, 4019, 4008, 3787, 3770, 3767, 8}"
476,1,173,0,0.4966237,"\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","Int[(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{(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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(a^4*d) - ((8*A + 6*B - 55*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((16*A + 12*B - 215*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((4*A + 3*B - 10*C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",6,6,41,0.1463,1,"{4084, 4019, 4008, 3998, 3770, 3794}"
477,1,148,0,0.4062982,"\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","Int[(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{(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}","\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,"((23*A - 2*B - 54*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) + ((8*A + 13*B + 36*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((6*A + B - 8*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",4,4,41,0.09756,1,"{4084, 4008, 4000, 3794}"
478,1,154,0,0.2604976,"\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","Int[(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{(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}","\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,"-((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) + ((8*A - B - 6*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3) + ((6*A + 8*B + 13*C)*Tan[c + d*x])/(105*d*(a^2 + a^2*Sec[c + d*x])^2) + ((6*A + 8*B + 13*C)*Tan[c + d*x])/(105*d*(a^4 + a^4*Sec[c + d*x]))","A",4,4,39,0.1026,1,"{4078, 4000, 3796, 3794}"
479,1,148,0,0.2839377,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^4} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^4,x]","-\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}","-\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,"(A*x)/a^4 - ((55*A - 6*B - 8*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (2*(80*A - 3*B - 4*C)*Tan[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((10*A - 3*B - 4*C)*Tan[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",5,4,33,0.1212,1,"{4052, 3922, 3919, 3794}"
480,1,176,0,0.5591753,"\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","Int[(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{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}","\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,"-(((4*A - B)*x)/a^4) + (2*(332*A - 80*B + 3*C)*Sin[c + d*x])/(105*a^4*d) - ((88*A - 25*B - 3*C)*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - ((4*A - B)*Sin[c + d*x])/(a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((12*A - 5*B - 2*C)*Sin[c + d*x])/(35*a*d*(a + a*Sec[c + d*x])^3)","A",7,5,39,0.1282,1,"{4084, 4020, 3787, 2637, 8}"
481,1,239,0,0.6988012,"\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","Int[(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{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}","-\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,"((21*A - 8*B + 2*C)*x)/(2*a^4) - (8*(216*A - 83*B + 20*C)*Sin[c + d*x])/(105*a^4*d) + ((21*A - 8*B + 2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*d) - ((129*A - 52*B + 10*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])^2) - (4*(216*A - 83*B + 20*C)*Cos[c + d*x]*Sin[c + d*x])/(105*a^4*d*(1 + Sec[c + d*x])) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(7*d*(a + a*Sec[c + d*x])^4) - ((2*A - B)*Cos[c + d*x]*Sin[c + d*x])/(5*a*d*(a + a*Sec[c + d*x])^3)","A",8,6,41,0.1463,1,"{4084, 4020, 3787, 2635, 8, 2637}"
482,1,239,0,0.5584603,"\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","Int[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{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}","\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,"(4*a*(99*A + 88*B + 80*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(99*A + 88*B + 80*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(11*B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(99*d*Sqrt[a + a*Sec[c + d*x]]) - (8*(99*A + 88*B + 80*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*C*Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(11*d) + (4*(99*A + 88*B + 80*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*a*d)","A",6,6,43,0.1395,1,"{4088, 4016, 3803, 3800, 4001, 3792}"
483,1,193,0,0.4731506,"\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","Int[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{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}","\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,"(2*a*(21*A + 18*B + 16*C)*Tan[c + d*x])/(45*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(9*B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) - (4*(21*A + 18*B + 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(9*d) + (2*(21*A + 18*B + 16*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*a*d)","A",5,5,43,0.1163,1,"{4088, 4016, 3800, 4001, 3792}"
484,1,147,0,0.4276482,"\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","Int[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{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}","\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,"(2*a*(35*A + 49*B + 27*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A - 14*B + 18*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(7*d) + (2*(7*B + C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*a*d)","A",4,4,43,0.09302,1,"{4088, 4010, 4001, 3792}"
485,1,104,0,0.2092868,"\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","Int[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{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}","\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,"(2*a*(15*A + 5*B + 7*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*a*d)","A",3,3,41,0.07317,1,"{4082, 4001, 3792}"
486,1,100,0,0.1523051,"\int \sqrt{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*Sqrt[a]*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*(3*B + C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",5,5,35,0.1429,1,"{4054, 3915, 3774, 203, 3792}"
487,1,98,0,0.2108908,"\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","Int[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{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}","\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,"(Sqrt[a]*(A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d - (a*(A - 2*C)*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,41,0.1220,1,"{4086, 3915, 3774, 203, 3792}"
488,1,117,0,0.2774875,"\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","Int[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{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}","\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[a]*(3*A + 4*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(A + 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,43,0.09302,1,"{4086, 4015, 3774, 203}"
489,1,163,0,0.3701841,"\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","Int[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{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}","\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,"(Sqrt[a]*(5*A + 6*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(5*A + 6*B + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(A + 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,43,0.1163,1,"{4086, 4015, 3805, 3774, 203}"
490,1,209,0,0.4541202,"\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","Int[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{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}","\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,"(Sqrt[a]*(35*A + 40*B + 48*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(35*A + 40*B + 48*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(35*A + 40*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(A + 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,43,0.1163,1,"{4086, 4015, 3805, 3774, 203}"
491,1,243,0,0.6946283,"\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","Int[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{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}","\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,"(2*a^2*(429*A + 374*B + 336*C)*Tan[c + d*x])/(495*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(99*A + 110*B + 84*C)*Sec[c + d*x]^3*Tan[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a*(429*A + 374*B + 336*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a*(11*B + 3*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*(429*A + 374*B + 336*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)","A",6,6,43,0.1395,1,"{4088, 4018, 4016, 3800, 4001, 3792}"
492,1,187,0,0.5158438,"\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","Int[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{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}","\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,"(8*a^2*(63*A + 57*B + 47*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(63*A + 57*B + 47*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*(63*A - 18*B + 22*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(9*d) + (2*(3*B + C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(21*a*d)","A",5,5,43,0.1163,1,"{4088, 4010, 4001, 3793, 3792}"
493,1,144,0,0.2834726,"\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","Int[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{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}","\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,"(8*a^2*(35*A + 21*B + 19*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(35*A + 21*B + 19*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*B - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*a*d)","A",4,4,41,0.09756,1,"{4082, 4001, 3793, 3792}"
494,1,142,0,0.2362399,"\int (a+a \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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^{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 (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}","\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^{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 (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^(3/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(15*A + 20*B + 12*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(5*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",6,6,35,0.1714,1,"{4054, 3917, 3915, 3774, 203, 3792}"
495,1,144,0,0.3104063,"\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","Int[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^2 (3 A-6 B-8 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","-\frac{a^2 (3 A-6 B-8 C) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{a^{3/2} (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 (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^(3/2)*(3*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (a^2*(3*A - 6*B - 8*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(3*A - 2*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,41,0.1463,1,"{4086, 3917, 3915, 3774, 203, 3792}"
496,1,157,0,0.4507593,"\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","Int[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^2 (5 A+4 B-8 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (5 A+4 B-8 C) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(7*A + 12*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(5*A + 4*B - 8*C)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(A - 4*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",5,5,43,0.1163,1,"{4086, 4018, 4015, 3774, 203}"
497,1,165,0,0.4811264,"\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","Int[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^2 (19 A+30 B+24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (19 A+30 B+24 C) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(11*A + 14*B + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(19*A + 30*B + 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(A + 2*B)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,43,0.1163,1,"{4086, 4017, 4015, 3774, 203}"
498,1,215,0,0.5860604,"\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","Int[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^2 (75 A+88 B+112 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (75 A+88 B+112 C) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(75*A + 88*B + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(75*A + 88*B + 112*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(39*A + 56*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(3*A + 8*B)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,6,43,0.1395,1,"{4086, 4017, 4015, 3805, 3774, 203}"
499,1,263,0,0.6726047,"\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","Int[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^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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 (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}","\frac{a^2 (133 A+150 B+176 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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 (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^(3/2)*(133*A + 150*B + 176*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(133*A + 150*B + 176*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(133*A + 150*B + 176*C)*Cos[c + d*x]*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(67*A + 90*B + 80*C)*Cos[c + d*x]^2*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(3*A + 10*B)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,6,43,0.1395,1,"{4086, 4017, 4015, 3805, 3774, 203}"
500,1,294,0,0.9048903,"\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","Int[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{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^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{2 a^3 (10439 A+9230 B+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}-\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}","\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^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{2 a^3 (10439 A+9230 B+8368 C) \tan (c+d x)}{6435 d \sqrt{a \sec (c+d x)+a}}-\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,"(2*a^3*(10439*A + 9230*B + 8368*C)*Tan[c + d*x])/(6435*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2717*A + 2522*B + 2224*C)*Sec[c + d*x]^3*Tan[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) - (4*a^2*(10439*A + 9230*B + 8368*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(45045*d) + (2*a^2*(143*A + 182*B + 136*C)*Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(1287*d) + (2*a*(10439*A + 9230*B + 8368*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(15015*d) + (2*a*(13*B + 5*C)*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)","A",7,6,43,0.1395,1,"{4088, 4018, 4016, 3800, 4001, 3792}"
501,1,229,0,0.5868019,"\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","Int[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{16 a^2 (165 A+143 B+125 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 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{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}","\frac{16 a^2 (165 A+143 B+125 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 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{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,"(64*a^3*(165*A + 143*B + 125*C)*Tan[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(165*A + 143*B + 125*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3465*d) + (2*a*(165*A + 143*B + 125*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(1155*d) + (2*(99*A - 22*B + 26*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*d) + (2*C*Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*d) + (2*(11*B + 5*C)*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*a*d)","A",6,5,43,0.1163,1,"{4088, 4010, 4001, 3793, 3792}"
502,1,184,0,0.3447,"\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","Int[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{16 a^2 (21 A+15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 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{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}","\frac{16 a^2 (21 A+15 B+13 C) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 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{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,"(64*a^3*(21*A + 15*B + 13*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(21*A + 15*B + 13*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*d) + (2*a*(21*A + 15*B + 13*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(105*d) + (2*(9*B - 2*C)*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*d) + (2*C*(a + a*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*a*d)","A",5,4,41,0.09756,1,"{4082, 4001, 3793, 3792}"
503,1,182,0,0.3243989,"\int (a+a \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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^{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 (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}","\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^{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 (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^(5/2)*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(245*A + 224*B + 160*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(35*A + 56*B + 40*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*a*(7*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + a*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",7,6,35,0.1714,1,"{4054, 3917, 3915, 3774, 203, 3792}"
504,1,184,0,0.4040803,"\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","Int[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^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^{5/2} (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 (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}","\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^{5/2} (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 (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^(5/2)*(5*A + 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d + (a^3*(15*A + 70*B + 64*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(15*A - 10*B - 16*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (a*(5*A - 2*C)*(a + a*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",7,6,41,0.1463,1,"{4086, 3917, 3915, 3774, 203, 3792}"
505,1,197,0,0.6310307,"\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","Int[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^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^{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 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}","\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^{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 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^(5/2)*(19*A + 20*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(27*A - 12*B - 56*C)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(A - 4*B - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d) - (a*(3*A - 4*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d)","A",6,5,43,0.1163,1,"{4086, 4018, 4015, 3774, 203}"
506,1,207,0,0.6583964,"\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","Int[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^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^{5/2} (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 (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}","\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^{5/2} (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 (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^(5/2)*(25*A + 38*B + 40*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(49*A + 54*B - 24*C)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(3*A + 2*B - 8*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*(5*A + 6*B)*Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",6,6,43,0.1395,1,"{4086, 4017, 4018, 4015, 3774, 203}"
507,1,215,0,0.6991863,"\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","Int[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^3 (299 A+392 B+432 C) \sin (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\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^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}","\frac{a^3 (299 A+392 B+432 C) \sin (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\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^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^(5/2)*(163*A + 200*B + 304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(299*A + 392*B + 432*C)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(17*A + 24*B + 16*C)*Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (a*(5*A + 8*B)*Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",6,5,43,0.1163,1,"{4086, 4017, 4015, 3774, 203}"
508,1,261,0,0.8102985,"\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","Int[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^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 (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}","\frac{a^3 (283 A+326 B+400 C) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 (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^(5/2)*(283*A + 326*B + 400*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(283*A + 326*B + 400*C)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(787*A + 950*B + 1040*C)*Cos[c + d*x]*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(79*A + 110*B + 80*C)*Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*(A + 2*B)*Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,43,0.1395,1,"{4086, 4017, 4015, 3805, 3774, 203}"
509,1,311,0,0.8948641,"\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","Int[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^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 (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}","\frac{a^3 (1015 A+1132 B+1304 C) \sin (c+d x)}{512 d \sqrt{a \sec (c+d x)+a}}+\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^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^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 (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^(5/2)*(1015*A + 1132*B + 1304*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1015*A + 1132*B + 1304*C)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1015*A + 1132*B + 1304*C)*Cos[c + d*x]*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(545*A + 628*B + 680*C)*Cos[c + d*x]^2*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(115*A + 156*B + 120*C)*Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(5*A + 12*B)*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (A*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)","A",8,6,43,0.1395,1,"{4086, 4017, 4015, 3805, 3774, 203}"
510,1,254,0,0.8566229,"\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","Int[(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]","\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}}","\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,"-((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (4*(147*A - 111*B + 143*C)*Tan[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(21*A - 3*B + 19*C)*Sec[c + d*x]^2*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(9*B - C)*Sec[c + d*x]^3*Tan[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^4*Tan[c + d*x])/(9*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(21*A - 93*B + 29*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(315*a*d)","A",7,6,43,0.1395,1,"{4088, 4021, 4010, 4001, 3795, 203}"
511,1,208,0,0.6370457,"\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","Int[(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]","\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}}","\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,"(Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - (4*(35*A - 49*B + 37*C)*Tan[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(7*B - C)*Sec[c + d*x]^2*Tan[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^3*Tan[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(35*A - 7*B + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(105*a*d)","A",6,6,43,0.1395,1,"{4088, 4021, 4010, 4001, 3795, 203}"
512,1,164,0,0.4519969,"\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","Int[(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{\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}}","-\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,"-((Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*(15*A - 10*B + 14*C)*Tan[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sec[c + d*x]^2*Tan[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]]) + (2*(5*B - C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(15*a*d)","A",5,5,43,0.1163,1,"{4088, 4010, 4001, 3795, 203}"
513,1,118,0,0.2253032,"\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","Int[(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{\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}","\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,"(Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*(3*B - 2*C)*Tan[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*C*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(3*a*d)","A",4,4,41,0.09756,1,"{4082, 4001, 3795, 203}"
514,1,118,0,0.1720128,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + a*Sec[c + d*x]],x]","-\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}}","-\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*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*C*Tan[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,35,0.1429,1,"{4054, 3920, 3774, 203, 3795}"
515,1,120,0,0.2393275,"\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","Int[(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{\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}}","\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 - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d)) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (A*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,41,0.1220,1,"{4086, 3920, 3774, 203, 3795}"
516,1,169,0,0.3972172,"\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","Int[(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]","\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}}","\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,"((7*A - 4*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((A - 4*B)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,43,0.1395,1,"{4086, 4022, 3920, 3774, 203, 3795}"
517,1,213,0,0.5885976,"\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","Int[(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{(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}}","\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,"-((9*A - 14*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((7*A - 2*B + 8*C)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 6*B)*Cos[c + d*x]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,43,0.1395,1,"{4086, 4022, 3920, 3774, 203, 3795}"
518,1,259,0,0.7768915,"\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","Int[(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{(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}}","-\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,"((107*A - 72*B + 112*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) - ((21*A - 56*B + 16*C)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + ((43*A - 8*B + 48*C)*Cos[c + d*x]*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) - ((A - 8*B)*Cos[c + d*x]^2*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,43,0.1395,1,"{4086, 4022, 3920, 3774, 203, 3795}"
519,1,277,0,0.8770864,"\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","Int[(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]","\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}}","\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,"((11*A - 15*B + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((455*A - 651*B + 799*C)*Tan[c + d*x])/(105*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((35*A - 63*B + 67*C)*Sec[c + d*x]^2*Tan[c + d*x])/(70*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((7*A - 7*B + 11*C)*Sec[c + d*x]^3*Tan[c + d*x])/(14*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((245*A - 273*B + 397*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(210*a^2*d)","A",7,6,43,0.1395,1,"{4084, 4021, 4010, 4001, 3795, 203}"
520,1,229,0,0.6825227,"\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","Int[(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]","-\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}}","-\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,"-((7*A - 11*B + 15*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((45*A - 65*B + 93*C)*Tan[c + d*x])/(15*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A - 5*B + 9*C)*Sec[c + d*x]^2*Tan[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((15*A - 35*B + 39*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(30*a^2*d)","A",6,6,43,0.1395,1,"{4084, 4021, 4010, 4001, 3795, 203}"
521,1,181,0,0.4764564,"\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","Int[(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]","\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}}","\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,"((3*A - 7*B + 11*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((3*A - 9*B + 13*C)*Tan[c + d*x])/(3*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((3*A - 3*B + 7*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(6*a^2*d)","A",5,5,43,0.1163,1,"{4084, 4010, 4001, 3795, 203}"
522,1,135,0,0.2462175,"\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","Int[(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{(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+5 C) \tan (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B+C) \tan (c+d x) \sec (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}","\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,"((A + 3*B - 7*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 5*C)*Tan[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,41,0.09756,1,"{4078, 4001, 3795, 203}"
523,1,131,0,0.1945705,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) - ((5*A - B - 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,5,35,0.1429,1,"{4052, 3920, 3774, 203, 3795}"
524,1,173,0,0.4167225,"\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","Int[(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{(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}}","\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,"-(((3*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d)) + ((9*A - 5*B + C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - B + C)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,41,0.1463,1,"{4084, 4022, 3920, 3774, 203, 3795}"
525,1,232,0,0.6226733,"\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","Int[(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]","\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}}","\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,"((19*A - 12*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((13*A - 9*B + 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((7*A - 6*B + 2*C)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((2*A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,6,43,0.1395,1,"{4084, 4022, 3920, 3774, 203, 3795}"
526,1,284,0,0.8340806,"\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","Int[(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{(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}}","-\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,"-((47*A - 38*B + 24*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*a^(3/2)*d) + ((17*A - 13*B + 9*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((21*A - 14*B + 12*C)*Sin[c + d*x])/(8*a*d*Sqrt[a + a*Sec[c + d*x]]) - ((13*A - 12*B + 6*C)*Cos[c + d*x]*Sin[c + d*x])/(12*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((5*A - 3*B + 3*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",9,6,43,0.1395,1,"{4084, 4022, 3920, 3774, 203, 3795}"
527,1,277,0,0.9016216,"\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","Int[(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]","\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{(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{(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}}","\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{(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{(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,"-((75*A - 163*B + 283*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^4*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((5*A - 13*B + 21*C)*Sec[c + d*x]^3*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((465*A - 985*B + 1729*C)*Tan[c + d*x])/(120*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((45*A - 85*B + 157*C)*Sec[c + d*x]^2*Tan[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]]) - ((195*A - 475*B + 787*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(240*a^3*d)","A",7,7,43,0.1628,1,"{4084, 4019, 4021, 4010, 4001, 3795, 203}"
528,1,227,0,0.695241,"\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","Int[(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]","\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}}","\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,"((19*A - 75*B + 163*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((A - 9*B + 17*C)*Sec[c + d*x]^2*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((21*A - 93*B + 197*C)*Tan[c + d*x])/(24*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((15*A - 39*B + 95*C)*Sqrt[a + a*Sec[c + d*x]]*Tan[c + d*x])/(48*a^3*d)","A",6,6,43,0.1395,1,"{4084, 4019, 4010, 4001, 3795, 203}"
529,1,179,0,0.4907199,"\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","Int[(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]","\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}}","\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,"((5*A + 19*B - 75*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^2*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((3*A + 5*B - 13*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 9*C)*Tan[c + d*x])/(4*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,43,0.1163,1,"{4084, 4008, 4001, 3795, 203}"
530,1,137,0,0.2727911,"\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","Int[(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{(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}}","\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,"((3*A + 5*B + 19*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((7*A + B - 9*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,41,0.09756,1,"{4078, 4000, 3795, 203}"
531,1,171,0,0.2778683,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"(2*A*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) - ((43*A - 3*B - 5*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Tan[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((11*A - 3*B - 5*C)*Tan[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,6,35,0.1714,1,"{4052, 3922, 3920, 3774, 203, 3795}"
532,1,217,0,0.610402,"\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","Int[(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{(35 A-11 B+3 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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}}","\frac{(35 A-11 B+3 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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,"-(((5*A - 2*B)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d)) + ((115*A - 43*B + 3*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((15*A - 7*B - C)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((35*A - 11*B + 3*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,41,0.1707,1,"{4084, 4020, 4022, 3920, 3774, 203, 3795}"
533,1,280,0,0.8688141,"\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","Int[(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]","-\frac{(63 A-35 B+11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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}}","-\frac{(63 A-35 B+11 C) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\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{(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,"((39*A - 20*B + 8*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(5/2)*d) - ((219*A - 115*B + 43*C)*ArcTan[(Sqrt[a]*Tan[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((19*A - 11*B + 3*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((63*A - 35*B + 11*C)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((31*A - 15*B + 7*C)*Cos[c + d*x]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",9,7,43,0.1628,1,"{4084, 4020, 4022, 3920, 3774, 203, 3795}"
534,1,217,0,0.256471,"\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","Int[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 (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}","\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*(5*A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(7*A + 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(5*A + 3*(B + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(7*A + 7*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a*(B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*a*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,7,41,0.1707,1,"{4076, 4047, 3768, 3771, 2641, 4046, 2639}"
535,1,181,0,0.2211797,"\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","Int[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 (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}","\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*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(5*A + 5*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a*(B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*a*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,7,41,0.1707,1,"{4076, 4047, 3768, 3771, 2639, 4046, 2641}"
536,1,143,0,0.2190376,"\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","Int[((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{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}","\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,"(2*a*(A - B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(3*A + 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*a*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,6,41,0.1463,1,"{4076, 4047, 3771, 2641, 4046, 2639}"
537,1,138,0,0.2092114,"\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","Int[((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{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}","\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,"(2*a*(A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*a*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,41,0.1463,1,"{4074, 4047, 3771, 2641, 4046, 2639}"
538,1,146,0,0.2167948,"\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","Int[((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{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)}","\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,"(2*a*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(A + B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,41,0.1463,1,"{4074, 4047, 3771, 2639, 4045, 2641}"
539,1,182,0,0.2443103,"\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","Int[((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{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)}","\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,"(2*a*(3*(A + B) + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*a*(5*A + 7*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(A + B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*a*(5*A + 7*(B + C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,41,0.1707,1,"{4074, 4047, 3769, 3771, 2641, 4045, 2639}"
540,1,215,0,0.2821044,"\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","Int[((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{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)}","\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,"(2*a*(7*A + 9*(B + C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*a*(5*(A + B) + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*a*(A + B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*a*(7*A + 9*(B + C))*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*a*(5*(A + B) + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,7,41,0.1707,1,"{4074, 4047, 3769, 3771, 2639, 4045, 2641}"
541,1,291,0,0.4953692,"\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","Int[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{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}","\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*a^2*(12*A + 9*B + 8*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(12*A + 9*B + 8*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(9*B + 4*C)*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d)","A",10,8,43,0.1860,1,"{4088, 4018, 3997, 3787, 3768, 3771, 2639, 2641}"
542,1,255,0,0.5055651,"\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","Int[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{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}","\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,"(-4*a^2*(5*A + 4*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^2*(5*A + 4*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*a^2*(35*A + 49*B + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(7*B + 4*C)*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d)","A",9,8,43,0.1860,1,"{4088, 4018, 3997, 3787, 3771, 2641, 3768, 2639}"
543,1,214,0,0.453359,"\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","Int[((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{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}","\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,"(-4*a^2*(5*B + 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(3*A + 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a^2*(15*A + 25*B + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (2*(5*B + 4*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",8,7,43,0.1628,1,"{4088, 4018, 3997, 3787, 3771, 2639, 2641}"
544,1,208,0,0.4457991,"\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","Int[((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{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)}}","-\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,"(4*a^2*(A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (4*a^2*(2*A + 3*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(A - 3*B - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(A - C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(3*d)","A",8,7,43,0.1628,1,"{4086, 4018, 3997, 3787, 3771, 2639, 2641}"
545,1,214,0,0.4930257,"\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","Int[((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{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)}","-\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,"(4*a^2*(4*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(A + 2*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*a^2*(7*A + 5*B - 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(4*A + 5*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4086, 4017, 3997, 3787, 3771, 2639, 2641}"
546,1,219,0,0.4969459,"\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","Int[((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{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)}","\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,"(4*a^2*(3*A + 4*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(4*A + 7*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))","A",8,7,43,0.1628,1,"{4086, 4017, 3996, 3787, 3771, 2639, 2641}"
547,1,255,0,0.5016626,"\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","Int[((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{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)}","\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,"(4*a^2*(8*A + 9*B + 12*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (4*a^2*(5*A + 6*B + 7*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(4*A + 9*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))","A",9,8,43,0.1860,1,"{4086, 4017, 3996, 3787, 3769, 3771, 2641, 2639}"
548,1,291,0,0.5237169,"\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","Int[((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{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)}","\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,"(4*a^2*(7*A + 8*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^2*(50*A + 55*B + 66*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a^2*(89*A + 121*B + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (4*a^2*(7*A + 8*B + 9*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (4*a^2*(50*A + 55*B + 66*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(4*A + 11*B)*(a^2 + a^2*Sec[c + d*x])*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2))","A",10,8,43,0.1860,1,"{4086, 4017, 3996, 3787, 3769, 3771, 2639, 2641}"
549,1,343,0,0.6994867,"\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","Int[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{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}","\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,"(-4*a^3*(21*A + 17*B + 15*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(21*A + 17*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(143*A + 121*B + 105*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(231*d) + (4*a^3*(264*A + 253*B + 210*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(1155*d) + (2*C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d) + (2*(11*B + 6*C)*Sec[c + d*x]^(5/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(99*a*d) + (2*(99*A + 143*B + 105*C)*Sec[c + d*x]^(5/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(693*d)","A",11,8,43,0.1860,1,"{4088, 4018, 3997, 3787, 3768, 3771, 2639, 2641}"
550,1,307,0,0.6455676,"\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","Int[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{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}","\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,"(-4*a^3*(27*A + 21*B + 17*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(27*A + 21*B + 17*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (4*a^3*(42*A + 41*B + 32*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(3*B + 2*C)*Sec[c + d*x]^(3/2)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(63*A + 99*B + 73*C)*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d)","A",10,8,43,0.1860,1,"{4088, 4018, 3997, 3787, 3771, 2641, 3768, 2639}"
551,1,271,0,0.6306525,"\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","Int[((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{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}","\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,"(-4*a^3*(5*A + 9*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(140*A + 147*B + 106*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + (2*(7*B + 6*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d) + (2*(5*A + 9*B + 7*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",9,7,43,0.1628,1,"{4088, 4018, 3997, 3787, 3771, 2639, 2641}"
552,1,271,0,0.6306594,"\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","Int[((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{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)}}","\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,"(4*a^3*(5*A - 5*B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (4*a^3*(5*A + 20*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) - (2*(5*A - 3*C)*Sqrt[Sec[c + d*x]]*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*a*d) - (2*(5*A - 5*B - 9*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",9,7,43,0.1628,1,"{4086, 4018, 3997, 3787, 3771, 2639, 2641}"
553,1,270,0,0.657914,"\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","Int[((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{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)}","-\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,"(4*a^3*(9*A + 5*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(3*A + 5*(B + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (4*a^3*(6*A - 5*B - 20*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(6*A + 5*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]]) - (2*(9*A + 5*B - 5*C)*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d)","A",9,8,43,0.1860,1,"{4086, 4017, 4018, 3997, 3787, 3771, 2639, 2641}"
554,1,271,0,0.6329893,"\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","Int[((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{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)}","-\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,"(4*a^3*(7*A + 9*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) - (4*a^3*(41*A + 42*B - 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(6*A + 7*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) + (2*(7*A + 9*B + 5*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",9,7,43,0.1628,1,"{4086, 4017, 3997, 3787, 3771, 2639, 2641}"
555,1,271,0,0.6468378,"\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","Int[((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{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)}","\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,"(4*a^3*(17*A + 21*B + 27*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(2*A + 3*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(21*a*d*Sec[c + d*x]^(5/2)) + (2*(73*A + 99*B + 63*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2))","A",9,7,43,0.1628,1,"{4086, 4017, 3996, 3787, 3771, 2639, 2641}"
556,1,307,0,0.6750964,"\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","Int[((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{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)}","\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,"(4*a^3*(15*A + 17*B + 21*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)) + (4*a^3*(105*A + 121*B + 143*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(6*A + 11*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(99*a*d*Sec[c + d*x]^(7/2)) + (2*(105*A + 143*B + 99*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2))","A",10,8,43,0.1860,1,"{4086, 4017, 3996, 3787, 3769, 3771, 2641, 2639}"
557,1,343,0,0.7177346,"\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","Int[((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{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)}","\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,"(4*a^3*(175*A + 195*B + 221*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (4*a^3*(95*A + 105*B + 121*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (20*a^3*(236*A + 273*B + 286*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (4*a^3*(175*A + 195*B + 221*C)*Sin[c + d*x])/(585*d*Sec[c + d*x]^(3/2)) + (4*a^3*(95*A + 105*B + 121*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2)) + (2*(6*A + 13*B)*(a^2 + a^2*Sec[c + d*x])^2*Sin[c + d*x])/(143*a*d*Sec[c + d*x]^(9/2)) + (2*(145*A + 195*B + 143*C)*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2))","A",11,8,43,0.1860,1,"{4086, 4017, 3996, 3787, 3769, 3771, 2639, 2641}"
558,1,250,0,0.2764204,"\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","Int[(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{(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}","-\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,"(-3*(5*A - 5*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((3*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + (3*(5*A - 5*B + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*a*d) - ((3*A - 5*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) + ((5*A - 5*B + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",9,6,43,0.1395,1,"{4084, 3787, 3768, 3771, 2641, 2639}"
559,1,205,0,0.2482902,"\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","Int[(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{(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}","-\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,"((A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((3*A - 3*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) - ((A - 3*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) + ((3*A - 3*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",8,6,43,0.1395,1,"{4084, 3787, 3768, 3771, 2639, 2641}"
560,1,162,0,0.2137081,"\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","Int[(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{(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}","-\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,"-(((A - B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((A + B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) + ((A - B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",7,6,43,0.1395,1,"{4084, 3787, 3771, 2641, 3768, 2639}"
561,1,133,0,0.2090643,"\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","Int[(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{(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}","-\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,"((3*A - B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*(a + a*Sec[c + d*x]))","A",6,5,43,0.1163,1,"{4084, 3787, 3771, 2639, 2641}"
562,1,174,0,0.2335656,"\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","Int[(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{(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}","\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,"-(((3*A - 3*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d)) + ((5*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((5*A - 3*B + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x]))","A",7,6,43,0.1395,1,"{4084, 3787, 3769, 3771, 2641, 2639}"
563,1,214,0,0.2537055,"\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","Int[(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{(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}","-\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,"(3*(7*A - 5*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) - ((5*A - 5*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a*d) + ((7*A - 5*B + 5*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - ((5*A - 5*B + 3*C)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x]))","A",8,6,43,0.1395,1,"{4084, 3787, 3769, 3771, 2639, 2641}"
564,1,250,0,0.2818342,"\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","Int[(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{(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}","-\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,"(-3*(7*A - 7*B + 5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a*d) + (5*(9*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a*d) + ((9*A - 7*B + 7*C)*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - ((7*A - 7*B + 5*C)*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) + (5*(9*A - 7*B + 7*C)*Sin[c + d*x])/(21*a*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x]))","A",9,6,43,0.1395,1,"{4084, 3787, 3769, 3771, 2641, 2639}"
565,1,251,0,0.4121101,"\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","Int[(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{(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}","-\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,"((A - 4*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((2*A - 5*B + 10*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((A - 4*B + 7*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((2*A - 5*B + 10*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d) - ((A - 4*B + 7*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",9,7,43,0.1628,1,"{4084, 4019, 3787, 3768, 3771, 2639, 2641}"
566,1,207,0,0.38067,"\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","Int[(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{(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}","\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,"((B - 4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + ((A + 2*B - 5*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((B - 4*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d) + ((A + 2*B - 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",8,7,43,0.1628,1,"{4084, 4019, 3787, 3771, 2641, 3768, 2639}"
567,1,173,0,0.3627179,"\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","Int[(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 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}","\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,"-(((A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((2*A + B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,6,43,0.1395,1,"{4084, 4019, 3787, 3771, 2639, 2641}"
568,1,184,0,0.364301,"\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","Int[(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{(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}","-\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,"((4*A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) - ((5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) - ((5*A - 2*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Sec[c + d*x])) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Sec[c + d*x])^2)","A",7,6,43,0.1395,1,"{4084, 4020, 3787, 3771, 2639, 2641}"
569,1,220,0,0.4029456,"\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","Int[(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{(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}","\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,"-(((7*A - 4*B + C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d)) + ((10*A - 5*B + 2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((10*A - 5*B + 2*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((7*A - 4*B + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2)","A",8,7,43,0.1628,1,"{4084, 4020, 3787, 3769, 3771, 2641, 2639}"
570,1,254,0,0.4253766,"\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","Int[(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{(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}","-\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,"((56*A - 35*B + 20*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^2*d) - (5*(3*A - 2*B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^2*d) + ((56*A - 35*B + 20*C)*Sin[c + d*x])/(15*a^2*d*Sec[c + d*x]^(3/2)) - (5*(3*A - 2*B + C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]]) - ((3*A - 2*B + C)*Sin[c + d*x])/(a^2*d*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2)","A",9,7,43,0.1628,1,"{4084, 4020, 3787, 3769, 3771, 2639, 2641}"
571,1,308,0,0.6211797,"\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","Int[(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{(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}","-\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,"((9*A - 49*B + 119*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((9*A - 49*B + 119*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) + ((3*A - 13*B + 33*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((B - 2*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*a*d*(a + a*Sec[c + d*x])^2) - ((9*A - 49*B + 119*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Sec[c + d*x]))","A",10,7,43,0.1628,1,"{4084, 4019, 3787, 3768, 3771, 2639, 2641}"
572,1,269,0,0.5841448,"\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","Int[(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{(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}","\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,"((A + 9*B - 49*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + 3*B - 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A + 9*B - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*a^3*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((2*A + 3*B - 8*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A + 3*B - 13*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",9,7,43,0.1628,1,"{4084, 4019, 3787, 3771, 2641, 3768, 2639}"
573,1,231,0,0.5488738,"\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","Int[(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{(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}","\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,"-((A - B - 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((A + B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((4*A + B - 6*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((A - B - 9*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,43,0.1395,1,"{4084, 4019, 3787, 3771, 2639, 2641}"
574,1,231,0,0.555442,"\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","Int[(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{(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}","\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,"-((9*A + B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((3*A + B + C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) + ((6*A - B - 4*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) + ((3*A + B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",8,7,43,0.1628,1,"{4084, 4019, 4020, 3787, 3771, 2639, 2641}"
575,1,241,0,0.5623112,"\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","Int[(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{(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}","-\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,"((49*A - 9*B - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Sec[c + d*x])^3) - ((8*A - 3*B - 2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Sec[c + d*x])^2) - ((13*A - 3*B - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Sec[c + d*x]))","A",8,6,43,0.1395,1,"{4084, 4020, 3787, 3771, 2639, 2641}"
576,1,274,0,0.5998192,"\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","Int[(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{(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}","\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,"-((119*A - 49*B + 9*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) + ((33*A - 13*B + 3*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + ((33*A - 13*B + 3*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3) - ((2*A - B)*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2) - ((119*A - 49*B + 9*C)*Sin[c + d*x])/(30*d*Sqrt[Sec[c + d*x]]*(a^3 + a^3*Sec[c + d*x]))","A",9,7,43,0.1628,1,"{4084, 4020, 3787, 3769, 3771, 2641, 2639}"
577,1,313,0,0.6386333,"\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","Int[(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{(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}","-\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,"(7*(33*A - 17*B + 7*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(10*a^3*d) - ((63*A - 33*B + 13*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(6*a^3*d) + (7*(33*A - 17*B + 7*C)*Sin[c + d*x])/(30*a^3*d*Sec[c + d*x]^(3/2)) - ((63*A - 33*B + 13*C)*Sin[c + d*x])/(6*a^3*d*Sqrt[Sec[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3) - ((12*A - 7*B + 2*C)*Sin[c + d*x])/(15*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2) - ((63*A - 33*B + 13*C)*Sin[c + d*x])/(10*d*Sec[c + d*x]^(3/2)*(a^3 + a^3*Sec[c + d*x]))","A",10,7,43,0.1628,1,"{4084, 4020, 3787, 3769, 3771, 2639, 2641}"
578,1,227,0,0.5183416,"\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","Int[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{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}","\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,"(Sqrt[a]*(48*A + 40*B + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a*(48*A + 40*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",6,5,45,0.1111,1,"{4088, 4016, 3803, 3801, 215}"
579,1,179,0,0.4227744,"\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","Int[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{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}","\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,"(Sqrt[a]*(8*A + 6*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a*(8*A + 6*B + 5*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(6*B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,45,0.1111,1,"{4088, 4016, 3803, 3801, 215}"
580,1,131,0,0.3384201,"\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","Int[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{\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}","\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,"(Sqrt[a]*(8*A + 4*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a*(4*B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",4,4,45,0.08889,1,"{4088, 4016, 3801, 215}"
581,1,119,0,0.3308948,"\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","Int[(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{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}","\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,"(Sqrt[a]*(2*B + C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a*(2*A - C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/d","A",4,4,45,0.08889,1,"{4088, 4015, 3801, 215}"
582,1,120,0,0.3207237,"\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","Int[(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{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}","\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,"(2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a*(A + 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",4,4,45,0.08889,1,"{4086, 4015, 3801, 215}"
583,1,129,0,0.3535313,"\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","Int[(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{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)}","\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,"(2*a*(7*A + 5*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]])","A",3,3,45,0.06667,1,"{4086, 4013, 3804}"
584,1,178,0,0.4257704,"\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","Int[(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{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)}","\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,"(2*a*(A + 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a*(24*A + 28*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",4,4,45,0.08889,1,"{4086, 4015, 3805, 3804}"
585,1,226,0,0.5100902,"\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","Int[(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{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)}","\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,"(2*a*(A + 9*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a*(16*A + 18*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",5,4,45,0.08889,1,"{4086, 4015, 3805, 3804}"
586,1,283,0,0.7481123,"\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","Int[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^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^{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 (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}","\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^{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 (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^(3/2)*(176*A + 150*B + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^2*(176*A + 150*B + 133*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 90*B + 67*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(240*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(10*B + 3*C)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",7,6,45,0.1333,1,"{4088, 4018, 4016, 3803, 3801, 215}"
587,1,233,0,0.6461212,"\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","Int[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^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^{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 (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}","\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^{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 (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^(3/2)*(112*A + 88*B + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^2*(112*A + 88*B + 75*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(64*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(48*A + 56*B + 39*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(96*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*B + 3*C)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",6,6,45,0.1333,1,"{4088, 4018, 4016, 3803, 3801, 215}"
588,1,181,0,0.548173,"\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","Int[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^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^{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 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}","\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^{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 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^(3/2)*(24*A + 14*B + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(2*B + C)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",5,5,45,0.1111,1,"{4088, 4018, 4016, 3801, 215}"
589,1,183,0,0.5248627,"\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","Int[((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^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^{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 (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}","\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^{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 (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^(3/2)*(8*A + 12*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^2*(8*A - 4*B - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (a*(4*B + 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",5,5,45,0.1111,1,"{4088, 4018, 4015, 3801, 215}"
590,1,177,0,0.522638,"\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","Int[((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^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^{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 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)}}","\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^{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 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^(3/2)*(2*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^2*(8*A + 6*B - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",5,5,45,0.1111,1,"{4086, 4018, 4015, 3801, 215}"
591,1,172,0,0.5120203,"\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","Int[((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{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^{3/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a (3 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)}","\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^{3/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a (3 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,"(2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",5,5,45,0.1111,1,"{4086, 4017, 4015, 3801, 215}"
592,1,181,0,0.465627,"\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","Int[((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{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)}","\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,"(8*a^2*(19*A + 21*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 21*B + 35*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(3*A + 7*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2))","A",4,4,45,0.08889,1,"{4086, 4013, 3809, 3804}"
593,1,232,0,0.6534024,"\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","Int[((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{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)}","\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,"(2*a^2*(52*A + 72*B + 63*C)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^2*(136*A + 156*B + 189*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 3*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",5,5,45,0.1111,1,"{4086, 4017, 4015, 3805, 3804}"
594,1,284,0,0.739475,"\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","Int[((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{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)}","\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,"(2*a^2*(84*A + 110*B + 99*C)*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(1155*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(336*A + 374*B + 429*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 11*B)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",6,5,45,0.1111,1,"{4086, 4017, 4015, 3805, 3804}"
595,1,333,0,0.9494989,"\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","Int[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 (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^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^{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 (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}","\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^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^{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 (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^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(512*d) + (a^3*(1304*A + 1132*B + 1015*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(512*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(768*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(680*A + 628*B + 545*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(480*d) + (a*(12*B + 5*C)*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(60*d) + (C*Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d)","A",8,6,45,0.1333,1,"{4088, 4018, 4016, 3803, 3801, 215}"
596,1,281,0,0.8584462,"\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","Int[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^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^{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 (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}","\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^{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 (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^(5/2)*(400*A + 326*B + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(128*d) + (a^3*(400*A + 326*B + 283*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(128*d*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1040*A + 950*B + 787*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(960*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 110*B + 79*C)*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + (a*(2*B + C)*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (C*Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,6,45,0.1333,1,"{4088, 4018, 4016, 3803, 3801, 215}"
597,1,233,0,0.7522731,"\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","Int[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^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^{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 (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}","\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^{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 (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^(5/2)*(304*A + 200*B + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(192*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (a*(8*B + 5*C)*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",6,5,45,0.1111,1,"{4088, 4018, 4016, 3801, 215}"
598,1,233,0,0.7256269,"\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","Int[((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^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^{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 (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}","\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^{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 (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^(5/2)*(40*A + 38*B + 25*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*d) + (a^3*(24*A - 54*B - 49*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 42*B + 31*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*(6*B + 5*C)*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",6,5,45,0.1111,1,"{4088, 4018, 4015, 3801, 215}"
599,1,233,0,0.7414832,"\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","Int[((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^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^{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 (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)}}","\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^{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 (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^(5/2)*(8*A + 20*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*d) + (a^3*(56*A + 12*B - 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 12*B - 21*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d) - (a*(4*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",6,5,45,0.1111,1,"{4086, 4018, 4015, 3801, 215}"
600,1,223,0,0.7192203,"\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","Int[((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^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{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{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)}","\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{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{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^(5/2)*(2*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (a^3*(64*A + 70*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A + 10*B - 15*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(A + B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",6,6,45,0.1333,1,"{4086, 4017, 4018, 4015, 3801, 215}"
601,1,222,0,0.6966767,"\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","Int[((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{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^{5/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\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)}","\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^{5/2} C \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\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,"(2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*a*(5*A + 7*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",6,5,45,0.1111,1,"{4086, 4017, 4015, 3801, 215}"
602,1,231,0,0.5552454,"\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","Int[((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{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)}","\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,"(64*a^3*(13*A + 15*B + 21*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 15*B + 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) + (2*a*(13*A + 15*B + 21*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(3/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(5*A + 9*B)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2))","A",5,4,45,0.08889,1,"{4086, 4013, 3809, 3804}"
603,1,284,0,0.8634858,"\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","Int[((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{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{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{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 (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)}","\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{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{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 (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,"(2*a^3*(1160*A + 1364*B + 1485*C)*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (4*a^3*(2840*A + 3212*B + 3795*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*a*(5*A + 11*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",6,5,45,0.1111,1,"{4086, 4017, 4015, 3805, 3804}"
604,1,334,0,0.9405318,"\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","Int[((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{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{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{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 (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)}","\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{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{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 (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,"(2*a^3*(2224*A + 2522*B + 2717*C)*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(15015*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^3*(8368*A + 9230*B + 10439*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*a*(5*A + 13*B)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2))","A",7,5,45,0.1111,1,"{4086, 4017, 4015, 3805, 3804}"
605,1,241,0,0.8035344,"\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","Int[(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{(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}}","\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,"-((8*A - 14*B + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((8*A - 2*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(8*d*Sqrt[a + a*Sec[c + d*x]]) + ((6*B - C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(12*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,45,0.1556,1,"{4088, 4021, 4023, 3808, 206, 3801, 215}"
606,1,195,0,0.5985306,"\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","Int[(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{\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}}","-\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,"((8*A - 4*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + ((4*B - C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*Sqrt[a + a*Sec[c + d*x]]) + (C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4088, 4021, 4023, 3808, 206, 3801, 215}"
607,1,141,0,0.4175668,"\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","Int[(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{\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}}","\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,"((2*B - C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4088, 4023, 3808, 206, 3801, 215}"
608,1,138,0,0.3882906,"\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","Int[(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{\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}","-\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*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4086, 4023, 3808, 206, 3801, 215}"
609,1,143,0,0.3614456,"\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","Int[(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{\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}}","\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,"(Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 3*B)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,45,0.08889,1,"{4086, 4013, 3808, 206}"
610,1,191,0,0.5706309,"\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","Int[(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{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}}","\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,"-((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d)) + (2*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(13*A - 5*B + 15*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,45,0.1111,1,"{4086, 4022, 4013, 3808, 206}"
611,1,237,0,0.7582921,"\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","Int[(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{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}}","-\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,"(Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(43*A - 91*B + 35*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,45,0.1111,1,"{4086, 4022, 4013, 3808, 206}"
612,1,152,0,0.472165,"\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","Int[(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{\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}}","\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,"((2*A*b + 2*a*B - b*B)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(Sqrt[a]*d) + (Sqrt[2]*(a - b)*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(Sqrt[a]*d) + (b*B*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,54,0.1111,1,"{4088, 4023, 3808, 206, 3801, 215}"
613,1,260,0,0.9062601,"\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","Int[(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{(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}}","-\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,"((8*A - 12*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((2*A - 6*B + 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((A - B + 2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,45,0.1556,1,"{4084, 4021, 4023, 3808, 206, 3801, 215}"
614,1,202,0,0.6035145,"\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","Int[(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{(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}}","\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,"((2*B - 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 3*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4084, 4021, 4023, 3808, 206, 3801, 215}"
615,1,149,0,0.4061623,"\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","Int[(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{(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}}","\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,"(2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2))","A",6,6,45,0.1333,1,"{4084, 4023, 3808, 206, 3801, 215}"
616,1,161,0,0.382554,"\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","Int[(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{(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}}","-\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,"-((7*A - 3*B - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*d*Sqrt[a + a*Sec[c + d*x]])","A",4,4,45,0.08889,1,"{4084, 4013, 3808, 206}"
617,1,213,0,0.5746403,"\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","Int[(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{(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}}","\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,"((11*A - 7*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((7*A - 3*B + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((19*A - 15*B + 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,45,0.1111,1,"{4084, 4022, 4013, 3808, 206}"
618,1,263,0,0.7529569,"\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","Int[(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{(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}}","-\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,"-((15*A - 11*B + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((9*A - 5*B + 5*C)*Sin[c + d*x])/(10*a*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B + 15*C)*Sin[c + d*x])/(30*a*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((147*A - 95*B + 75*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,5,45,0.1111,1,"{4084, 4022, 4013, 3808, 206}"
619,1,254,0,0.8248558,"\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","Int[(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{(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{(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{(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}}","\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{(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{(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,"((2*B - 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((A + 7*B - 15*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - 11*B + 35*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4084, 4019, 4021, 4023, 3808, 206, 3801, 215}"
620,1,201,0,0.5962847,"\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","Int[(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{(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}}","\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,"(2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B - 11*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",7,7,45,0.1556,1,"{4084, 4019, 4023, 3808, 206, 3801, 215}"
621,1,163,0,0.4082287,"\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","Int[(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{(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}}","\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,"((19*A + 5*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - B - 7*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2))","A",4,4,45,0.08889,1,"{4084, 4012, 3808, 206}"
622,1,211,0,0.5896,"\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","Int[(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{(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{(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{(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}}","\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{(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{(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,"-((75*A - 19*B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(16*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,45,0.1111,1,"{4084, 4020, 4013, 3808, 206}"
623,1,261,0,0.8020001,"\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","Int[(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{(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{(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{(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}}","-\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{(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{(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,"((163*A - 75*B + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sin[c + d*x])/(16*a*d*Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((95*A - 39*B + 15*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((299*A - 147*B + 27*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4084, 4020, 4022, 4013, 3808, 206}"
624,1,313,0,0.9900285,"\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","Int[(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{(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{(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{(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}}","\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{(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{(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,"-((283*A - 163*B + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B + 5*C)*Sin[c + d*x])/(16*a*d*Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)) + ((157*A - 85*B + 45*C)*Sin[c + d*x])/(80*a^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B + 195*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((2671*A - 1495*B + 735*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,45,0.1333,1,"{4084, 4020, 4022, 4013, 3808, 206}"
625,1,446,0,0.7226435,"\int (a+a \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(3*C*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(5*d) + (3*Sqrt[2]*A*AppellF1[7/6, 1/2, 1, 13/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(7*d*Sqrt[1 - Sec[c + d*x]]) + (3*(5*B + 2*C)*(a + a*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(10*d*(1 + Sec[c + d*x])) - (3^(3/4)*(5*B + 2*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(10*2^(1/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",10,10,35,0.2857,1,"{4054, 3924, 3779, 3778, 136, 3828, 3827, 50, 63, 225}"
626,1,390,0,0.4516483,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(1/3),x]","\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}}","\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*Tan[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(1/3)) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) - (3^(3/4)*(2*B - C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(1/3)*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",9,9,35,0.2571,1,"{4054, 3924, 3779, 3778, 136, 3828, 3827, 63, 225}"
627,1,402,0,0.4690289,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{4/3}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(4/3),x]","\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}}","\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,"(-3*(A - B + C)*Tan[c + d*x])/(5*d*(a + a*Sec[c + d*x])^(4/3)) + (3*Sqrt[2]*A*AppellF1[1/6, 1/2, 1, 7/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(A - B - 4*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(5*2^(1/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",9,9,35,0.2571,1,"{4052, 3924, 3779, 3778, 136, 3828, 3827, 63, 225}"
628,1,466,0,0.5184514,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{7/3}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(7/3),x]","-\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}}","-\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,"(-3*(A - B + C)*Tan[c + d*x])/(11*d*(a + a*Sec[c + d*x])^(7/3)) - (3*(4*A - 4*B - 7*C)*Tan[c + d*x])/(55*a^2*d*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) - (3*Sqrt[2]*A*AppellF1[-5/6, 1/2, 1, 1/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(5*a^2*d*Sqrt[1 - Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)) + (3^(3/4)*(4*A - 4*B - 7*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(55*2^(1/3)*a^2*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",10,10,35,0.2857,1,"{4052, 3924, 3779, 3778, 136, 3828, 3827, 51, 63, 225}"
629,1,839,0,1.0677914,"\int (a+a \sec (c+d x))^{4/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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)}","\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,"(3*a*(7*B + 4*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(28*d) + (3*Sqrt[2]*a*A*AppellF1[11/6, 1/2, 1, 17/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(11*d*Sqrt[1 - Sec[c + d*x]]) + (3*C*(a + a*Sec[c + d*x])^(4/3)*Tan[c + d*x])/(7*d) - (15*(1 + Sqrt[3])*a*(7*B + 4*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(28*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (15*3^(1/4)*a*(7*B + 4*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(14*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (5*3^(3/4)*(1 - Sqrt[3])*a*(7*B + 4*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(28*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",12,12,35,0.3429,1,"{4054, 3924, 3779, 3778, 136, 3828, 3827, 50, 63, 308, 225, 1881}"
630,1,786,0,0.8446912,"\int \sqrt[3]{a+a \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(3*C*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*(4*B + C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(4*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*3^(1/4)*(4*B + C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(4*B + C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(4*2^(2/3)*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",11,11,35,0.3143,1,"{4054, 3924, 3779, 3778, 136, 3828, 3827, 63, 308, 225, 1881}"
631,1,803,0,0.8875936,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(2/3),x]","\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)}","\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,"(-3*(A - B + C)*Tan[c + d*x])/(d*(a + a*Sec[c + d*x])^(2/3)) + (3*Sqrt[2]*A*AppellF1[5/6, 1/2, 1, 11/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(5*a*d*Sqrt[1 - Sec[c + d*x]]) - (3*(1 + Sqrt[3])*(A - B + 2*C)*(a + a*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(a*d*(1 + Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(A - B + 2*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(A - B + 2*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(a + a*Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(2^(2/3)*a*d*(1 - Sec[c + d*x])*(1 + Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",11,11,35,0.3143,1,"{4052, 3924, 3779, 3778, 136, 3828, 3827, 63, 308, 225, 1881}"
632,1,856,0,0.9738709,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+a \sec (c+d x))^{5/3}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + a*Sec[c + d*x])^(5/3),x]","\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}}","\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,"(-3*(A - B + C)*Tan[c + d*x])/(7*d*(a + a*Sec[c + d*x])^(5/3)) - (3*(2*A - 2*B - 5*C)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)) - (3*Sqrt[2]*A*AppellF1[-1/6, 1/2, 1, 5/6, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*Tan[c + d*x])/(a*d*Sqrt[1 - Sec[c + d*x]]*(a + a*Sec[c + d*x])^(2/3)) - (3*(1 + Sqrt[3])*(2*A - 2*B - 5*C)*(1 + Sec[c + d*x])^(1/3)*Tan[c + d*x])/(7*a*d*(a + a*Sec[c + d*x])^(2/3)*(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))) + (3*2^(1/3)*3^(1/4)*(2*A - 2*B - 5*C)*EllipticE[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)]) + (3^(3/4)*(1 - Sqrt[3])*(2*A - 2*B - 5*C)*EllipticF[ArcCos[(2^(1/3) - (1 - Sqrt[3])*(1 + Sec[c + d*x])^(1/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))], (2 + Sqrt[3])/4]*(1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3))*Sqrt[(2^(2/3) + 2^(1/3)*(1 + Sec[c + d*x])^(1/3) + (1 + Sec[c + d*x])^(2/3))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2]*Tan[c + d*x])/(7*2^(2/3)*a*d*(1 - Sec[c + d*x])*(a + a*Sec[c + d*x])^(2/3)*Sqrt[-(((1 + Sec[c + d*x])^(1/3)*(2^(1/3) - (1 + Sec[c + d*x])^(1/3)))/(2^(1/3) - (1 + Sqrt[3])*(1 + Sec[c + d*x])^(1/3))^2)])","A",12,12,35,0.3429,1,"{4052, 3924, 3779, 3778, 136, 3828, 3827, 51, 63, 308, 225, 1881}"
633,1,259,0,0.6124146,"\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","Int[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]","\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)}","\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,"(C*Sec[c + d*x]^(1 + m)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + m + n)) + (2^(3/2 + n)*(C*n + B*(1 + m + n))*AppellF1[1/2, 1 - m, -1/2 - n, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n)) + (2^(1/2 + n)*(C*(m - n) + A*(1 + m + n) - B*(1 + m + n))*AppellF1[1/2, 1 - m, 1/2 - n, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/(d*(1 + m + n))","A",8,5,41,0.1220,1,"{4088, 4023, 3828, 3825, 133}"
634,1,258,0,0.5554728,"\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","Int[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 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}","\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,"(A*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n) + ((A*n + B*(1 + n) - C*(1 + n))*Hypergeometric2F1[1/2 - n, -n, 1 - n, (-2*Sec[c + d*x])/(1 - Sec[c + d*x])]*Sec[c + d*x]^(1 - n)*((1 + Sec[c + d*x])/(1 - Sec[c + d*x]))^(1/2 - n)*(a + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*n*(1 + n)*(1 + Sec[c + d*x])) + (2^(3/2 + n)*C*AppellF1[1/2, 1 + n, -1/2 - n, 3/2, 1 - Sec[c + d*x], (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - n)*(a + a*Sec[c + d*x])^n*Tan[c + d*x])/d","A",8,6,45,0.1333,1,"{4086, 4023, 3828, 3825, 132, 133}"
635,1,38,0,0.9814811,"\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","Int[((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)+a)^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 + a*Sec[c + d*x])^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n)","A",16,6,102,0.05882,1,"{4023, 3828, 3825, 132, 133, 4086}"
636,1,171,0,0.2635874,"\int (a+a \sec (c+d x))^m \left(B-C+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + a*Sec[c + d*x])^m*(B - C + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(Sqrt[2]*(B - C)*AppellF1[3/2 + m, 1/2, 1, 5/2 + m, (1 + Sec[c + d*x])/2, 1 + Sec[c + d*x]]*(1 + Sec[c + d*x])*(a + a*Sec[c + d*x])^m*Tan[c + d*x])/(d*(3 + 2*m)*Sqrt[1 - Sec[c + d*x]]) + (2^(3/2 + m)*C*Hypergeometric2F1[1/2, -1/2 - m, 3/2, (1 - Sec[c + d*x])/2]*(1 + Sec[c + d*x])^(-1/2 - m)*(a + a*Sec[c + d*x])^m*Tan[c + d*x])/d","A",8,8,36,0.2222,1,"{4041, 3924, 3779, 3778, 136, 3828, 3827, 69}"
637,1,140,0,0.1749578,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (b*(5*A + 4*C)*Tan[c + d*x])/(5*d) + (a*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (b*(5*A + 4*C)*Tan[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4077, 4047, 3767, 4046, 3768, 3770}"
638,1,117,0,0.1630164,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(b*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (b*(4*A + 3*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (b*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,31,0.2258,1,"{4077, 4047, 3768, 3770, 4046, 3767, 8}"
639,1,86,0,0.1029994,"\int \sec (c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a*(2*A + C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(3*A + 2*C)*Tan[c + d*x])/(3*d) + (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,29,0.2069,1,"{4077, 4047, 3767, 8, 4046, 3770}"
640,1,58,0,0.0536912,"\int (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(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{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}","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 + (b*(2*A + 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",5,4,23,0.1739,1,"{4049, 3770, 3767, 8}"
641,1,42,0,0.0960225,"\int \cos (c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*Sin[c + d*x])/d + (b*C*Tan[c + d*x])/d","A",5,5,29,0.1724,1,"{4077, 4047, 8, 4045, 3770}"
642,1,58,0,0.1225368,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(A + 2*C)*x)/2 + (b*C*ArcTanh[Sin[c + d*x]])/d + (A*b*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",5,5,31,0.1613,1,"{4075, 4047, 8, 4045, 3770}"
643,1,77,0,0.1409668,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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)","\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,"(b*(A + 2*C)*x)/2 + (a*(2*A + 3*C)*Sin[c + d*x])/(3*d) + (A*b*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{4075, 4047, 2637, 4045, 8}"
644,1,95,0,0.1727791,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a*(3*A + 4*C)*x)/8 + (b*(A + C)*Sin[c + d*x])/d + (a*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (A*b*Sin[c + d*x]^3)/(3*d)","A",7,6,31,0.1935,1,"{4075, 4047, 2635, 8, 4044, 3013}"
645,1,131,0,0.1727932,"\int \cos ^5(c+d x) (a+b \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","-\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)","-\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,"(b*(3*A + 4*C)*x)/8 + (a*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (b*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - (a*(4*A + 5*C)*Sin[c + d*x]^3)/(15*d)","A",7,6,31,0.1935,1,"{4075, 4047, 2633, 4045, 2635, 8}"
646,1,226,0,0.5029383,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\frac{\left(2 a^2 b^2 (5 A+3 C)+a^4 C+2 b^4 (5 A+4 C)\right) \tan (c+d x)}{15 b^2 d}+\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{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}","\frac{\left(2 a^2 b^2 (5 A+3 C)+a^4 C+2 b^4 (5 A+4 C)\right) \tan (c+d x)}{15 b^2 d}+\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{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,"(a*b*(4*A + 3*C)*ArcTanh[Sin[c + d*x]])/(4*d) + ((a^4*C + 2*a^2*b^2*(5*A + 3*C) + 2*b^4*(5*A + 4*C))*Tan[c + d*x])/(15*b^2*d) + (a*(20*A*b^2 + 2*a^2*C + 13*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(60*b*d) + ((a^2*C + 2*b^2*(5*A + 4*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(30*b^2*d) - (a*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(10*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*b*d)","A",8,8,33,0.2424,1,"{4093, 4082, 4002, 3997, 3787, 3770, 3767, 8}"
647,1,170,0,0.3083345,"\int \sec (c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"((4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(12*A*b^2 - a^2*C + 8*b^2*C)*Tan[c + d*x])/(6*b*d) - ((2*a^2*C - 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (a*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)","A",7,7,31,0.2258,1,"{4083, 4002, 3997, 3787, 3770, 3767, 8}"
648,1,103,0,0.1393012,"\int (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"a^2*A*x + (a*b*(2*A + C)*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^2 + 2*(a^2 + b^2)*C)*Tan[c + d*x])/(3*d) + (a*b*C*Sec[c + d*x]*Tan[c + d*x])/(3*d) + (C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,5,25,0.2000,1,"{4057, 4048, 3770, 3767, 8}"
649,1,109,0,0.164549,"\int \cos (c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"2*a*A*b*x + ((2*A*b^2 + (2*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (2*a*b*(A - C)*Tan[c + d*x])/d - (b^2*(2*A - C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,5,31,0.1613,1,"{4095, 4048, 3770, 3767, 8}"
650,1,103,0,0.2896069,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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*A*b^2 + a^2*(A + 2*C))*x)/2 + (2*a*b*C*ArcTanh[Sin[c + d*x]])/d + (a*A*b*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(A - 2*C)*Tan[c + d*x])/(2*d)","A",6,6,33,0.1818,1,"{4095, 4076, 4047, 8, 4045, 3770}"
651,1,112,0,0.2912471,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"a*b*(A + 2*C)*x + (b^2*C*ArcTanh[Sin[c + d*x]])/d + ((2*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + (a*A*b*Cos[c + d*x]*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{4095, 4074, 4047, 8, 4045, 3770}"
652,1,145,0,0.3819236,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"((4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x)/8 + (2*a*b*(2*A + 3*C)*Sin[c + d*x])/(3*d) + ((2*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*b*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)","A",6,6,33,0.1818,1,"{4095, 4074, 4047, 2637, 4045, 8}"
653,1,161,0,0.3995492,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","-\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)","-\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,"(a*b*(3*A + 4*C)*x)/4 + ((a^2 + b^2)*(4*A + 5*C)*Sin[c + d*x])/(5*d) + (a*b*(3*A + 4*C)*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*A*b*Cos[c + d*x]^3*Sin[c + d*x])/(10*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - ((2*A*b^2 + a^2*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)","A",8,7,33,0.2121,1,"{4095, 4074, 4047, 2635, 8, 4044, 3013}"
654,1,306,0,0.7192355,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\frac{a \left(a^2 b^2 (30 A+17 C)+2 a^4 C+24 b^4 (5 A+4 C)\right) \tan (c+d x)}{60 b^2 d}+\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{\left(12 a^2 b^2 (5 A+3 C)+4 a^4 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}","\frac{a \left(a^2 b^2 (30 A+17 C)+2 a^4 C+24 b^4 (5 A+4 C)\right) \tan (c+d x)}{60 b^2 d}+\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{\left(12 a^2 b^2 (5 A+3 C)+4 a^4 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,"(b*(6*a^2*(4*A + 3*C) + b^2*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) + (a*(2*a^4*C + 24*b^4*(5*A + 4*C) + a^2*b^2*(30*A + 17*C))*Tan[c + d*x])/(60*b^2*d) + ((4*a^4*C + 12*a^2*b^2*(5*A + 3*C) + 15*b^4*(6*A + 5*C))*Sec[c + d*x]*Tan[c + d*x])/(240*b*d) + (a*(30*A*b^2 + 2*a^2*C + 21*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b^2*d) + ((2*a^2*C + 5*b^2*(6*A + 5*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b^2*d) - (a*C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(15*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(6*b*d)","A",9,8,33,0.2424,1,"{4093, 4082, 4002, 3997, 3787, 3770, 3767, 8}"
655,1,234,0,0.4883043,"\int \sec (c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\frac{\left(-4 a^2 b^2 (20 A+13 C)+3 a^4 C-4 b^4 (5 A+4 C)\right) \tan (c+d x)}{30 b d}+\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{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}","-\frac{\left(-4 a^2 b^2 (20 A+13 C)+3 a^4 C-4 b^4 (5 A+4 C)\right) \tan (c+d x)}{30 b d}+\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{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,"(a*(4*a^2*(2*A + C) + 3*b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) - ((3*a^4*C - 4*b^4*(5*A + 4*C) - 4*a^2*b^2*(20*A + 13*C))*Tan[c + d*x])/(30*b*d) + (a*(100*A*b^2 - 6*a^2*C + 71*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) - ((3*a^2*C - 4*b^2*(5*A + 4*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) - (a*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)","A",8,7,31,0.2258,1,"{4083, 4002, 3997, 3787, 3770, 3767, 8}"
656,1,167,0,0.3132527,"\int (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),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}+a^3 A x+\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}","\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}+a^3 A x+\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,"a^3*A*x + (b*(12*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (a*(6*A*b^2 + (a^2 + 4*b^2)*C)*Tan[c + d*x])/(2*d) + (b*(2*a^2*C + b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (a*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,6,25,0.2400,1,"{4057, 4056, 4048, 3770, 3767, 8}"
657,1,167,0,0.3120662,"\int \cos (c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"3*a^2*A*b*x + (a*(6*A*b^2 + 2*a^2*C + 3*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d - (b*(a^2*(6*A - 8*C) - b^2*(3*A + 2*C))*Tan[c + d*x])/(3*d) - (a*b^2*(6*A - 5*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",7,6,31,0.1935,1,"{4095, 4056, 4048, 3770, 3767, 8}"
658,1,168,0,0.3931035,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a*(6*A*b^2 + a^2*(A + 2*C))*x)/2 + (b*(2*A*b^2 + (6*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (3*A*b*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - (3*a*b^2*(3*A - 2*C)*Tan[c + d*x])/(2*d) - (b^3*(4*A - C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,6,33,0.1818,1,"{4095, 4094, 4048, 3770, 3767, 8}"
659,1,163,0,0.5236041,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(b*(2*A*b^2 + 3*a^2*(A + 2*C))*x)/2 + (3*a*b^2*C*ArcTanh[Sin[c + d*x]])/d + (a*(3*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + (A*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) - (b^3*(5*A - 6*C)*Tan[c + d*x])/(6*d)","A",7,7,33,0.2121,1,"{4095, 4094, 4076, 4047, 8, 4045, 3770}"
660,1,182,0,0.5586593,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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 b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{4 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}","\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 b \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{4 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,"(a*(12*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x)/8 + (b^3*C*ArcTanh[Sin[c + d*x]])/d + (b*(A*b^2 + a^2*(4*A + 6*C))*Sin[c + d*x])/(2*d) + (a*(2*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (A*b*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d)","A",7,7,33,0.2121,1,"{4095, 4094, 4074, 4047, 8, 4045, 3770}"
661,1,218,0,0.650911,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(b*(4*b^2*(A + 2*C) + 3*a^2*(3*A + 4*C))*x)/8 + (a*(15*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Sin[c + d*x])/(15*d) + (3*b*(2*A*b^2 + 5*a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a*(3*A*b^2 + 2*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + (3*A*b*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d)","A",7,7,33,0.2121,1,"{4095, 4094, 4074, 4047, 2637, 4045, 8}"
662,1,257,0,0.7523508,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*x)/16 + (b*(9*a^2*(4*A + 5*C) + b^2*(11*A + 15*C))*Sin[c + d*x])/(15*d) + (a*(6*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(6*A*b^2 + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + (A*b*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) - (b*(A*b^2 + 3*a^2*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)","A",9,8,33,0.2424,1,"{4095, 4094, 4074, 4047, 2635, 8, 4044, 3013}"
663,1,381,0,0.9827168,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{\left(a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)+2 a^6 C+8 b^6 (7 A+6 C)\right) \tan (c+d x)}{105 b^2 d}+\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(3 a^2 b^2 (14 A+9 C)+2 a^4 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(12 a^2 b^2 (7 A+4 C)+4 a^4 C+b^4 (406 A+333 C)\right) \tan (c+d x) \sec (c+d x)}{420 b 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}","\frac{\left(a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)+2 a^6 C+8 b^6 (7 A+6 C)\right) \tan (c+d x)}{105 b^2 d}+\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(3 a^2 b^2 (14 A+9 C)+2 a^4 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(12 a^2 b^2 (7 A+4 C)+4 a^4 C+b^4 (406 A+333 C)\right) \tan (c+d x) \sec (c+d x)}{420 b 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,"(a*b*(b^2*(6*A + 5*C) + a^2*(8*A + 6*C))*ArcTanh[Sin[c + d*x]])/(4*d) + ((2*a^6*C + 8*b^6*(7*A + 6*C) + a^4*b^2*(42*A + 23*C) + 8*a^2*b^4*(49*A + 39*C))*Tan[c + d*x])/(105*b^2*d) + (a*(4*a^4*C + 12*a^2*b^2*(7*A + 4*C) + b^4*(406*A + 333*C))*Sec[c + d*x]*Tan[c + d*x])/(420*b*d) + ((2*a^4*C + 8*b^4*(7*A + 6*C) + 3*a^2*b^2*(14*A + 9*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(210*b^2*d) + (a*(42*A*b^2 + 2*a^2*C + 31*b^2*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(210*b^2*d) + ((a^2*C + 3*b^2*(7*A + 6*C))*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(105*b^2*d) - (a*C*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(21*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(7*b*d)","A",10,8,33,0.2424,1,"{4093, 4082, 4002, 3997, 3787, 3770, 3767, 8}"
664,1,310,0,0.7047788,"\int \sec (c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\frac{a \left(-a^2 b^2 (190 A+121 C)+4 a^4 C-32 b^4 (5 A+4 C)\right) \tan (c+d x)}{60 b d}+\frac{\left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+b^4 (6 A+5 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\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{\left(-2 a^2 b^2 (130 A+89 C)+8 a^4 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}","-\frac{a \left(-a^2 b^2 (190 A+121 C)+4 a^4 C-32 b^4 (5 A+4 C)\right) \tan (c+d x)}{60 b d}+\frac{\left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+b^4 (6 A+5 C)\right) \tanh ^{-1}(\sin (c+d x))}{16 d}-\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{\left(-2 a^2 b^2 (130 A+89 C)+8 a^4 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,"((8*a^4*(2*A + C) + 12*a^2*b^2*(4*A + 3*C) + b^4*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) - (a*(4*a^4*C - 32*b^4*(5*A + 4*C) - a^2*b^2*(190*A + 121*C))*Tan[c + d*x])/(60*b*d) - ((8*a^4*C - 15*b^4*(6*A + 5*C) - 2*a^2*b^2*(130*A + 89*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + (a*(70*A*b^2 - 4*a^2*C + 53*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) - ((4*a^2*C - 5*b^2*(6*A + 5*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) - (a*C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (C*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)","A",9,7,31,0.2258,1,"{4083, 4002, 3997, 3787, 3770, 3767, 8}"
665,1,227,0,0.4782014,"\int (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{\left(a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right) \tan (c+d x)}{15 d}+\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}+a^4 A x+\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}","\frac{\left(a^2 b^2 (85 A+56 C)+6 a^4 C+2 b^4 (5 A+4 C)\right) \tan (c+d x)}{15 d}+\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}+a^4 A x+\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,"a^4*A*x + (a*b*(4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((6*a^4*C + 2*b^4*(5*A + 4*C) + a^2*b^2*(85*A + 56*C))*Tan[c + d*x])/(15*d) + (a*b*(40*A*b^2 + 6*a^2*C + 29*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(30*d) + ((3*a^2*C + b^2*(5*A + 4*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(15*d) + (a*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)","A",8,6,25,0.2400,1,"{4057, 4056, 4048, 3770, 3767, 8}"
666,1,229,0,0.4927176,"\int \cos (c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),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{\left(24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 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}+4 a^3 A b x-\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}","-\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{\left(24 a^2 b^2 (2 A+C)+8 a^4 C+b^4 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 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}+4 a^3 A b x-\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,"4*a^3*A*b*x + ((8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/d - (a*b*(a^2*(12*A - 19*C) - 8*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) - (b^2*(a^2*(24*A - 26*C) - 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (a*b*(12*A - 7*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",8,6,31,0.1935,1,"{4095, 4056, 4048, 3770, 3767, 8}"
667,1,219,0,0.6050498,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\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{b^2 (15 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^2}{6 d}-\frac{a b^3 (9 A-4 C) \tan (c+d x) \sec (c+d x)}{3 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}","-\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{b^2 (15 A-2 C) \tan (c+d x) (a+b \sec (c+d x))^2}{6 d}-\frac{a b^3 (9 A-4 C) \tan (c+d x) \sec (c+d x)}{3 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*(12*A*b^2 + a^2*(A + 2*C))*x)/2 + (2*a*b*(2*A*b^2 + (2*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/d + (2*A*b*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - (b^2*(a^2*(39*A - 34*C) - 2*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) - (a*b^3*(9*A - 4*C)*Sec[c + d*x]*Tan[c + d*x])/(3*d) - (b^2*(15*A - 2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(6*d)","A",8,7,33,0.2121,1,"{4095, 4094, 4056, 4048, 3770, 3767, 8}"
668,1,251,0,0.7542705,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"2*a*b*(2*A*b^2 + a^2*(A + 2*C))*x + (b^2*(2*A*b^2 + (12*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((6*A*b^2 + a^2*(2*A + 3*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d) + (2*A*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - (2*a*b*(b^2*(11*A - 6*C) + a^2*(2*A + 3*C))*Tan[c + d*x])/(3*d) - (b^2*(3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(6*d)","A",8,6,33,0.1818,1,"{4095, 4094, 4048, 3770, 3767, 8}"
669,1,246,0,0.8465021,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 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}","\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(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 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,"((8*A*b^4 + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x)/8 + (4*a*b^3*C*ArcTanh[Sin[c + d*x]])/d + (a*b*(12*A*b^2 + a^2*(23*A + 36*C))*Sin[c + d*x])/(12*d) + ((4*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) + (A*b*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) - (b^2*(2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Tan[c + d*x])/(24*d)","A",8,7,33,0.2121,1,"{4095, 4094, 4076, 4047, 8, 4045, 3770}"
670,1,250,0,0.8844527,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\frac{\left(a^2 b^2 (56 A+85 C)+2 a^4 (4 A+5 C)+6 A b^4\right) \sin (c+d x)}{15 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{A b \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{5 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}","\frac{\left(a^2 b^2 (56 A+85 C)+2 a^4 (4 A+5 C)+6 A b^4\right) \sin (c+d x)}{15 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{A b \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{5 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,"(a*b*(4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x)/2 + (b^4*C*ArcTanh[Sin[c + d*x]])/d + ((6*A*b^4 + 2*a^4*(4*A + 5*C) + a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(15*d) + (a*b*(6*A*b^2 + a^2*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(30*d) + ((3*A*b^2 + a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + (A*b*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(5*d)","A",8,7,33,0.2121,1,"{4095, 4094, 4074, 4047, 8, 4045, 3770}"
671,1,298,0,1.0372588,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","\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(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right) \sin (c+d x) \cos (c+d x)}{240 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{1}{16} x \left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 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}","\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(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+24 A b^4\right) \sin (c+d x) \cos (c+d x)}{240 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{1}{16} x \left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 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,"((8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*x)/16 + (4*a*b*(5*b^2*(2*A + 3*C) + 2*a^2*(4*A + 5*C))*Sin[c + d*x])/(15*d) + ((24*A*b^4 + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*b*(4*A*b^2 + a^2*(39*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + ((12*A*b^2 + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(120*d) + (2*A*b*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(6*d)","A",8,7,33,0.2121,1,"{4095, 4094, 4074, 4047, 2637, 4045, 8}"
672,1,339,0,1.1513188,"\int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^7*(a + b*Sec[c + d*x])^4*(A + C*Sec[c + d*x]^2),x]","-\frac{\left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+4 A b^4\right) \sin ^3(c+d x)}{105 d}+\frac{\left(3 a^2 b^2 (162 A+203 C)+12 a^4 (6 A+7 C)+b^4 (74 A+105 C)\right) \sin (c+d x)}{105 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{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}","-\frac{\left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+4 A b^4\right) \sin ^3(c+d x)}{105 d}+\frac{\left(3 a^2 b^2 (162 A+203 C)+12 a^4 (6 A+7 C)+b^4 (74 A+105 C)\right) \sin (c+d x)}{105 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{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,"(a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*x)/4 + ((12*a^4*(6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^2*(162*A + 203*C))*Sin[c + d*x])/(105*d) + (a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(4*d) + (a*b*(6*A*b^2 + a^2*(103*A + 126*C))*Cos[c + d*x]^3*Sin[c + d*x])/(210*d) + ((2*A*b^2 + a^2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(35*d) + (2*A*b*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(21*d) + (A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) - ((4*A*b^4 + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sin[c + d*x]^3)/(105*d)","A",10,8,33,0.2424,1,"{4095, 4094, 4074, 4047, 2635, 8, 4044, 3013}"
673,1,158,0,0.28609,"\int (a+b \sec (c+d x))^3 \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^3*(a^2 - b^2*Sec[c + d*x]^2),x]","\frac{a b^2 \left(5 a^2-4 b^2\right) \tan (c+d x)}{2 d}+\frac{b \left(-8 a^2 b^2+24 a^4-3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^3 \left(2 a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+a^5 x-\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}","\frac{a b^2 \left(5 a^2-4 b^2\right) \tan (c+d x)}{2 d}+\frac{b \left(-8 a^2 b^2+24 a^4-3 b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^3 \left(2 a^2-3 b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+a^5 x-\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,"a^5*x + (b*(24*a^4 - 8*a^2*b^2 - 3*b^4)*ArcTanh[Sin[c + d*x]])/(8*d) + (a*b^2*(5*a^2 - 4*b^2)*Tan[c + d*x])/(2*d) + (b^3*(2*a^2 - 3*b^2)*Sec[c + d*x]*Tan[c + d*x])/(8*d) - (a*b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(4*d) - (b^2*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",8,7,30,0.2333,1,"{4042, 3918, 4056, 4048, 3770, 3767, 8}"
674,1,106,0,0.1723362,"\int (a+b \sec (c+d x))^2 \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^2*(a^2 - b^2*Sec[c + d*x]^2),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}+a^4 x-\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}","\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}+a^4 x-\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 + (a*b*(2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/d + (b^2*(a^2 - 2*b^2)*Tan[c + d*x])/(3*d) - (a*b^3*Sec[c + d*x]*Tan[c + d*x])/(3*d) - (b^2*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",7,6,30,0.2000,1,"{4042, 3918, 4048, 3770, 3767, 8}"
675,1,75,0,0.08655,"\int (a+b \sec (c+d x)) \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])*(a^2 - b^2*Sec[c + d*x]^2),x]","\frac{b \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 x-\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}","\frac{b \left(2 a^2-b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 x-\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 + (b*(2*a^2 - b^2)*ArcTanh[Sin[c + d*x]])/(2*d) - (a*b^2*Tan[c + d*x])/(2*d) - (b^2*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",6,5,28,0.1786,1,"{4042, 3918, 3770, 3767, 8}"
676,1,186,0,0.6452324,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\left(3 a^2 C+b^2 (3 A+2 C)\right) \tan (c+d x)}{3 b^3 d}-\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{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}","\frac{\left(3 a^2 C+b^2 (3 A+2 C)\right) \tan (c+d x)}{3 b^3 d}-\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{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,"-(a*(2*A*b^2 + (2*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) + (2*a^2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*a^2*C + b^2*(3*A + 2*C))*Tan[c + d*x])/(3*b^3*d) - (a*C*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)","A",8,8,33,0.2424,1,"{4103, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
677,1,137,0,0.3808631,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}","\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,"((2*a^2*C + b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*a*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) - (a*C*Tan[c + d*x])/(b^2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",7,7,33,0.2121,1,"{4093, 4082, 3998, 3770, 3831, 2659, 208}"
678,1,95,0,0.1929002,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}","\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,"-((a*C*ArcTanh[Sin[c + d*x]])/(b^2*d)) + (2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Tan[c + d*x])/(b*d)","A",6,6,31,0.1935,1,"{4083, 3998, 3770, 3831, 2659, 208}"
679,1,88,0,0.1489255,"\int \frac{A+C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","-\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}","-\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,"(A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(b*d) - (2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d)","A",6,6,25,0.2400,1,"{4051, 3770, 3919, 3831, 2659, 208}"
680,1,86,0,0.177109,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}","\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*x)/a^2) + (2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (A*Sin[c + d*x])/(a*d)","A",5,5,31,0.1613,1,"{4105, 3919, 3831, 2659, 208}"
681,1,126,0,0.3872216,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","-\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(\frac{2 A b^2}{a^2}+A+2 C\right)}{2 a}-\frac{A b \sin (c+d x)}{a^2 d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a 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 b \sin (c+d x)}{a^2 d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d}",1,"((A + (2*A*b^2)/a^2 + 2*C)*x)/(2*a) - (2*b*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - (A*b*Sin[c + d*x])/(a^2*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",6,6,33,0.1818,1,"{4105, 4104, 3919, 3831, 2659, 208}"
682,1,173,0,0.605461,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\left(a^2 (2 A+3 C)+3 A b^2\right) \sin (c+d x)}{3 a^3 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(\frac{2 A b^2}{a^2}+A+2 C\right)}{2 a^2}-\frac{A b \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}","\frac{\left(a^2 (2 A+3 C)+3 A b^2\right) \sin (c+d x)}{3 a^3 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{A b \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"-(b*(A + (2*A*b^2)/a^2 + 2*C)*x)/(2*a^2) + (2*b^2*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((3*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*a^3*d) - (A*b*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)","A",7,6,33,0.1818,1,"{4105, 4104, 3919, 3831, 2659, 208}"
683,1,232,0,0.9271477,"\int \frac{\cos ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{a+b \sec (c+d x)} \, dx","Int[(Cos[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","-\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{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{x \left(4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right)}{8 a^5}-\frac{A b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}","-\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{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{x \left(4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+8 A b^4\right)}{8 a^5}-\frac{A b \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"((8*A*b^4 + 4*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x)/(8*a^5) - (2*b^3*(A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - (b*(3*A*b^2 + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*d) + ((4*A*b^2 + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - (A*b*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",8,6,33,0.1818,1,"{4105, 4104, 3919, 3831, 2659, 208}"
684,1,271,0,0.8611299,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","-\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{\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{2 a \left(a^2 A b^2-4 a^2 b^2 C+3 a^4 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}}-\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{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{\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{2 a \left(a^2 A b^2-4 a^2 b^2 C+3 a^4 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}}-\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)}",1,"((2*A*b^2 + (6*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a*(a^2*A*b^2 - 2*A*b^4 + 3*a^4*C - 4*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) - (a*(A*b^2 + 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 + 3*a^2*C - b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,8,33,0.2424,1,"{4099, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
685,1,153,0,0.4847772,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","-\frac{2 \left(3 a^2 b^2 C-2 a^4 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{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 a C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}","-\frac{2 \left(3 a^2 b^2 C-2 a^4 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{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 a C \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}",1,"(-2*a*C*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*(A*b^4 - 2*a^4*C + 3*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Tan[c + d*x])/(b^2*d) + (a*(A*b^2 + a^2*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,7,33,0.2121,1,"{4091, 4082, 3998, 3770, 3831, 2659, 208}"
686,1,135,0,0.2723988,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(b^2*d) + (2*a*(A*b^2 - a^2*C + 2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 + a^2*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,31,0.1935,1,"{4081, 3998, 3770, 3831, 2659, 208}"
687,1,125,0,0.2267772,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","-\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}","-\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,"(A*x)/a^2 - (2*b*(2*a^2*A - A*b^2 + a^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,25,0.2000,1,"{4061, 3919, 3831, 2659, 208}"
688,1,171,0,0.4339615,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","-\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{2 \left(3 a^2 A b^2+a^4 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)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\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 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{2 \left(3 a^2 A b^2+a^4 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)}{a^3 d (a-b)^{3/2} (a+b)^{3/2}}+\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 A b x}{a^3}",1,"(-2*A*b*x)/a^3 + (2*(3*a^2*A*b^2 - 2*A*b^4 + a^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b^2 - a^2*(A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,31,0.1935,1,"{4101, 4104, 3919, 3831, 2659, 208}"
689,1,256,0,0.860479,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","\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)}-\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{2 b \left(4 a^2 A b^2-a^2 b^2 C+2 a^4 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{\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{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)}-\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{2 b \left(4 a^2 A b^2-a^2 b^2 C+2 a^4 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{\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}",1,"((6*A*b^2 + a^2*(A + 2*C))*x)/(2*a^4) - (2*b*(4*a^2*A*b^2 - 3*A*b^4 + 2*a^4*C - a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(3*A*b^2 - a^2*(2*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - a^2*(A - 2*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,6,33,0.1818,1,"{4101, 4104, 3919, 3831, 2659, 208}"
690,1,326,0,1.2552883,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^2} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","-\frac{\left(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+12 A b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\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{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(5 a^2 A b^2-2 a^2 b^2 C+3 a^4 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}}+\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(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+12 A b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}-\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{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(5 a^2 A b^2-2 a^2 b^2 C+3 a^4 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}}+\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}",1,"-((b*(4*A*b^2 + a^2*(A + 2*C))*x)/a^5) + (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C - 2*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((12*A*b^4 - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) + (b*(2*A*b^2 - a^2*(A - C))*Cos[c + d*x]*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((4*A*b^2 - a^2*(A - 3*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,6,33,0.1818,1,"{4101, 4104, 3919, 3831, 2659, 208}"
691,1,381,0,1.62036,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","-\frac{a \left(a^2 b^2 (2 A-21 C)+12 a^4 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(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(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 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}}-\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(7 a^2 b^2 C-4 a^4 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(a^2 b^2 (A-10 C)+6 a^4 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(a^2 b^2 (2 A-21 C)+12 a^4 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(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(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+12 a^6 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}}-\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(7 a^2 b^2 C-4 a^4 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(a^2 b^2 (A-10 C)+6 a^4 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}",1,"((2*A*b^2 + (12*a^2 + b^2)*C)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) - (a*(a^2*b^2*(2*A - 21*C) - b^4*(5*A - 6*C) + 12*a^4*C)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) + ((a^2*b^2*(A - 10*C) - b^4*(4*A - C) + 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 - 4*a^4*C + 7*a^2*b^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",9,9,33,0.2727,1,"{4099, 4098, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
692,1,271,0,1.0221257,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\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{\left(a^2 b^4 (A+12 C)-15 a^4 b^2 C+6 a^6 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{\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{a \left(a^2 b^2 (A+6 C)-3 a^4 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{3 a C \tanh ^{-1}(\sin (c+d x))}{b^4 d}","\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{\left(a^2 b^4 (A+12 C)-15 a^4 b^2 C+6 a^6 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{\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{a \left(a^2 b^2 (A+6 C)-3 a^4 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{3 a C \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(-3*a*C*ArcTanh[Sin[c + d*x]])/(b^4*d) + ((2*A*b^6 + 6*a^6*C - 15*a^4*b^2*C + a^2*b^4*(A + 12*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 + 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,8,33,0.2424,1,"{4099, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
693,1,212,0,0.5942748,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","-\frac{a \left(C \left(-5 a^2 b^2+2 a^4+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{\left(a^2 b^2 (A+6 C)-3 a^4 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(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{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}","-\frac{a \left(C \left(-5 a^2 b^2+2 a^4+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{\left(a^2 b^2 (A+6 C)-3 a^4 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(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{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"(C*ArcTanh[Sin[c + d*x]])/(b^3*d) - (a*(3*A*b^4 + (2*a^4 - 5*a^2*b^2 + 6*b^4)*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 + a^2*C)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((2*A*b^4 - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,33,0.2121,1,"{4091, 4080, 3998, 3770, 3831, 2659, 208}"
694,1,177,0,0.3213873,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","\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}","\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,"((a^2*(2*A + C) + b^2*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((A*b^2 + a^2*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(3*A*b^2 - a^2*C + 4*b^2*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,31,0.1935,1,"{4081, 4003, 12, 3831, 2659, 208}"
695,1,202,0,0.445455,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\frac{b \left(5 a^2 A b^2-3 a^4 (2 A+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)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-a^2 b^2 (5 A+2 C)+a^4 (-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{\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{A x}{a^3}","\frac{b \left(5 a^2 A b^2-3 a^4 (2 A+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)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-a^2 b^2 (5 A+2 C)+a^4 (-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{\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{A x}{a^3}",1,"(A*x)/a^3 + (b*(5*a^2*A*b^2 - 2*A*b^4 - 3*a^4*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((2*A*b^4 - a^4*C - a^2*b^2*(5*A + 2*C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,25,0.2400,1,"{4061, 4060, 3919, 3831, 2659, 208}"
696,1,266,0,0.9629699,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","-\frac{\left(11 a^2 A b^2+a^4 (-(2 A-3 C))-6 A b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-2 a^6 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)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-a^2 b^2 (6 A+C)-2 a^4 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(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{3 A b x}{a^4}","-\frac{\left(11 a^2 A b^2+a^4 (-(2 A-3 C))-6 A b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-2 a^6 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)}{a^4 d (a-b)^{5/2} (a+b)^{5/2}}-\frac{\left(-a^2 b^2 (6 A+C)-2 a^4 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(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{3 A b x}{a^4}",1,"(-3*A*b*x)/a^4 - ((15*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - a^4*b^2*(12*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((11*a^2*A*b^2 - 6*A*b^4 - a^4*(2*A - 3*C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(6*A + C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,31,0.2258,1,"{4101, 4100, 4104, 3919, 3831, 2659, 208}"
697,1,369,0,1.6145074,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^3} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","-\frac{b \left(-a^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-a^2 b^2 (10 A-C)+a^4 (A-4 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(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 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}}+\frac{\left(7 a^2 A b^2+3 a^4 C-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^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^2 b^2 (21 A-2 C)+a^4 (6 A-5 C)+12 A b^4\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\left(-a^2 b^2 (10 A-C)+a^4 (A-4 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(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+6 a^6 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}}+\frac{\left(7 a^2 A b^2+3 a^4 C-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^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}",1,"((12*A*b^2 + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - (b*(12*A*b^4 + a^4*(6*A - 5*C) - a^2*b^2*(21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 + 3*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,7,33,0.2121,1,"{4101, 4100, 4104, 3919, 3831, 2659, 208}"
698,1,378,0,1.8111969,"\int \frac{\sec ^4(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^4*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\frac{\left(5 A b^4-C \left(-23 a^2 b^2+12 a^4+6 b^4\right)\right) \tan (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^6 (3 A+20 C)+28 a^6 b^2 C-35 a^4 b^4 C-8 a^8 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{\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(a^2 b^2 (2 A+9 C)-4 a^4 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(3 a^2 b^4 (A+4 C)-11 a^4 b^2 C+4 a^6 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{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}","-\frac{\left(5 A b^4-C \left(-23 a^2 b^2+12 a^4+6 b^4\right)\right) \tan (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(a^2 b^6 (3 A+20 C)+28 a^6 b^2 C-35 a^4 b^4 C-8 a^8 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{\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(a^2 b^2 (2 A+9 C)-4 a^4 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(3 a^2 b^4 (A+4 C)-11 a^4 b^2 C+4 a^6 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{4 a C \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"(-4*a*C*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 - (12*a^4 - 23*a^2*b^2 + 6*b^4)*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,9,33,0.2727,1,"{4099, 4098, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
699,1,313,0,1.2514229,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{a \left(a^2 b^4 (A-8 C)+7 a^4 b^2 C-2 a^6 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(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{\left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+9 a^6 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{a \left(a^2 b^2 (3 A+8 C)-3 a^4 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{C \tanh ^{-1}(\sin (c+d x))}{b^4 d}","\frac{a \left(a^2 b^4 (A-8 C)+7 a^4 b^2 C-2 a^6 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(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{\left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+9 a^6 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{a \left(a^2 b^2 (3 A+8 C)-3 a^4 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{C \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(C*ArcTanh[Sin[c + d*x]])/(b^4*d) + (a*(a^2*b^4*(A - 8*C) - 2*a^6*C + 7*a^4*b^2*C + 4*b^6*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a*(2*A*b^4 - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((4*A*b^6 + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,8,33,0.2424,1,"{4099, 4090, 4080, 3998, 3770, 3831, 2659, 208}"
700,1,261,0,0.6729037,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","-\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 b^2 (2 A-5 C)+2 a^4 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(a^2 b^2 (2 A+9 C)-4 a^4 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}+\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{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 b^2 (2 A-5 C)+2 a^4 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(a^2 b^2 (2 A+9 C)-4 a^4 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}+\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}",1,"-((b*(b^2*(A + 2*C) + a^2*(4*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) + (a*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*b^2*(2*A - 5*C) + 2*a^4*C + b^4*(13*A + 18*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,7,33,0.2121,1,"{4091, 4080, 4003, 12, 3831, 2659, 208}"
701,1,252,0,0.5534129,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\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{\left(-a^2 b^2 (11 A+10 C)+a^4 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))}-\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{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{\left(-a^2 b^2 (11 A+10 C)+a^4 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))}-\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}",1,"(a*(a^2*(2*A + C) + b^2*(3*A + 4*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) - ((A*b^2 + a^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a*(5*A*b^2 - a^2*C + 6*b^2*C)*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,31,0.1935,1,"{4081, 4003, 12, 3831, 2659, 208}"
702,1,292,0,0.9367198,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4,x]","-\frac{b \left(-a^4 b^2 (8 A-C)+7 a^2 A b^4+4 a^6 (2 A+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)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4-2 a^6 C-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))}-\frac{\left(-a^2 b^2 (8 A+3 C)-2 a^4 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{\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{A x}{a^4}","-\frac{b \left(-a^4 b^2 (8 A-C)+7 a^2 A b^4+4 a^6 (2 A+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)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4-2 a^6 C-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))}-\frac{\left(-a^2 b^2 (8 A+3 C)-2 a^4 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{\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{A x}{a^4}",1,"(A*x)/a^4 - (b*(7*a^2*A*b^4 - 2*A*b^6 - a^4*b^2*(8*A - C) + 4*a^6*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((A*b^2 + a^2*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((3*A*b^4 - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,25,0.2400,1,"{4061, 4060, 3919, 3831, 2659, 208}"
703,1,367,0,1.8235719,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{\left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-24 A b^6\right) \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-2 a^8 C+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}}-\frac{\left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^6 C-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(-a^2 b^2 (9 A+2 C)-3 a^4 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^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{4 A b x}{a^5}","\frac{\left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-24 A b^6\right) \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-2 a^8 C+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}}-\frac{\left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^6 C-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(-a^2 b^2 (9 A+2 C)-3 a^4 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^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{4 A b x}{a^5}",1,"(-4*A*b*x)/a^5 - ((35*a^4*A*b^4 - 28*a^2*A*b^6 + 8*A*b^8 - 2*a^8*C - a^6*b^2*(20*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((68*a^2*A*b^4 - 24*A*b^6 + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((4*A*b^4 - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,7,31,0.2258,1,"{4101, 4100, 4104, 3919, 3831, 2659, 208}"
704,1,513,0,2.3450961,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^4} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^4,x]","\frac{b \left(a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}-\frac{\left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 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(-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b 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{\left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 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))}-\frac{\left(-a^2 b^2 (10 A+C)-4 a^4 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(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{b \left(a^4 b^2 (146 A-17 C)-a^2 b^4 (167 A-6 C)+a^6 (-(24 A-26 C))+60 A b^6\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}-\frac{\left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 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(-a^2 b^7 (69 A-2 C)+7 a^4 b^5 (12 A-C)-8 a^6 b^3 (5 A-C)-8 a^8 b 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{\left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+12 a^6 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))}-\frac{\left(-a^2 b^2 (10 A+C)-4 a^4 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(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}",1,"((20*A*b^2 + a^2*(A + 2*C))*x)/(2*a^6) + ((20*A*b^9 - a^2*b^7*(69*A - 2*C) - 8*a^6*b^3*(5*A - C) + 7*a^4*b^5*(12*A - C) - 8*a^8*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + (b*(60*A*b^6 - a^6*(24*A - 26*C) + a^4*b^2*(146*A - 17*C) - a^2*b^4*(167*A - 6*C))*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 + a^2*C)*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*A*b^4 - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((20*A*b^6 - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,7,33,0.2121,1,"{4101, 4100, 4104, 3919, 3831, 2659, 208}"
705,1,17,0,0.0442866,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[(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",3,2,30,0.06667,1,"{4042, 3770}"
706,1,52,0,0.1307435,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,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}}","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,"x - (4*b*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*d)","A",5,5,30,0.1667,1,"{4042, 3919, 3831, 2659, 208}"
707,1,107,0,0.205855,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","-\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}","-\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,"x/a - (2*b*(3*a^2 - b^2)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*(a - b)^(3/2)*(a + b)^(3/2)*d) + (2*b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,30,0.2000,1,"{4042, 3923, 3919, 3831, 2659, 208}"
708,1,162,0,0.3395606,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4,x]","-\frac{2 b \left(-2 a^2 b^2+4 a^4+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}}+\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(-2 a^2 b^2+4 a^4+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}}+\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}",1,"x/a^2 - (2*b*(4*a^4 - 2*a^2*b^2 + b^4)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(4*a^2 - b^2)*Tan[c + d*x])/(a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,30,0.2333,1,"{4042, 3923, 4060, 3919, 3831, 2659, 208}"
709,1,467,0,1.3048325,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","-\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(12 a^2 b C+16 a^3 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 (a-b) \sqrt{a+b} \left(6 a^2 b^2 (7 A+4 C)+16 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 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}","-\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(12 a^2 b C+16 a^3 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 (a-b) \sqrt{a+b} \left(6 a^2 b^2 (7 A+4 C)+16 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 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,"(2*(a - b)*Sqrt[a + b]*(16*a^4*C + 6*a^2*b^2*(7*A + 4*C) - 21*b^4*(9*A + 7*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) + (2*(a - b)*Sqrt[a + 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) + (2*a*(21*A*b^2 + 8*a^2*C + 13*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) - (2*(6*a^2*C - 7*b^2*(9*A + 7*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*a*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)","A",7,7,35,0.2000,1,"{4097, 4102, 4092, 4082, 4005, 3832, 4004}"
710,1,375,0,0.7724561,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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-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{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{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}","\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-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{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{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,"(-2*a*(a - b)*Sqrt[a + b]*(35*A*b^2 + 8*a^2*C + 19*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) - (2*(a - b)*Sqrt[a + b]*(35*A*b^2 + (8*a^2 + 6*a*b + 25*b^2)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(8*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) - (8*a*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*b*d)","A",6,6,35,0.1714,1,"{4093, 4082, 4002, 4005, 3832, 4004}"
711,1,308,0,0.5051706,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*(a - b)*Sqrt[a + b]*(2*a^2*C - 3*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(a - b)*Sqrt[a + b]*(15*A*b + 2*a*C + 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) - (4*a*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)","A",5,5,33,0.1515,1,"{4083, 4002, 4005, 3832, 4004}"
712,1,355,0,0.3674597,"\int \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(-2*a*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*Sqrt[a + b]*(3*A*b - (a - b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,27,0.2222,1,"{4057, 4058, 3921, 3784, 3832, 4004}"
713,1,352,0,0.3791906,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"((a - b)*Sqrt[a + b]*(A - 2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(A*b + 2*(a - b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,33,0.1818,1,"{4095, 4058, 3921, 3784, 3832, 4004}"
714,1,411,0,0.6810026,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(A*b + 2*a*(A + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(A*b^2 - 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + (A*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,35,0.2000,1,"{4095, 4104, 4058, 3921, 3784, 3832, 4004}"
715,1,502,0,1.0609894,"\int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"-((a - b)*Sqrt[a + b]*(3*A*b^2 - 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*b*d) + (Sqrt[a + b]*(2*a*A*b - 3*A*b^2 + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*d) - (b*Sqrt[a + b]*(A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^3*d) - ((3*A*b^2 - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^2*d) + (A*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,7,35,0.2000,1,"{4095, 4104, 4058, 3921, 3784, 3832, 4004}"
716,1,587,0,1.5753167,"\int \cos ^4(c+d x) \sqrt{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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{\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(-4 a^2 b (7 A+12 C)-24 a^3 (3 A+4 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-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(8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 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{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}","\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{\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(-4 a^2 b (7 A+12 C)-24 a^3 (3 A+4 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-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(8 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 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{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,"((a - b)*Sqrt[a + b]*(15*A*b^2 + 4*a^2*(7*A + 12*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^3*d) - (Sqrt[a + b]*(10*a*A*b^2 - 15*A*b^3 - 24*a^3*(3*A + 4*C) - 4*a^2*b*(7*A + 12*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^3*d) + (Sqrt[a + b]*(5*A*b^4 + 8*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^4*d) + (b*(15*A*b^2 + 4*a^2*(7*A + 12*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^3*d) - ((5*A*b^2 - 12*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a^2*d) + (A*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + (A*Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d)","A",9,7,35,0.2000,1,"{4095, 4104, 4058, 3921, 3784, 3832, 4004}"
717,1,550,0,1.919181,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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{2 \left(a^2 b^2 (33 A+19 C)+8 a^4 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(6 a^2 b^2 (11 A+8 C)+12 a^3 b C+16 a^4 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{4 a (a-b) \sqrt{a+b} \left(3 a^2 b^2 (11 A+6 C)+8 a^4 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 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}","\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{2 \left(a^2 b^2 (33 A+19 C)+8 a^4 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(6 a^2 b^2 (11 A+8 C)+12 a^3 b C+16 a^4 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{4 a (a-b) \sqrt{a+b} \left(3 a^2 b^2 (11 A+6 C)+8 a^4 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 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,"(4*a*(a - b)*Sqrt[a + b]*(8*a^4*C + 3*a^2*b^2*(11*A + 6*C) - b^4*(451*A + 348*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1155*b^5*d) + (2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1155*b^4*d) + (2*(8*a^4*C + 25*b^4*(11*A + 9*C) + a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1155*b^3*d) + (4*a*(132*A*b^2 - 3*a^2*C + 101*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1155*b^2*d) + (2*(a^2*C + 3*b^2*(11*A + 9*C))*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(231*b*d) + (2*a*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(33*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)","A",8,8,35,0.2286,1,"{4097, 4096, 4102, 4092, 4082, 4005, 3832, 4004}"
718,1,454,0,1.0497185,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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(6 a^2 b C+8 a^3 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{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (21 A+11 C)+8 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^4 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}","\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(6 a^2 b C+8 a^3 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{2 (a-b) \sqrt{a+b} \left(3 a^2 b^2 (21 A+11 C)+8 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^4 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,"(-2*(a - b)*Sqrt[a + b]*(8*a^4*C + 21*b^4*(9*A + 7*C) + 3*a^2*b^2*(21*A + 11*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*a*(63*A*b^2 + 8*a^2*C + 39*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(8*a^2*C + 7*b^2*(9*A + 7*C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) - (8*a*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)","A",7,6,35,0.1714,1,"{4093, 4082, 4002, 4005, 3832, 4004}"
719,1,374,0,0.7609993,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(-4*a*(a - b)*Sqrt[a + b]*(70*A*b^2 - 3*a^2*C + 41*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(105*a*A*b - 35*A*b^2 + 6*a^2*C + 57*a*b*C - 25*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) - (2*(6*a^2*C - 5*b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) - (4*a*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)","A",6,5,33,0.1515,1,"{4083, 4002, 4005, 3832, 4004}"
720,1,415,0,0.5763131,"\int (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(-2*(a - b)*Sqrt[a + b]*(a^2*C + b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^2*d) - (2*Sqrt[a + b]*(a^2*C - 2*a*b*(5*A + 2*C) + b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b*d) - (2*a*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*a*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",7,7,27,0.2593,1,"{4057, 4056, 4058, 3921, 3784, 3832, 4004}"
721,1,408,0,0.5745123,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a*(a - b)*Sqrt[a + b]*(3*A - 8*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(a*b*(3*A - 8*C) + 6*a^2*C + 2*b^2*(3*A + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (3*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*A - 2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,33,0.2121,1,"{4095, 4056, 4058, 3921, 3784, 3832, 4004}"
722,1,414,0,0.6754026,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","-\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}","-\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,"((a - b)*Sqrt[a + b]*(5*A - 8*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (Sqrt[a + b]*(2*a*A + 5*A*b + 16*a*C - 8*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(3*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (3*A*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",7,7,35,0.2000,1,"{4095, 4094, 4058, 3921, 3784, 3832, 4004}"
723,1,504,0,1.1779782,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"((a - b)*Sqrt[a + b]*(3*A*b^2 + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(16*a^2*A + 14*a*A*b + 3*A*b^2 + 24*a^2*C + 48*a*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) + (b*Sqrt[a + b]*(A*b^2 - 12*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((3*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + (A*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",8,8,35,0.2286,1,"{4095, 4094, 4104, 4058, 3921, 3784, 3832, 4004}"
724,1,583,0,1.5402543,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","-\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{\sqrt{a+b} \left(a^2 (52 A b+80 b C)+8 a^3 (3 A+4 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{(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(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 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}","-\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{\sqrt{a+b} \left(a^2 (52 A b+80 b C)+8 a^3 (3 A+4 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{(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(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 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,"-((a - b)*Sqrt[a + b]*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + (Sqrt[a + b]*(2*a*A*b^2 - 3*A*b^3 + 8*a^3*(3*A + 4*C) + a^2*(52*A*b + 80*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) - (Sqrt[a + b]*(3*A*b^4 + 24*a^2*b^2*(A + 2*C) + 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^3*d) - (b*(3*A*b^2 - 4*a^2*(13*A + 20*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(64*a^2*d) + ((A*b^2 + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*a*d) + (A*b*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",9,8,35,0.2286,1,"{4095, 4094, 4104, 4058, 3921, 3784, 3832, 4004}"
725,1,650,0,2.7703939,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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(-15 a^2 b^2 (715 A+543 C)+90 a^4 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(5 a^2 b^2 (143 A+79 C)+120 a^4 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(10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)+180 a^4 b C+240 a^5 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 (a-b) \sqrt{a+b} \left(10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)+240 a^6 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 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}","\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(-15 a^2 b^2 (715 A+543 C)+90 a^4 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(5 a^2 b^2 (143 A+79 C)+120 a^4 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(10 a^3 b^2 (143 A+94 C)+15 a^2 b^3 (1573 A+1175 C)+180 a^4 b C+240 a^5 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 (a-b) \sqrt{a+b} \left(10 a^4 b^2 (143 A+76 C)-3 a^2 b^4 (13299 A+10223 C)+240 a^6 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 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,"(2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^5*d) + (2*(a - b)*Sqrt[a + 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(45045*b^4*d) + (2*a*(120*a^4*C + 5*a^2*b^2*(143*A + 79*C) + b^4*(23309*A + 18973*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(45045*b^3*d) - (2*(90*a^4*C - 539*b^4*(13*A + 11*C) - 15*a^2*b^2*(715*A + 543*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(45045*b^2*d) + (2*a*(2717*A*b^2 + 15*a^2*C + 2209*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9009*b*d) + (2*(15*a^2*C + 11*b^2*(13*A + 11*C))*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(1287*d) + (10*a*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(143*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(13*d)","A",9,8,35,0.2286,1,"{4097, 4096, 4102, 4092, 4082, 4005, 3832, 4004}"
726,1,534,0,1.4896339,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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(3 a^2 b^2 (33 A+19 C)+8 a^4 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-b) \sqrt{a+b} \left(3 a^2 b^2 (33 A+19 C)+6 a^3 b C+8 a^4 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{2 a (a-b) \sqrt{a+b} \left(3 a^2 b^2 (33 A+17 C)+8 a^4 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{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}","\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(3 a^2 b^2 (33 A+19 C)+8 a^4 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-b) \sqrt{a+b} \left(3 a^2 b^2 (33 A+19 C)+6 a^3 b C+8 a^4 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{2 a (a-b) \sqrt{a+b} \left(3 a^2 b^2 (33 A+17 C)+8 a^4 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{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,"(-2*a*(a - b)*Sqrt[a + b]*(8*a^4*C + 3*a^2*b^2*(33*A + 17*C) + 3*b^4*(319*A + 247*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^4*d) - (2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(693*b^3*d) + (2*(8*a^4*C + 15*b^4*(11*A + 9*C) + 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(693*b^2*d) + (2*a*(99*A*b^2 + 8*a^2*C + 67*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(693*b^2*d) + (2*(8*a^2*C + 9*b^2*(11*A + 9*C))*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) - (8*a*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)","A",8,6,35,0.1714,1,"{4093, 4082, 4002, 4005, 3832, 4004}"
727,1,454,0,1.0320836,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","-\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(15 a^2 b (21 A+11 C)+10 a^3 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 (a-b) \sqrt{a+b} \left(-3 a^2 b^2 (161 A+93 C)+10 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^3 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}","-\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(15 a^2 b (21 A+11 C)+10 a^3 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 (a-b) \sqrt{a+b} \left(-3 a^2 b^2 (161 A+93 C)+10 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^3 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,"(2*(a - b)*Sqrt[a + b]*(10*a^4*C - 21*b^4*(9*A + 7*C) - 3*a^2*b^2*(161*A + 93*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (4*a*(84*A*b^2 - 5*a^2*C + 57*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) - (2*(10*a^2*C - 7*b^2*(9*A + 7*C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) - (4*a*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)","A",7,5,33,0.1515,1,"{4083, 4002, 4005, 3832, 4004}"
728,1,481,0,0.8310597,"\int (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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 \sqrt{a+b} \left(-9 a^2 b (7 A+3 C)+3 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 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 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}","\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 \sqrt{a+b} \left(-9 a^2 b (7 A+3 C)+3 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 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 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,"(-2*a*(a - b)*Sqrt[a + b]*(49*A*b^2 + 3*a^2*C + 29*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b^2*d) - (2*Sqrt[a + b]*(3*a^3*C - 9*a^2*b*(7*A + 3*C) - b^3*(7*A + 5*C) + a*b^2*(49*A + 29*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(21*b*d) - (2*a^2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(3*a^2*C + b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(21*d) + (2*a*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",8,7,27,0.2593,1,"{4057, 4056, 4058, 3921, 3784, 3832, 4004}"
729,1,478,0,0.8095964,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\frac{\sqrt{a+b} \left(a^2 b (15 A-46 C)+30 a^3 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{(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{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}","\frac{\sqrt{a+b} \left(a^2 b (15 A-46 C)+30 a^3 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{(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{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,"((a - b)*Sqrt[a + b]*(a^2*(15*A - 46*C) - 6*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (Sqrt[a + b]*(a^2*b*(15*A - 46*C) + 30*a^3*C - 6*b^3*(5*A + 3*C) + 2*a*b^2*(45*A + 17*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (5*a*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d - (a*b*(15*A - 16*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",8,7,33,0.2121,1,"{4095, 4056, 4058, 3921, 3784, 3832, 4004}"
730,1,463,0,0.9132314,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(a*(a - b)*Sqrt[a + b]*(27*A - 56*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*d) + (Sqrt[a + b]*(a*b*(27*A - 56*C) + 8*b^2*(3*A + C) + 6*a^2*(A + 12*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*d) - (Sqrt[a + b]*(15*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (5*A*b*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d) - (b^2*(21*A - 8*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(12*d)","A",8,8,35,0.2286,1,"{4095, 4094, 4056, 4058, 3921, 3784, 3832, 4004}"
731,1,507,0,1.202008,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"((a - b)*Sqrt[a + b]*(3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(16*a^2*A + 26*a*A*b + 33*A*b^2 + 24*a^2*C + 144*a*b*C - 48*b^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (5*b*Sqrt[a + b]*(A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (5*A*b*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",8,7,35,0.2000,1,"{4095, 4094, 4058, 3921, 3784, 3832, 4004}"
732,1,587,0,1.6330366,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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{\sqrt{a+b} \left(4 a^2 b (71 A+108 C)+24 a^3 (3 A+4 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-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(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 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{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}","\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{\sqrt{a+b} \left(4 a^2 b (71 A+108 C)+24 a^3 (3 A+4 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-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(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 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{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,"((a - b)*Sqrt[a + b]*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(15*A*b^3 + 24*a^3*(3*A + 4*C) + 4*a^2*b*(71*A + 108*C) + 2*a*b^2*(59*A + 192*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(5*A*b^4 - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + (b*(15*A*b^2 + 4*a^2*(71*A + 108*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((5*A*b^2 + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + (5*A*b*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",9,8,35,0.2286,1,"{4095, 4094, 4104, 4058, 3921, 3784, 3832, 4004}"
733,1,403,0,0.5697007,"\int (a+b \sec (c+d x))^{3/2} \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(3/2)*(a^2 - b^2*Sec[c + d*x]^2),x]","\frac{2 \sqrt{a+b} \left(-4 a^2 b+10 a^3-4 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) \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 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^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}","\frac{2 \sqrt{a+b} \left(-4 a^2 b+10 a^3-4 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) \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 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^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,"(-2*(a - b)*Sqrt[a + b]*(4*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*d) + (2*Sqrt[a + b]*(10*a^3 - 4*a^2*b - 4*a*b^2 + 3*b^3)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*d) - (2*a^3*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*a*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*d) - (2*b^2*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",8,8,32,0.2500,1,"{4042, 3918, 4056, 4058, 3921, 3784, 3832, 4004}"
734,1,353,0,0.4029346,"\int \sqrt{a+b \sec (c+d x)} \left(a^2-b^2 \sec ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*(a^2 - b^2*Sec[c + d*x]^2),x]","\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}","\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,"(2*a*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) + (2*Sqrt[a + b]*(3*a^2 + a*b - b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*a^2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*b^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,32,0.2188,1,"{4042, 3918, 4058, 3921, 3784, 3832, 4004}"
735,1,393,0,0.9079022,"\int \frac{\sec ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\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(-12 a^2 b C+48 a^3 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{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{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}","\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(-12 a^2 b C+48 a^3 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{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{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,"(4*a*(a - b)*Sqrt[a + b]*(35*A*b^2 + 24*a^2*C + 22*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) + (2*Sqrt[a + 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(24*a^2*C + 5*b^2*(7*A + 5*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) - (12*a*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)","A",6,6,35,0.1714,1,"{4103, 4092, 4082, 4005, 3832, 4004}"
736,1,320,0,0.5719928,"\int \frac{\sec ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","-\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{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{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}","-\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{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{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,"(-2*(a - b)*Sqrt[a + b]*(8*a^2*C + 3*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) - (2*Sqrt[a + b]*(8*a^2*C - 2*a*b*C + 3*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (8*a*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)","A",5,5,35,0.1429,1,"{4093, 4082, 4005, 3832, 4004}"
737,1,253,0,0.3224253,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\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}","\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,"(4*a*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) + (2*Sqrt[a + b]*(3*A*b + (2*a + b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)","A",4,4,33,0.1212,1,"{4083, 4005, 3832, 4004}"
738,1,313,0,0.2341258,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],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 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}","-\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,"(-2*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) - (2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",5,5,27,0.1852,1,"{4059, 3921, 3784, 3832, 4004}"
739,1,352,0,0.3952252,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\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}","\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,"(A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*(A*b + 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",6,6,33,0.1818,1,"{4105, 4058, 3921, 3784, 3832, 4004}"
740,1,411,0,0.6401743,"\int \frac{\cos ^2(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","-\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{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{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a 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{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{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"(-3*A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + (A*(2*a - 3*b)*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(3*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - (3*A*b*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)","A",7,7,35,0.2000,1,"{4105, 4104, 4058, 3921, 3784, 3832, 4004}"
741,1,506,0,1.0276514,"\int \frac{\cos ^3(c+d x) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(Cos[c + d*x]^3*(A + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\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{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{5 A b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 a 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{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{5 A b \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d}",1,"((a - b)*Sqrt[a + b]*(15*A*b^2 + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*b*d) - (Sqrt[a + b]*(10*a*A*b - 15*A*b^2 - 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^3*d) + (b*Sqrt[a + b]*(5*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^4*d) + ((15*A*b^2 + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^3*d) - (5*A*b*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a^2*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",8,7,35,0.2000,1,"{4105, 4104, 4058, 3921, 3784, 3832, 4004}"
742,1,460,0,1.080913,"\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","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","-\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(12 a^2 b C+16 a^3 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}}-\frac{2 \left(2 a^2 b^2 (5 A-4 C)+16 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(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(12 a^2 b C+16 a^3 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}}-\frac{2 \left(2 a^2 b^2 (5 A-4 C)+16 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}}",1,"(-2*(2*a^2*b^2*(5*A - 4*C) + 16*a^4*C - b^4*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^5*Sqrt[a + b]*d) - (2*(16*a^3*C + 12*a^2*b*C + 2*a*b^2*(5*A + 2*C) + b^3*(5*A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(5*b^4*Sqrt[a + b]*d) - (2*(A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*a*(5*A*b^2 + 8*a^2*C - 3*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 + 6*a^2*C - b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",6,6,35,0.1714,1,"{4099, 4092, 4082, 4005, 3832, 4004}"
743,1,327,0,0.6752957,"\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","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\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 \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 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 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d}","\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 \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 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 C \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d}",1,"(2*a*(3*A*b^2 + 8*a^2*C - 5*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*(3*A*b^2 + (8*a^2 + 6*a*b + b^2)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) + (2*a*(A*b^2 + a^2*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)","A",5,5,35,0.1429,1,"{4091, 4082, 4005, 3832, 4004}"
744,1,279,0,0.3913998,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"(-2*(A*b^2 + 2*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(A*b - (2*a + b)*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",4,4,33,0.1212,1,"{4081, 4005, 3832, 4004}"
745,1,381,0,0.4212508,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(2*(A*b^2 + a^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b^2*Sqrt[a + b]*d) - (2*(A*b - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",6,6,27,0.2222,1,"{4061, 4058, 3921, 3784, 3832, 4004}"
746,1,431,0,0.6588952,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","-\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{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{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}","-\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{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{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"-(((3*A*b^2 - a^2*(A - 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d)) + ((a*A*b + 3*A*b^2 + 2*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + (3*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (A*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) - (b*(3*A*b^2 - a^2*(A - 2*C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,33,0.2121,1,"{4105, 4060, 4058, 3921, 3784, 3832, 4004}"
747,1,501,0,1.0122917,"\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","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(3/2),x]","\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{\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{5 A b \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}","\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{\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{5 A b \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"((15*A*b^2 - a^2*(7*A - 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - ((5*a*A*b + 15*A*b^2 - 2*a^2*(A - 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(15*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - (5*A*b*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(15*A*b^2 - a^2*(7*A - 8*C))*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,35,0.2286,1,"{4105, 4104, 4060, 4058, 3921, 3784, 3832, 4004}"
748,1,488,0,1.3011533,"\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","Int[(Sec[c + d*x]^3*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","-\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(5 a^2 b^2 C-3 a^4 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(2 a^2 b^2 (A-8 C)+12 a^3 b C+16 a^4 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)}+\frac{4 a \left(a^2 b^2 (A-14 C)+8 a^4 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{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(5 a^2 b^2 C-3 a^4 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(2 a^2 b^2 (A-8 C)+12 a^3 b C+16 a^4 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)}+\frac{4 a \left(a^2 b^2 (A-14 C)+8 a^4 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)}",1,"(4*a*(a^2*b^2*(A - 14*C) - b^4*(3*A - 4*C) + 8*a^4*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^5*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 + a^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (4*a*(2*A*b^4 - 3*a^4*C + 5*a^2*b^2*C)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 + 2*a^2*C - b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",6,6,35,0.1714,1,"{4099, 4090, 4082, 4005, 3832, 4004}"
749,1,408,0,0.8213864,"\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","Int[(Sec[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(a^2 b^2 (A+9 C)-5 a^4 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 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(6 a^2 b C+8 a^3 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)}+\frac{2 \left(a^2 b^2 (A+15 C)-8 a^4 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(a^2 b^2 (A+9 C)-5 a^4 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 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(6 a^2 b C+8 a^3 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)}+\frac{2 \left(a^2 b^2 (A+15 C)-8 a^4 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)}",1,"(2*(3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(3*b^3*(A - C) + 8*a^3*C + 6*a^2*b*C - a*b^2*(A + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 - 5*a^4*C + a^2*b^2*(A + 9*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,35,0.1429,1,"{4091, 4080, 4005, 3832, 4004}"
750,1,378,0,0.6937505,"\int \frac{\sec (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Sec[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"(-4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*C + 3*a*b*(A + C) - b^2*(A + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (4*a*(2*A*b^2 - a^2*C + 3*b^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,33,0.1515,1,"{4081, 4003, 4005, 3832, 4004}"
751,1,517,0,0.7732284,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 \left(-a^2 b^2 (7 A+3 C)+a^4 (-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^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(6 a^2 A b+3 a^2 b C+a^3 (-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}}-\frac{2 \left(-a^2 b^2 (7 A+3 C)+a^4 (-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 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 b^2 (7 A+3 C)+a^4 (-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^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(6 a^2 A b+3 a^2 b C+a^3 (-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}}-\frac{2 \left(-a^2 b^2 (7 A+3 C)+a^4 (-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 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}",1,"(-2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(6*a^2*A*b - a*A*b^2 - 3*A*b^3 - a^3*C + 3*a^2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(3*A*b^4 - a^4*C - a^2*b^2*(7*A + 3*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,27,0.2593,1,"{4061, 4060, 4058, 3921, 3784, 3832, 4004}"
752,1,559,0,1.1900124,"\int \frac{\cos (c+d x) \left(A+C \sec ^2(c+d x)\right)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(Cos[c + d*x]*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{b \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-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{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{\left(21 a^2 A b^2+a^3 b (3 A-2 C)+6 a^4 C-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{\left(26 a^2 A b^2+a^4 (-(3 A-8 C))-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{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{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}","-\frac{b \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-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{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{\left(21 a^2 A b^2+a^3 b (3 A-2 C)+6 a^4 C-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{\left(26 a^2 A b^2+a^4 (-(3 A-8 C))-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{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{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"-((26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) + ((21*a^2*A*b^2 - 5*a*A*b^3 - 15*A*b^4 + a^3*b*(3*A - 2*C) + 6*a^4*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) + (5*A*b*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (A*Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(5*A*b^2 - a^2*(3*A - 2*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - a^4*(3*A - 8*C))*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",8,7,33,0.2121,1,"{4105, 4060, 4058, 3921, 3784, 3832, 4004}"
753,1,645,0,1.5586925,"\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","Int[(Cos[c + d*x]^2*(A + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{b^2 \left(-2 a^2 b^2 (85 A-12 C)+a^4 (33 A-56 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{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(-3 a^2 b^2 (45 A-8 C)-a^3 (27 A b-8 b C)+6 a^4 (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{\left(-2 a^2 b^2 (85 A-12 C)+a^4 (33 A-56 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{\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{7 A b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}","-\frac{b^2 \left(-2 a^2 b^2 (85 A-12 C)+a^4 (33 A-56 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{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(-3 a^2 b^2 (45 A-8 C)-a^3 (27 A b-8 b C)+6 a^4 (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{\left(-2 a^2 b^2 (85 A-12 C)+a^4 (33 A-56 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{\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{7 A b \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}",1,"-((105*A*b^4 + a^4*(33*A - 56*C) - 2*a^2*b^2*(85*A - 12*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*Sqrt[a + b]*(a^2 - b^2)*d) + ((35*a*A*b^3 + 105*A*b^4 + 6*a^4*(A - 8*C) - 3*a^2*b^2*(45*A - 8*C) - a^3*(27*A*b - 8*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*a^4*Sqrt[a + b]*(a^2 - b^2)*d) - (Sqrt[a + b]*(35*A*b^2 + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^5*d) - (7*A*b*Sin[c + d*x])/(4*a^2*d*(a + b*Sec[c + d*x])^(3/2)) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*(a + b*Sec[c + d*x])^(3/2)) + (b^2*(35*A*b^2 - a^2*(27*A - 8*C))*Tan[c + d*x])/(12*a^3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b^2*(105*A*b^4 + a^4*(33*A - 56*C) - 2*a^2*b^2*(85*A - 12*C))*Tan[c + d*x])/(12*a^4*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,8,35,0.2286,1,"{4105, 4104, 4060, 4058, 3921, 3784, 3832, 4004}"
754,1,626,0,1.2756994,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]","-\frac{2 \left(-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-3 a^6 C-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(-a^2 b^2 (13 A+5 C)-3 a^4 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(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(a^3 b^2 (13 A+5 C)+36 a^2 A b^3-3 a^4 b (15 A+8 C)+3 a^5 C-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}-\frac{2 \left(-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-3 a^6 C-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 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(-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-3 a^6 C-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(-a^2 b^2 (13 A+5 C)-3 a^4 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(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(a^3 b^2 (13 A+5 C)+36 a^2 A b^3-3 a^4 b (15 A+8 C)+3 a^5 C-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}-\frac{2 \left(-29 a^4 b^2 (2 A+C)+41 a^2 A b^4-3 a^6 C-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 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}",1,"(-2*(41*a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*b^2*Sqrt[a + b]*(a^2 - b^2)^2*d) + (2*(36*a^2*A*b^3 - 5*a*A*b^4 - 15*A*b^5 + 3*a^5*C + a^3*b^2*(13*A + 5*C) - 3*a^4*b*(15*A + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*a^3*b*Sqrt[a + b]*(a^2 - b^2)^2*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (2*(A*b^2 + a^2*C)*Tan[c + d*x])/(5*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(5/2)) - (2*(5*A*b^4 - 3*a^4*C - a^2*b^2*(13*A + 5*C))*Tan[c + d*x])/(15*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(41*a^2*A*b^4 - 15*A*b^6 - 3*a^6*C - 29*a^4*b^2*(2*A + C))*Tan[c + d*x])/(15*a^3*(a^2 - b^2)^3*d*Sqrt[a + b*Sec[c + d*x]])","A",8,7,27,0.2593,1,"{4061, 4060, 4058, 3921, 3784, 3832, 4004}"
755,1,303,0,0.2891656,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(a^2 - b^2*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],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}} 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}","\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,"(2*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*a*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d","A",7,7,32,0.2188,1,"{4042, 3916, 3784, 12, 3837, 3832, 4004}"
756,1,200,0,0.1641501,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),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}","-\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,"(-2*Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d","A",4,4,32,0.1250,1,"{4042, 3921, 3784, 3832}"
757,1,338,0,0.4039179,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","\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}","\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,"(4*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) - (4*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(Sqrt[a + b]*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (4*b^2*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,32,0.2188,1,"{4042, 3923, 4058, 3921, 3784, 3832, 4004}"
758,1,445,0,0.6263792,"\int \frac{a^2-b^2 \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx","Int[(a^2 - b^2*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]","\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}","\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,"(2*(11*a^2 - 3*b^2)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*(9*a^2 - 2*a*b - 3*b^2)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a*(a - b)*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (4*b^2*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(11*a^2 - 3*b^2)*Tan[c + d*x])/(3*a*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,32,0.2500,1,"{4042, 3923, 4060, 4058, 3921, 3784, 3832, 4004}"
759,1,145,0,0.2514069,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),x]","\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}","\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,"(2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*A*b*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A*b^2 + a^2*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)","A",8,7,35,0.2000,1,"{4107, 3849, 2805, 3787, 3771, 2639, 2641}"
760,1,213,0,0.5664291,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","-\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)}}","-\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,"(-2*A*b*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",11,11,37,0.2973,1,"{4109, 3859, 2807, 2805, 3862, 3856, 2655, 2653, 3858, 2663, 2661}"
761,0,0,0,0.3103478,"\int (a+b \sec (c+d x))^{2/3} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(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,"(Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -5/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(2/3), x]","A",0,0,0,0,-1,"{}"
762,0,0,0,0.2919748,"\int \sqrt[3]{a+b \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[(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,"(Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(1/3), x]","A",0,0,0,0,-1,"{}"
763,0,0,0,0.2946585,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3),x]","\int \frac{A+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} 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,"(Sqrt[2]*C*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(-1/3), x]","A",0,0,0,0,-1,"{}"
764,0,0,0,0.2945171,"\int \frac{A+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Int[(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,"(Sqrt[2]*C*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) - (Sqrt[2]*a*C*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(-2/3), x]","A",0,0,0,0,-1,"{}"
765,1,145,0,0.2031105,"\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","Int[Sec[c + d*x]^3*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(3*(b*B + a*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((5*a*B + 4*b*C)*Tan[c + d*x])/(5*d) + (3*(b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((b*B + a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*a*B + 4*b*C)*Tan[c + d*x]^3)/(15*d)","A",8,6,38,0.1579,1,"{4072, 3997, 3787, 3767, 3768, 3770}"
766,1,114,0,0.1851974,"\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","Int[Sec[c + d*x]^2*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"((4*a*B + 3*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((b*B + a*C)*Tan[c + d*x])/d + ((4*a*B + 3*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((b*B + a*C)*Tan[c + d*x]^3)/(3*d)","A",7,6,38,0.1579,1,"{4072, 3997, 3787, 3768, 3770, 3767}"
767,1,93,0,0.1533763,"\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","Int[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 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}","\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,"((b*B + a*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*a*B + 2*b*C)*Tan[c + d*x])/(3*d) + ((b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",7,7,36,0.1944,1,"{4072, 3997, 3787, 3767, 8, 3768, 3770}"
768,1,61,0,0.0658929,"\int (a+b \sec (c+d x)) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"((2*a*B + b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((b*B + a*C)*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,4,30,0.1333,1,"{4048, 3770, 3767, 8}"
769,1,35,0,0.0736953,"\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","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{(a C+b B) \tanh ^{-1}(\sin (c+d x))}{d}+a B x+\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 + a*C)*ArcTanh[Sin[c + d*x]])/d + (b*C*Tan[c + d*x])/d","A",5,5,36,0.1389,1,"{4072, 3914, 3767, 8, 3770}"
770,1,35,0,0.1017633,"\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","Int[Cos[c + d*x]^2*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","x (a C+b B)+\frac{a B \sin (c+d x)}{d}+\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 + a*C)*x + (b*C*ArcTanh[Sin[c + d*x]])/d + (a*B*Sin[c + d*x])/d","A",4,3,38,0.07895,1,"{4072, 3996, 3770}"
771,1,52,0,0.1415578,"\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","Int[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"((a*B + 2*b*C)*x)/2 + ((b*B + a*C)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",5,5,38,0.1316,1,"{4072, 3996, 3787, 2637, 8}"
772,1,84,0,0.1663299,"\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","Int[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"((b*B + a*C)*x)/2 + ((2*a*B + 3*b*C)*Sin[c + d*x])/(3*d) + ((b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",6,6,38,0.1579,1,"{4072, 3996, 3787, 2635, 8, 2637}"
773,1,105,0,0.1778227,"\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","Int[Cos[c + d*x]^5*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\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}","-\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,"((3*a*B + 4*b*C)*x)/8 + ((b*B + a*C)*Sin[c + d*x])/d + ((3*a*B + 4*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((b*B + a*C)*Sin[c + d*x]^3)/(3*d)","A",7,6,38,0.1579,1,"{4072, 3996, 3787, 2633, 2635, 8}"
774,1,136,0,0.1998664,"\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","Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(3*(b*B + a*C)*x)/8 + ((4*a*B + 5*b*C)*Sin[c + d*x])/(5*d) + (3*(b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((b*B + a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a*B + 5*b*C)*Sin[c + d*x]^3)/(15*d)","A",8,6,38,0.1579,1,"{4072, 3996, 3787, 2635, 8, 2633}"
775,1,198,0,0.3517827,"\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","Int[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{\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}","\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,"((4*a^2*B + 3*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*b^2*C + 5*a*(2*b*B + a*C))*Tan[c + d*x])/(5*d) + ((4*a^2*B + 3*b^2*B + 6*a*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + (b*(5*b*B + 6*a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(20*d) + (b*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])*Tan[c + d*x])/(5*d) + ((4*b^2*C + 5*a*(2*b*B + a*C))*Tan[c + d*x]^3)/(15*d)","A",8,7,40,0.1750,1,"{4072, 4026, 4047, 3767, 4046, 3768, 3770}"
776,1,179,0,0.3475559,"\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","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\left(4 a^2 b B+a^3 (-C)+8 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 b 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{(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}","\frac{\left(4 a^2 b B+a^3 (-C)+8 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 b 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{(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,"((8*a*b*B + 4*a^2*C + 3*b^2*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a^2*b*B + 4*b^3*B - a^3*C + 8*a*b^2*C)*Tan[c + d*x])/(6*b*d) + ((8*a*b*B - 2*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)","A",8,8,38,0.2105,1,"{4072, 4010, 4002, 3997, 3787, 3770, 3767, 8}"
777,1,116,0,0.1464605,"\int (a+b \sec (c+d x))^2 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"((2*a^2*B + b^2*B + 2*a*b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (2*(3*a*b*B + a^2*C + b^2*C)*Tan[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,5,32,0.1562,1,"{4056, 4048, 3770, 3767, 8}"
778,1,86,0,0.1419748,"\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","Int[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 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}","\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,"a^2*B*x + ((4*a*b*B + 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(2*b*B + 3*a*C)*Tan[c + d*x])/(2*d) + (b*C*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",6,5,38,0.1316,1,"{4072, 3918, 3770, 3767, 8}"
779,1,60,0,0.176471,"\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","Int[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)}{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}","\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)*x + (b*(b*B + 2*a*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*B*Sin[c + d*x])/d + (b^2*C*Tan[c + d*x])/d","A",6,5,40,0.1250,1,"{4072, 4024, 3770, 3767, 8}"
780,1,80,0,0.2515978,"\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","Int[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{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}","\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,"((a^2*B + 2*b^2*B + 4*a*b*C)*x)/2 + (b^2*C*ArcTanh[Sin[c + d*x]])/d + (a*(2*b*B + a*C)*Sin[c + d*x])/d + (a^2*B*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",6,6,40,0.1500,1,"{4072, 4024, 4047, 8, 4045, 3770}"
781,1,107,0,0.2886757,"\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","Int[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{\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}","\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,"((2*a*b*B + a^2*C + 2*b^2*C)*x)/2 + ((2*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x])/(3*d) + (a*(2*b*B + a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a^2*B*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",6,6,40,0.1500,1,"{4072, 4024, 4047, 2637, 4045, 8}"
782,1,136,0,0.3205462,"\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","Int[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{\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}","\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,"((3*a^2*B + 4*b^2*B + 8*a*b*C)*x)/8 + ((2*a*b*B + a^2*C + b^2*C)*Sin[c + d*x])/d + ((3*a^2*B + 4*b^2*B + 8*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*B*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - (a*(2*b*B + a*C)*Sin[c + d*x]^3)/(3*d)","A",8,7,40,0.1750,1,"{4072, 4024, 4047, 2635, 8, 4044, 3013}"
783,1,180,0,0.3381058,"\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","Int[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{\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}","-\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,"((6*a*b*B + 3*a^2*C + 4*b^2*C)*x)/8 + ((4*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x])/(5*d) + ((6*a*b*B + 3*a^2*C + 4*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*b*B + a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a^2*B*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a^2*B + 5*b^2*B + 10*a*b*C)*Sin[c + d*x]^3)/(15*d)","A",8,7,40,0.1750,1,"{4072, 4024, 4047, 2633, 4045, 2635, 8}"
784,1,278,0,0.608875,"\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","Int[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{\left(15 a^2 b B+5 a^3 C+12 a b^2 C+4 b^3 B\right) \tan ^3(c+d x)}{15 d}+\frac{\left(15 a^2 b B+5 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{5 d}+\frac{\left(18 a^2 b C+8 a^3 B+18 a b^2 B+5 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{16 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(18 a^2 b C+8 a^3 B+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}","\frac{\left(15 a^2 b B+5 a^3 C+12 a b^2 C+4 b^3 B\right) \tan ^3(c+d x)}{15 d}+\frac{\left(15 a^2 b B+5 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{5 d}+\frac{\left(18 a^2 b C+8 a^3 B+18 a b^2 B+5 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{16 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(18 a^2 b C+8 a^3 B+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,"((8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*ArcTanh[Sin[c + d*x]])/(16*d) + ((15*a^2*b*B + 4*b^3*B + 5*a^3*C + 12*a*b^2*C)*Tan[c + d*x])/(5*d) + ((8*a^3*B + 18*a*b^2*B + 18*a^2*b*C + 5*b^3*C)*Sec[c + d*x]*Tan[c + d*x])/(16*d) + (b*(18*a*b*B + 14*a^2*C + 5*b^2*C)*Sec[c + d*x]^3*Tan[c + d*x])/(24*d) + (b^2*(3*b*B + 4*a*C)*Sec[c + d*x]^4*Tan[c + d*x])/(15*d) + (b*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(6*d) + ((15*a^2*b*B + 4*b^3*B + 5*a^3*C + 12*a*b^2*C)*Tan[c + d*x]^3)/(15*d)","A",9,8,40,0.2000,1,"{4072, 4026, 4076, 4047, 3767, 4046, 3768, 3770}"
785,1,252,0,0.498703,"\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","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\left(52 a^2 b^2 C+15 a^3 b B-3 a^4 C+60 a b^3 B+16 b^4 C\right) \tan (c+d x)}{30 b d}+\frac{\left(12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 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(30 a^2 b B-6 a^3 C+71 a b^2 C+45 b^3 B\right) \tan (c+d x) \sec (c+d x)}{120 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}","\frac{\left(52 a^2 b^2 C+15 a^3 b B-3 a^4 C+60 a b^3 B+16 b^4 C\right) \tan (c+d x)}{30 b d}+\frac{\left(12 a^2 b B+4 a^3 C+9 a b^2 C+3 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 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(30 a^2 b B-6 a^3 C+71 a b^2 C+45 b^3 B\right) \tan (c+d x) \sec (c+d x)}{120 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,"((12*a^2*b*B + 3*b^3*B + 4*a^3*C + 9*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 52*a^2*b^2*C + 16*b^4*C)*Tan[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + 71*a*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((15*a*b*B - 3*a^2*C + 16*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)","A",9,8,38,0.2105,1,"{4072, 4010, 4002, 3997, 3787, 3770, 3767, 8}"
786,1,180,0,0.2620397,"\int (a+b \sec (c+d x))^3 \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\left(16 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 d}+\frac{\left(12 a^2 b C+8 a^3 B+12 a b^2 B+3 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{8 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{(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}","\frac{\left(16 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 d}+\frac{\left(12 a^2 b C+8 a^3 B+12 a b^2 B+3 b^3 C\right) \tanh ^{-1}(\sin (c+d x))}{8 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{(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,"((8*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 3*b^3*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Tan[c + d*x])/(6*d) + (b*(20*a*b*B + 6*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,5,32,0.1562,1,"{4056, 4048, 3770, 3767, 8}"
787,1,137,0,0.2401193,"\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","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),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(6 a^2 b B+2 a^3 C+3 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 B x+\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}","\frac{b \left(8 a^2 C+9 a b B+2 b^2 C\right) \tan (c+d x)}{3 d}+\frac{\left(6 a^2 b B+2 a^3 C+3 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 B x+\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,"a^3*B*x + ((6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b*(9*a*b*B + 8*a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*d) + (b^2*(3*b*B + 5*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",7,6,38,0.1579,1,"{4072, 3918, 4048, 3770, 3767, 8}"
788,1,131,0,0.2911895,"\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","Int[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{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}","-\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,"a^2*(3*b*B + a*C)*x + (b*(6*a*b*B + 6*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (a*B*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (b*(2*a^2*B - b^2*B - 3*a*b*C)*Tan[c + d*x])/d - (b^2*(2*a*B - b*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,6,40,0.1500,1,"{4072, 4025, 4048, 3770, 3767, 8}"
789,1,124,0,0.4042873,"\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","Int[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{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}","\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,"(a*(a^2*B + 6*b^2*B + 6*a*b*C)*x)/2 + (b^2*(b*B + 3*a*C)*ArcTanh[Sin[c + d*x]])/d + (a^2*(2*b*B + a*C)*Sin[c + d*x])/d + (a*B*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(a*B - 2*b*C)*Tan[c + d*x])/(2*d)","A",7,7,40,0.1750,1,"{4072, 4025, 4076, 4047, 8, 4045, 3770}"
790,1,145,0,0.4148001,"\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","Int[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 \left(2 a^2 B+9 a b C+8 b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(3 a^2 b B+a^3 C+6 a b^2 C+2 b^3 B\right)+\frac{a^2 (3 a C+5 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\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}","\frac{a \left(2 a^2 B+9 a b C+8 b^2 B\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(3 a^2 b B+a^3 C+6 a b^2 C+2 b^3 B\right)+\frac{a^2 (3 a C+5 b B) \sin (c+d x) \cos (c+d x)}{6 d}+\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,"((3*a^2*b*B + 2*b^3*B + a^3*C + 6*a*b^2*C)*x)/2 + (b^3*C*ArcTanh[Sin[c + d*x]])/d + (a*(2*a^2*B + 8*b^2*B + 9*a*b*C)*Sin[c + d*x])/(3*d) + (a^2*(5*b*B + 3*a*C)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (a*B*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)","A",7,7,40,0.1750,1,"{4072, 4025, 4074, 4047, 8, 4045, 3770}"
791,1,179,0,0.4912157,"\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","Int[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{\left(6 a^2 b B+2 a^3 C+9 a b^2 C+3 b^3 B\right) \sin (c+d x)}{3 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{1}{8} x \left(12 a^2 b C+3 a^3 B+12 a b^2 B+8 b^3 C\right)+\frac{a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}","\frac{\left(6 a^2 b B+2 a^3 C+9 a b^2 C+3 b^3 B\right) \sin (c+d x)}{3 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{1}{8} x \left(12 a^2 b C+3 a^3 B+12 a b^2 B+8 b^3 C\right)+\frac{a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"((3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*x)/8 + ((6*a^2*b*B + 3*b^3*B + 2*a^3*C + 9*a*b^2*C)*Sin[c + d*x])/(3*d) + (a*(3*a^2*B + 10*b^2*B + 12*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(3*b*B + 2*a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (a*B*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)","A",7,7,40,0.1750,1,"{4072, 4025, 4074, 4047, 2637, 4045, 8}"
792,1,221,0,0.5480438,"\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","Int[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{a \left(4 a^2 B+15 a b C+12 b^2 B\right) \sin ^3(c+d x)}{15 d}+\frac{\left(15 a^2 b C+4 a^3 B+14 a b^2 B+5 b^3 C\right) \sin (c+d x)}{5 d}+\frac{\left(9 a^2 b B+3 a^3 C+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(9 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right)+\frac{a^2 (5 a C+7 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a B \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 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{\left(15 a^2 b C+4 a^3 B+14 a b^2 B+5 b^3 C\right) \sin (c+d x)}{5 d}+\frac{\left(9 a^2 b B+3 a^3 C+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(9 a^2 b B+3 a^3 C+12 a b^2 C+4 b^3 B\right)+\frac{a^2 (5 a C+7 b B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a B \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}",1,"((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*x)/8 + ((4*a^3*B + 14*a*b^2*B + 15*a^2*b*C + 5*b^3*C)*Sin[c + d*x])/(5*d) + ((9*a^2*b*B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a^2*(7*b*B + 5*a*C)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (a*B*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - (a*(4*a^2*B + 12*b^2*B + 15*a*b*C)*Sin[c + d*x]^3)/(15*d)","A",9,8,40,0.2000,1,"{4072, 4025, 4074, 4047, 2635, 8, 4044, 3013}"
793,1,187,0,0.7165445,"\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","Int[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","-\frac{\left(-3 a^2 C+3 a b B-2 b^2 C\right) \tan (c+d x)}{3 b^3 d}+\frac{\left(2 a^2+b^2\right) (b B-a C) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\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{(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}","-\frac{\left(-3 a^2 C+3 a b B-2 b^2 C\right) \tan (c+d x)}{3 b^3 d}+\frac{\left(2 a^2+b^2\right) (b B-a C) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\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{(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^2 + b^2)*(b*B - a*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a^3*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) - ((3*a*b*B - 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)","A",9,9,40,0.2250,1,"{4072, 4033, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
794,1,143,0,0.4503332,"\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","Int[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","-\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{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{(b B-a C) \tan (c+d x)}{b^2 d}+\frac{C \tan (c+d x) \sec (c+d x)}{2 b d}","-\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{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{(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,"-((2*a*b*B - 2*a^2*C - b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^3*d) + (2*a^2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Tan[c + d*x])/(b^2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",8,8,40,0.2000,1,"{4072, 4033, 4082, 3998, 3770, 3831, 2659, 208}"
795,1,98,0,0.265515,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}","\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,"((b*B - a*C)*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*a*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Tan[c + d*x])/(b*d)","A",8,8,38,0.2105,1,"{4072, 4010, 12, 3789, 3770, 3831, 2659, 208}"
796,1,76,0,0.1199252,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","\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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(b*d) + (2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b*Sqrt[a + b]*d)","A",6,6,32,0.1875,1,"{4050, 3770, 12, 3831, 2659, 208}"
797,1,67,0,0.1635372,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}}","\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*x)/a - (2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)","A",5,5,38,0.1316,1,"{4072, 3919, 3831, 2659, 208}"
798,1,90,0,0.2259046,"\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","Int[(Cos[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}","\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)*x)/a^2) + (2*b*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (B*Sin[c + d*x])/(a*d)","A",6,6,40,0.1500,1,"{4072, 4034, 12, 3783, 2659, 208}"
799,1,134,0,0.4789188,"\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","Int[(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 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{x \left(a^2 B-2 a b C+2 b^2 B\right)}{2 a^3}-\frac{(b B-a C) \sin (c+d x)}{a^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 a 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{x \left(a^2 B-2 a b C+2 b^2 B\right)}{2 a^3}-\frac{(b B-a C) \sin (c+d x)}{a^2 d}+\frac{B \sin (c+d x) \cos (c+d x)}{2 a d}",1,"((a^2*B + 2*b^2*B - 2*a*b*C)*x)/(2*a^3) - (2*b^2*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - ((b*B - a*C)*Sin[c + d*x])/(a^2*d) + (B*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",7,7,40,0.1750,1,"{4072, 4034, 4104, 3919, 3831, 2659, 208}"
800,1,178,0,0.7043634,"\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","Int[(Cos[c + d*x]^4*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\frac{\left(2 a^2 B-3 a b C+3 b^2 B\right) \sin (c+d x)}{3 a^3 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{x \left(a^2+2 b^2\right) (b B-a C)}{2 a^4}-\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{B \sin (c+d x) \cos ^2(c+d x)}{3 a d}","\frac{\left(2 a^2 B-3 a b C+3 b^2 B\right) \sin (c+d x)}{3 a^3 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{x \left(a^2+2 b^2\right) (b B-a C)}{2 a^4}-\frac{(b B-a C) \sin (c+d x) \cos (c+d x)}{2 a^2 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)*x)/(2*a^4) + (2*b^3*(b*B - a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((2*a^2*B + 3*b^2*B - 3*a*b*C)*Sin[c + d*x])/(3*a^3*d) - ((b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (B*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)","A",8,7,40,0.1750,1,"{4072, 4034, 4104, 3919, 3831, 2659, 208}"
801,1,272,0,0.8842209,"\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","Int[(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(2 a^2 b B-3 a^3 C+2 a b^2 C-b^3 B\right) \tan (c+d x)}{b^3 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{2 a^2 \left(2 a^2 b B-3 a^3 C+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}}+\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(2 a^2 b B-3 a^3 C+2 a b^2 C-b^3 B\right) \tan (c+d x)}{b^3 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{2 a^2 \left(2 a^2 b B-3 a^3 C+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}}+\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)}",1,"-((4*a*b*B - 6*a^2*C - b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) + (2*a^2*(2*a^2*b*B - 3*b^3*B - 3*a^3*C + 4*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - 3*a^3*C + 2*a*b^2*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) - ((2*a*b*B - 3*a^2*C + b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) + (a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,9,40,0.2250,1,"{4072, 4029, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
802,1,164,0,0.6228205,"\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","Int[(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{2 a \left(a^2 b B-2 a^3 C+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{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{(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}","-\frac{2 a \left(a^2 b B-2 a^3 C+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{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{(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}",1,"((b*B - 2*a*C)*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*a*(a^2*b*B - 2*b^3*B - 2*a^3*C + 3*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Tan[c + d*x])/(b^2*d) - (a^2*(b*B - a*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,8,40,0.2000,1,"{4072, 4028, 4082, 3998, 3770, 3831, 2659, 208}"
803,1,131,0,0.3193538,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","-\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}","-\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,"(C*ArcTanh[Sin[c + d*x]])/(b^2*d) - (2*(b^3*B + a^3*C - 2*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) + (a*(b*B - a*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,7,38,0.1842,1,"{4072, 4009, 3998, 3770, 3831, 2659, 208}"
804,1,100,0,0.1257228,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","\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))}","\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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*(a + b)^(3/2)*d) - ((b*B - a*C)*Tan[c + d*x])/((a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,32,0.1562,1,"{4060, 12, 3831, 2659, 208}"
805,1,124,0,0.2738717,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^2,x]","-\frac{2 \left(2 a^2 b B+a^3 (-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}}+\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(2 a^2 b B+a^3 (-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}}+\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}",1,"(B*x)/a^2 - (2*(2*a^2*b*B - b^3*B - a^3*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b*(b*B - a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,38,0.1579,1,"{4072, 3923, 3919, 3831, 2659, 208}"
806,1,180,0,0.6329147,"\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","Int[(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{\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{2 b \left(3 a^2 b B-2 a^3 C+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}}+\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{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{2 b \left(3 a^2 b B-2 a^3 C+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}}+\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{x (2 b B-a C)}{a^3}",1,"-(((2*b*B - a*C)*x)/a^3) + (2*b*(3*a^2*b*B - 2*b^3*B - 2*a^3*C + a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((a^2*B - 2*b^2*B + a*b*C)*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + (b*(b*B - a*C)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,7,40,0.1750,1,"{4072, 4030, 4104, 3919, 3831, 2659, 208}"
807,1,261,0,0.9309861,"\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","Int[(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{\left(2 a^2 b B+a^3 (-C)+2 a b^2 C-3 b^3 B\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}+\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{2 b^2 \left(4 a^2 b B-3 a^3 C+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}}+\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(2 a^2 b B+a^3 (-C)+2 a b^2 C-3 b^3 B\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}+\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{2 b^2 \left(4 a^2 b B-3 a^3 C+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}}+\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}",1,"((a^2*B + 6*b^2*B - 4*a*b*C)*x)/(2*a^4) - (2*b^2*(4*a^2*b*B - 3*b^3*B - 3*a^3*C + 2*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*a^2*b*B - 3*b^3*B - a^3*C + 2*a*b^2*C)*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) + ((a^2*B - 3*b^2*B + 2*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + (b*(b*B - a*C)*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,7,40,0.1750,1,"{4072, 4030, 4104, 3919, 3831, 2659, 208}"
808,1,289,0,1.4248213,"\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","Int[(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{\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 \left(-5 a^2 b^3 B+15 a^3 b^2 C+2 a^4 b B-6 a^5 C-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{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{a^2 \left(a^2 b B-3 a^3 C+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{(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}","-\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 \left(-5 a^2 b^3 B+15 a^3 b^2 C+2 a^4 b B-6 a^5 C-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{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{a^2 \left(a^2 b B-3 a^3 C+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{(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"((b*B - 3*a*C)*ArcTanh[Sin[c + d*x]])/(b^4*d) - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) - ((a*b*B - 3*a^2*C + 2*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + (a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a^2*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",9,9,40,0.2250,1,"{4072, 4029, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
809,1,220,0,0.7507993,"\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","Int[(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{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-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{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(a^2 b B-3 a^3 C+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{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}","\frac{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-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{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(a^2 b B-3 a^3 C+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{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"(C*ArcTanh[Sin[c + d*x]])/(b^3*d) + ((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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) - (a^2*(b*B - a*C)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (a*(a^2*b*B - 4*b^3*B - 3*a^3*C + 6*a*b^2*C)*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,8,40,0.2000,1,"{4072, 4028, 4080, 3998, 3770, 3831, 2659, 208}"
810,1,180,0,0.3706158,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","-\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{\left(a^2 b B+a^3 C-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))}+\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^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{\left(a^2 b B+a^3 C-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))}+\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}",1,"-(((3*a*b*B - a^2*C - 2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) + (a*(b*B - a*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - 4*a*b^2*C)*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,38,0.1842,1,"{4072, 4009, 4003, 12, 3831, 2659, 208}"
811,1,164,0,0.2649858,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\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}","\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*a^2*B + b^2*B - 3*a*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d) - ((b*B - a*C)*Tan[c + d*x])/(2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*a*b*B - a^2*C - 2*b^2*C)*Tan[c + d*x])/(2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,5,32,0.1562,1,"{4060, 12, 3831, 2659, 208}"
812,1,205,0,0.580986,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^3,x]","-\frac{\left(-5 a^2 b^3 B-a^3 b^2 C+6 a^4 b B-2 a^5 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}}+\frac{b \left(5 a^2 b B-3 a^3 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 (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 x}{a^3}","-\frac{\left(-5 a^2 b^3 B-a^3 b^2 C+6 a^4 b B-2 a^5 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}}+\frac{b \left(5 a^2 b B-3 a^3 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 (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 x}{a^3}",1,"(B*x)/a^3 - ((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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b*(b*B - a*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(5*a^2*b*B - 2*b^3*B - 3*a^3*C)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,38,0.1842,1,"{4072, 3923, 4060, 3919, 3831, 2659, 208}"
813,1,290,0,1.5329925,"\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","Int[(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{\left(-11 a^2 b^2 B+5 a^3 b C+2 a^4 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(-15 a^2 b^3 B+5 a^3 b^2 C+12 a^4 b B-6 a^5 C-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}}+\frac{b \left(6 a^2 b B-4 a^3 C+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{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{x (3 b B-a C)}{a^4}","\frac{\left(-11 a^2 b^2 B+5 a^3 b C+2 a^4 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(-15 a^2 b^3 B+5 a^3 b^2 C+12 a^4 b B-6 a^5 C-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}}+\frac{b \left(6 a^2 b B-4 a^3 C+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{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{x (3 b B-a C)}{a^4}",1,"-(((3*b*B - a*C)*x)/a^4) + (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((2*a^4*B - 11*a^2*b^2*B + 6*b^4*B + 5*a^3*b*C - 2*a*b^3*C)*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + (b*(b*B - a*C)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b*(6*a^2*b*B - 3*b^3*B - 4*a^3*C + a*b^2*C)*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,8,40,0.2000,1,"{4072, 4030, 4100, 4104, 3919, 3831, 2659, 208}"
814,1,485,0,1.4536636,"\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","Int[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{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(12 a^2 b B-8 a^3 C-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(12 a^2 b (2 B-C)-16 a^3 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(-24 a^2 b^2 C+24 a^3 b B-16 a^4 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}","\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(12 a^2 b B-8 a^3 C-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(12 a^2 b (2 B-C)-16 a^3 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(-24 a^2 b^2 C+24 a^3 b B-16 a^4 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*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) + 18*a*b^2*(B - 2*C) + 12*a^2*b*(2*B - C) - 16*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(12*a^2*b*B - 75*b^3*B - 8*a^3*C - 13*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(9*a*b*B - 6*a^2*C + 49*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B + a*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)","A",8,8,42,0.1905,1,"{4072, 4031, 4102, 4092, 4082, 4005, 3832, 4004}"
815,1,397,0,1.0026088,"\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","Int[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{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(14 a^2 b B-8 a^3 C-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}","\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(14 a^2 b B-8 a^3 C-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*(a - b)*Sqrt[a + b]*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - 19*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*B - 25*C) + 2*a*b*(7*B - 3*C) - 8*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(7*a*b*B - 4*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) + (2*(7*b*B + a*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*d)","A",7,7,42,0.1667,1,"{4072, 4031, 4092, 4082, 4005, 3832, 4004}"
816,1,314,0,0.6018435,"\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","Int[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\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}","-\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,"(-2*(a - b)*Sqrt[a + b]*(5*a*b*B - 2*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) - (2*(a - b)*Sqrt[a + b]*(5*b*B - 2*a*C - 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)","A",6,6,40,0.1500,1,"{4072, 4010, 4002, 4005, 3832, 4004}"
817,1,256,0,0.2887722,"\int \sqrt{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\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}","-\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*(a - b)*Sqrt[a + b]*(3*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*(a - b)*Sqrt[a + b]*(3*B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",5,5,34,0.1471,1,"{4056, 4058, 12, 3832, 4004}"
818,1,320,0,0.3611085,"\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","Int[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 \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}","\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*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(b*(B - C) + a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d","A",6,6,40,0.1500,1,"{4072, 3916, 3784, 4005, 3832, 4004}"
819,1,344,0,0.4636401,"\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","Int[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} (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}","\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,"((a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(B + 2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(b*B + 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",7,7,42,0.1667,1,"{4072, 4032, 4058, 3921, 3784, 3832, 4004}"
820,1,429,0,0.8052792,"\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","Int[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{\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}","-\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,"((a - b)*Sqrt[a + b]*(b*B + 4*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*b*d) + (Sqrt[a + b]*(b*B + 2*a*(B + 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) - (Sqrt[a + b]*(4*a^2*B - b^2*B + 4*a*b*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((b*B + 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (B*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",8,8,42,0.1905,1,"{4072, 4032, 4104, 4058, 3921, 3784, 3832, 4004}"
821,1,573,0,1.9810827,"\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","Int[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 \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(88 a^2 b B-48 a^3 C-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(-144 a^2 b^2 C+88 a^3 b B-48 a^4 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(6 a^2 b^2 (11 B-24 C)+4 a^3 b (22 B-9 C)-48 a^4 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(363 a^2 b^3 B-108 a^3 b^2 C+88 a^4 b B-48 a^5 C+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}","-\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(88 a^2 b B-48 a^3 C-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(-144 a^2 b^2 C+88 a^3 b B-48 a^4 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(6 a^2 b^2 (11 B-24 C)+4 a^3 b (22 B-9 C)-48 a^4 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(363 a^2 b^3 B-108 a^3 b^2 C+88 a^4 b B-48 a^5 C+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)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^5*d) - (2*(a - b)*Sqrt[a + b]*(3*a*b^3*(143*B - 471*C) - 3*b^4*(539*B - 225*C) + 6*a^2*b^2*(11*B - 24*C) + 4*a^3*b*(22*B - 9*C) - 48*a^4*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) + (2*(88*a^3*b*B + 429*a*b^3*B - 48*a^4*C - 144*a^2*b^2*C + 675*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^3*d) + (2*(88*a^2*b*B + 539*b^3*B - 48*a^3*C - 204*a*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^3*d) - (2*(44*a*b*B - 24*a^2*C - 81*b^2*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^3*d) + (2*(11*b*B - 6*a*C)*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(11*b*d)","A",9,8,42,0.1905,1,"{4072, 4033, 4092, 4082, 4002, 4005, 3832, 4004}"
822,1,475,0,1.2238818,"\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","Int[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]","-\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(18 a^2 b B-8 a^3 C-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(-6 a^2 b (3 B-C)+8 a^3 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(-33 a^2 b^2 C+18 a^3 b B-8 a^4 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}","-\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(18 a^2 b B-8 a^3 C-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(-6 a^2 b (3 B-C)+8 a^3 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(-33 a^2 b^2 C+18 a^3 b B-8 a^4 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,"(2*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) - 3*a*b^2*(57*B - 13*C) - 6*a^2*b*(3*B - C) + 8*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 39*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) - (2*(18*a*b*B - 8*a^2*C - 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)","A",8,7,42,0.1667,1,"{4072, 4033, 4082, 4002, 4005, 3832, 4004}"
823,1,387,0,0.8530354,"\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","Int[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{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(21 a^2 b B-6 a^3 C+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}","\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(21 a^2 b B-6 a^3 C+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)*Sqrt[a + b]*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 82*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) - (2*(a - b)*Sqrt[a + b]*(a*b*(21*B - 57*C) - b^2*(63*B - 25*C) - 6*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)","A",7,6,40,0.1500,1,"{4072, 4010, 4002, 4005, 3832, 4004}"
824,1,312,0,0.4677491,"\int (a+b \sec (c+d x))^{3/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","-\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}","-\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)*Sqrt[a + b]*(20*a*b*B + 3*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(a - b)*Sqrt[a + b]*(15*a*B - 5*b*B - 3*a*C + 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",6,5,34,0.1471,1,"{4056, 4058, 12, 3832, 4004}"
825,1,380,0,0.5293851,"\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","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(-2*(a - b)*Sqrt[a + b]*(3*b*B + 4*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (2*Sqrt[a + b]*(a*b*(6*B - 4*C) - b^2*(3*B - C) + 3*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*a*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,40,0.1750,1,"{4072, 3918, 4058, 3921, 3784, 3832, 4004}"
826,1,361,0,0.5480506,"\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","Int[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{\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}","\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,"((a - b)*Sqrt[a + b]*(a*B - 2*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(2*b*(B - C) + a*(B + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Sqrt[a + b]*(3*b*B + 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",7,7,42,0.1667,1,"{4072, 4025, 4058, 3921, 3784, 3832, 4004}"
827,1,428,0,0.8743805,"\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","Int[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{\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}","-\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]*(5*b*B + 4*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(2*a*B + 5*b*B + 4*a*C + 8*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*B + 3*b^2*B + 12*a*b*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + ((5*b*B + 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",8,8,42,0.1905,1,"{4072, 4025, 4104, 4058, 3921, 3784, 3832, 4004}"
828,1,520,0,1.3245258,"\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","Int[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{\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(12 a^2 b B+8 a^3 C+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}","\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(12 a^2 b B+8 a^3 C+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,"((a - b)*Sqrt[a + b]*(16*a^2*B + 3*b^2*B + 30*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(16*a^2*B + 14*a*b*B + 3*b^2*B + 12*a^2*C + 30*a*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) - (Sqrt[a + b]*(12*a^2*b*B - b^3*B + 8*a^3*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((16*a^2*B + 3*b^2*B + 30*a*b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + ((7*b*B + 6*a*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) + (a*B*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",9,8,42,0.1905,1,"{4072, 4025, 4104, 4058, 3921, 3784, 3832, 4004}"
829,1,565,0,1.8112272,"\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","Int[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]","-\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(110 a^2 b B-40 a^3 C-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(-285 a^2 b^2 C+110 a^3 b B-40 a^4 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(-15 a^2 b^2 (121 B-19 C)-a^3 b (110 B-30 C)+40 a^4 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(-3069 a^2 b^3 B-255 a^3 b^2 C+110 a^4 b B-40 a^5 C-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}","-\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(110 a^2 b B-40 a^3 C-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(-285 a^2 b^2 C+110 a^3 b B-40 a^4 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(-15 a^2 b^2 (121 B-19 C)-a^3 b (110 B-30 C)+40 a^4 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(-3069 a^2 b^3 B-255 a^3 b^2 C+110 a^4 b B-40 a^5 C-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,"(2*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(a - b)*Sqrt[a + b]*(6*a*b^3*(209*B - 505*C) - 3*b^4*(539*B - 225*C) - a^3*b*(110*B - 30*C) - 15*a^2*b^2*(121*B - 19*C) + 40*a^4*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^3*d) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 285*a^2*b^2*C - 675*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 335*a*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^2*d) - (2*(22*a*b*B - 8*a^2*C - 81*b^2*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)","A",9,7,42,0.1667,1,"{4072, 4033, 4082, 4002, 4005, 3832, 4004}"
830,1,469,0,1.1956174,"\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","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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(45 a^2 b B-10 a^3 C+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(15 a^2 b (3 B-11 C)-10 a^3 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(279 a^2 b^2 C+45 a^3 b B-10 a^4 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}","\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(45 a^2 b B-10 a^3 C+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(15 a^2 b (3 B-11 C)-10 a^3 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(279 a^2 b^2 C+45 a^3 b B-10 a^4 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,"(-2*(a - b)*Sqrt[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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(a - b)*Sqrt[a + b]*(3*b^3*(25*B - 49*C) - 6*a*b^2*(60*B - 19*C) + 15*a^2*b*(3*B - 11*C) - 10*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 114*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) + (2*(45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)","A",8,6,40,0.1500,1,"{4072, 4010, 4002, 4005, 3832, 4004}"
831,1,384,0,0.6776141,"\int (a+b \sec (c+d x))^{5/2} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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(161 a^2 b B+15 a^3 C+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}","\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(161 a^2 b B+15 a^3 C+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,"(-2*(a - b)*Sqrt[a + b]*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 145*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(a - b)*Sqrt[a + b]*(b^2*(63*B - 25*C) - 8*a*b*(7*B - 15*C) + 15*a^2*(7*B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b*d) + (2*(56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",7,5,34,0.1471,1,"{4056, 4058, 12, 3832, 4004}"
832,1,442,0,0.7153489,"\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","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{2 \sqrt{a+b} \left(a^2 b (45 B-23 C)+15 a^3 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 (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 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}","\frac{2 \sqrt{a+b} \left(a^2 b (45 B-23 C)+15 a^3 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 (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 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,"(-2*(a - b)*Sqrt[a + b]*(35*a*b*B + 23*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (2*Sqrt[a + b]*(a^2*b*(45*B - 23*C) - a*b^2*(35*B - 17*C) + b^3*(5*B - 9*C) + 15*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (2*a^2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*(5*b*B + 8*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",8,8,40,0.2000,1,"{4072, 3918, 4056, 4058, 3921, 3784, 3832, 4004}"
833,1,433,0,0.777233,"\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","Int[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]","\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}","\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,"((a - b)*Sqrt[a + b]*(3*a^2*B - 6*b^2*B - 14*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(2*a*b*(9*B - 7*C) - 2*b^2*(3*B - C) + 3*a^2*(B + 6*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (a*Sqrt[a + b]*(5*b*B + 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (a*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*a*B - 2*b*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",8,8,42,0.1905,1,"{4072, 4025, 4056, 4058, 3921, 3784, 3832, 4004}"
834,1,450,0,0.8997704,"\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","Int[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{\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}","\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,"((a - b)*Sqrt[a + b]*(9*a*b*B + 4*a^2*C - 8*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(8*b^2*(B - C) + 2*a^2*(B + 2*C) + 3*a*b*(3*B + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(4*a^2*B + 15*b^2*B + 20*a*b*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + (a*(7*b*B + 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",8,8,42,0.1905,1,"{4072, 4025, 4094, 4058, 3921, 3784, 3832, 4004}"
835,1,518,0,1.2742936,"\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","Int[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{\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(20 a^2 b B+8 a^3 C+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}","\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(20 a^2 b B+8 a^3 C+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,"((a - b)*Sqrt[a + b]*(16*a^2*B + 33*b^2*B + 54*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(4*a^2*(4*B + 3*C) + 3*b^2*(11*B + 16*C) + a*b*(26*B + 54*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (Sqrt[a + b]*(20*a^2*b*B + 5*b^3*B + 8*a^3*C + 30*a*b^2*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((16*a^2*B + 33*b^2*B + 54*a*b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*(3*b*B + 2*a*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (a*B*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",9,9,42,0.2143,1,"{4072, 4025, 4094, 4104, 4058, 3921, 3784, 3832, 4004}"
836,1,617,0,1.8278189,"\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","Int[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]","\frac{\left(284 a^2 b B+128 a^3 C+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{\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{\sqrt{a+b} \left(4 a^2 b (71 B+52 C)+8 a^3 (9 B+16 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(284 a^2 b B+128 a^3 C+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(120 a^2 b^2 B+160 a^3 b C+48 a^4 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}","\frac{\left(284 a^2 b B+128 a^3 C+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{\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{\sqrt{a+b} \left(4 a^2 b (71 B+52 C)+8 a^3 (9 B+16 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(284 a^2 b B+128 a^3 C+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(120 a^2 b^2 B+160 a^3 b C+48 a^4 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,"((a - b)*Sqrt[a + b]*(284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*b*d) + (Sqrt[a + b]*(15*b^3*B + 8*a^3*(9*B + 16*C) + 4*a^2*b*(71*B + 52*C) + 2*a*b^2*(59*B + 132*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) - (Sqrt[a + b]*(48*a^4*B + 120*a^2*b^2*B - 5*b^4*B + 160*a^3*b*C + 40*a*b^3*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + ((284*a^2*b*B + 15*b^3*B + 128*a^3*C + 264*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((36*a^2*B + 59*b^2*B + 104*a*b*C)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*d) + (a*(11*b*B + 8*a*C)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (a*B*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",10,9,42,0.2143,1,"{4072, 4025, 4094, 4104, 4058, 3921, 3784, 3832, 4004}"
837,1,411,0,1.0448485,"\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","Int[(Sec[c + d*x]^3*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","-\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 \sqrt{a+b} \left(4 a^2 b (14 B+3 C)-48 a^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 (a-b) \sqrt{a+b} \left(56 a^2 b B-48 a^3 C-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 (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}","-\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 \sqrt{a+b} \left(4 a^2 b (14 B+3 C)-48 a^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 (a-b) \sqrt{a+b} \left(56 a^2 b B-48 a^3 C-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 (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,"(-2*(a - b)*Sqrt[a + b]*(56*a^2*b*B + 63*b^3*B - 48*a^3*C - 44*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) - (2*Sqrt[a + b]*(b^3*(63*B - 25*C) - 48*a^3*C + 4*a^2*b*(14*B + 3*C) - 2*a*b^2*(7*B + 22*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) - (2*(28*a*b*B - 24*a^2*C - 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) + (2*(7*b*B - 6*a*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)","A",7,7,42,0.1667,1,"{4072, 4033, 4092, 4082, 4005, 3832, 4004}"
838,1,329,0,0.68511,"\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","Int[(Sec[c + d*x]^2*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","\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 (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 (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}","\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 (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 (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,"(2*(a - b)*Sqrt[a + b]*(10*a*b*B - 8*a^2*C - 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) + (2*Sqrt[a + b]*(b^2*(5*B - 9*C) - 8*a^2*C + 2*a*b*(5*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)","A",6,6,42,0.1429,1,"{4072, 4033, 4082, 4005, 3832, 4004}"
839,1,261,0,0.436528,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],x]","-\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 (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}","-\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 (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,"(-2*(a - b)*Sqrt[a + b]*(3*b*B - 2*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) - (2*Sqrt[a + b]*(3*b*B - 2*a*C - b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)","A",5,5,40,0.1250,1,"{4072, 4010, 4005, 3832, 4004}"
840,1,210,0,0.1737934,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],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}","\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*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) + (2*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d)","A",4,4,34,0.1176,1,"{4058, 12, 3832, 4004}"
841,1,208,0,0.2085049,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/Sqrt[a + b*Sec[c + d*x]],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}","\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,"(2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",4,4,40,0.1000,1,"{4072, 3921, 3784, 3832}"
842,1,348,0,0.4993788,"\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","Int[(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{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}","\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,"((a - b)*Sqrt[a + b]*B*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*B*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (Sqrt[a + b]*(b*B - 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (B*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",7,7,42,0.1667,1,"{4072, 4034, 4059, 3921, 3784, 3832, 4004}"
843,1,471,0,1.2729509,"\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","Int[(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]","\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(20 a^2 b B-24 a^3 C+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(4 a^2 b (10 B-9 C)-48 a^3 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(24 a^2 b^2 C+40 a^3 b B-48 a^4 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}}","\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(20 a^2 b B-24 a^3 C+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(4 a^2 b (10 B-9 C)-48 a^3 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(24 a^2 b^2 C+40 a^3 b B-48 a^4 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,"(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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^5*Sqrt[a + b]*d) + (2*(b^3*(5*B - 9*C) + 4*a^2*b*(10*B - 9*C) + 6*a*b^2*(5*B - 2*C) - 48*a^3*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*Sqrt[a + b]*d) + (2*a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 24*a^3*C + 9*a*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^3*(a^2 - b^2)*d) - (2*(5*a*b*B - 6*a^2*C + b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",7,7,42,0.1667,1,"{4072, 4029, 4092, 4082, 4005, 3832, 4004}"
844,1,329,0,0.8102448,"\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","Int[(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]","-\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(6 a^2 b B-8 a^3 C+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}","-\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(6 a^2 b B-8 a^3 C+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,"(-2*(6*a^2*b*B - 3*b^3*B - 8*a^3*C + 5*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) - (2*(2*a + b)*(3*b*B - 4*a*C - b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) - (2*a^2*(b*B - a*C)*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)","A",6,6,42,0.1429,1,"{4072, 4028, 4082, 4005, 3832, 4004}"
845,1,275,0,0.5049065,"\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","Int[(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{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}}","\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,"(2*(a*b*B - 2*a^2*C + b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(b*(B - C) - 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*a*(b*B - a*C)*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,40,0.1250,1,"{4072, 4009, 4005, 3832, 4004}"
846,1,254,0,0.3043759,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"(-2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) + (2*(B + C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*Sqrt[a + b]*d) - (2*(b*B - a*C)*Tan[c + d*x])/((a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,34,0.1471,1,"{4060, 4058, 12, 3832, 4004}"
847,1,376,0,0.5188131,"\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","Int[(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{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}}","\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,"(2*(b*B - a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*(b*B - a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b*(b*B - a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,40,0.1750,1,"{4072, 3923, 4058, 3921, 3784, 3832, 4004}"
848,1,427,0,0.79253,"\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","Int[(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 \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{\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 \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}","\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{\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 \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((a^2*B - 3*b^2*B + 2*a*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + ((3*b*B + a*(B - 2*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*Sqrt[a + b]*d) + (Sqrt[a + b]*(3*b*B - 2*a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (B*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(a^2*B - 3*b^2*B + 2*a*b*C)*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,42,0.1905,1,"{4072, 4034, 4061, 4058, 3921, 3784, 3832, 4004}"
849,1,509,0,1.5747854,"\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","Int[(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]","\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 a^2 \left(3 a^2 b B-6 a^3 C+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(-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 \left(2 a^2 b^2 (3 B+8 C)+a^3 b (8 B-12 C)-16 a^4 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(-15 a^2 b^3 B+28 a^3 b^2 C+8 a^4 b B-16 a^5 C-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}}","\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 a^2 \left(3 a^2 b B-6 a^3 C+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(-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 \left(2 a^2 b^2 (3 B+8 C)+a^3 b (8 B-12 C)-16 a^4 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(-15 a^2 b^3 B+28 a^3 b^2 C+8 a^4 b B-16 a^5 C-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,"(-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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^5*(a + b)^(3/2)*d) - (2*(a^3*b*(8*B - 12*C) - 9*a*b^3*(B - C) - b^4*(3*B - C) - 16*a^4*C + 2*a^2*b^2*(3*B + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(b*B - a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a^2*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C)*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(a*b*B - 2*a^2*C + b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",7,7,42,0.1667,1,"{4072, 4029, 4090, 4082, 4005, 3832, 4004}"
850,1,417,0,1.0049732,"\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","Int[(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]","-\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(2 a^2 b B-5 a^3 C+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(2 a^2 b (B-3 C)-8 a^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(15 a^2 b^2 C+2 a^3 b B-8 a^4 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}}","-\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(2 a^2 b B-5 a^3 C+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(2 a^2 b (B-3 C)-8 a^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(15 a^2 b^2 C+2 a^3 b B-8 a^4 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,"(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)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^4*(a + b)^(3/2)*d) + (2*(2*a^2*b*(B - 3*C) - 3*b^3*(B - C) - 8*a^3*C + 3*a*b^2*(B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) - (2*a^2*(b*B - a*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*a*(2*a^2*b*B - 6*b^3*B - 5*a^3*C + 9*a*b^2*C)*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",6,6,42,0.1429,1,"{4072, 4028, 4080, 4005, 3832, 4004}"
851,1,387,0,0.7345758,"\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","Int[(Sec[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(a^2 b B+2 a^3 C-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 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(a^2 b B+2 a^3 C-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}}","\frac{2 \left(a^2 b B+2 a^3 C-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 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(a^2 b B+2 a^3 C-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,"(2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*C - 3*b^2*(B + C) + a*b*(B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(b*B - a*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 6*a*b^2*C)*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",6,6,40,0.1500,1,"{4072, 4009, 4003, 4005, 3832, 4004}"
852,1,353,0,0.5210923,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","-\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}}","-\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,"(-2*(4*a*b*B - a^2*C - 3*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(3*a*B - b*B + a*C - 3*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b*(a + b)^(3/2)*d) - (2*(b*B - a*C)*Tan[c + d*x])/(3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*a*b*B - a^2*C - 3*b^2*C)*Tan[c + d*x])/(3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",6,5,34,0.1471,1,"{4060, 4058, 12, 3832, 4004}"
853,1,495,0,0.8568093,"\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","Int[(Cos[c + d*x]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 b \left(7 a^2 b B-4 a^3 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 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(6 a^2 b B+a^2 b C-3 a^3 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}}+\frac{2 \left(7 a^2 b B-4 a^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}} 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 \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 \left(7 a^2 b B-4 a^3 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 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(6 a^2 b B+a^2 b C-3 a^3 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}}+\frac{2 \left(7 a^2 b B-4 a^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}} 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 \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}",1,"(2*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*(6*a^2*b*B - a*b^2*B - 3*b^3*B - 3*a^3*C + a^2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b*(a + b)^(3/2)*d) - (2*Sqrt[a + b]*B*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b*(b*B - a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b*(7*a^2*b*B - 3*b^3*B - 4*a^3*C)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,40,0.2000,1,"{4072, 3923, 4060, 4058, 3921, 3784, 3832, 4004}"
854,1,446,0,0.8281845,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{7/2}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(7/2),x]","-\frac{2 \left(23 a^2 b B-3 a^3 C-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^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(23 a^2 b B-3 a^3 C-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}}","-\frac{2 \left(23 a^2 b B-3 a^3 C-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^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(23 a^2 b B-3 a^3 C-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,"(-2*(23*a^2*b*B + 9*b^3*B - 3*a^3*C - 29*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*(a - b)^2*b^2*(a + b)^(5/2)*d) + (2*(3*a^2*(5*B + C) - 8*a*b*(B + 3*C) + b^2*(9*B + 5*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*Sqrt[a + b]*(a^2 - b^2)^2*d) - (2*(b*B - a*C)*Tan[c + d*x])/(5*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(5/2)) - (2*(8*a*b*B - 3*a^2*C - 5*b^2*C)*Tan[c + d*x])/(15*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(23*a^2*b*B + 9*b^3*B - 3*a^3*C - 29*a*b^2*C)*Tan[c + d*x])/(15*(a^2 - b^2)^3*d*Sqrt[a + b*Sec[c + d*x]])","A",7,5,34,0.1471,1,"{4060, 4058, 12, 3832, 4004}"
855,1,101,0,0.2837403,"\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","Int[(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 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)}","\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*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a + b)*d)","A",6,6,42,0.1429,1,"{4072, 4038, 3771, 2641, 3849, 2805}"
856,1,138,0,0.5142828,"\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","Int[(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 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)}}","\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*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,44,0.1818,1,"{4072, 4036, 3858, 2663, 2661, 3859, 2807, 2805}"
857,1,229,0,0.2674418,"\int (a+b \sec (c+d x))^{2/3} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(2/3)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}}","\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,"(Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -5/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3))","A",8,5,34,0.1471,1,"{4062, 12, 3834, 139, 138}"
858,1,229,0,0.2469308,"\int \sqrt[3]{a+b \sec (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(1/3)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}}}","\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,"(Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3))","A",8,5,34,0.1471,1,"{4062, 12, 3834, 139, 138}"
859,1,226,0,0.2404612,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(1/3),x]","\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}}","\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,"(Sqrt[2]*C*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3))","A",8,5,34,0.1471,1,"{4062, 12, 3834, 139, 138}"
860,1,226,0,0.2439714,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(2/3),x]","\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}}}","\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,"(Sqrt[2]*C*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3))","A",8,5,34,0.1471,1,"{4062, 12, 3834, 139, 138}"
861,1,165,0,0.2323189,"\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","Int[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{\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}","\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,"((4*a*A + 3*b*B + 3*a*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((5*A*b + 5*a*B + 4*b*C)*Tan[c + d*x])/(5*d) + ((4*a*A + 3*b*B + 3*a*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((b*B + a*C)*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (b*C*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((5*A*b + 5*a*B + 4*b*C)*Tan[c + d*x]^3)/(15*d)","A",7,6,39,0.1538,1,"{4076, 4047, 3767, 4046, 3768, 3770}"
862,1,137,0,0.2043387,"\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","Int[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{\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}","\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,"((4*A*b + 4*a*B + 3*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) + ((3*a*A + 2*b*B + 2*a*C)*Tan[c + d*x])/(3*d) + ((4*A*b + 4*a*B + 3*b*C)*Sec[c + d*x]*Tan[c + d*x])/(8*d) + ((b*B + a*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + (b*C*Sec[c + d*x]^3*Tan[c + d*x])/(4*d)","A",7,7,39,0.1795,1,"{4076, 4047, 3768, 3770, 4046, 3767, 8}"
863,1,101,0,0.1382196,"\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","Int[Sec[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"((b*B + a*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*A*b + 3*a*B + 2*b*C)*Tan[c + d*x])/(3*d) + ((b*B + a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (b*C*Sec[c + d*x]^2*Tan[c + d*x])/(3*d)","A",6,6,37,0.1622,1,"{4076, 4047, 3767, 8, 4046, 3770}"
864,1,69,0,0.0706864,"\int (a+b \sec (c+d x)) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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 + ((2*A*b + 2*a*B + b*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((b*B + a*C)*Tan[c + d*x])/d + (b*C*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",5,4,31,0.1290,1,"{4048, 3770, 3767, 8}"
865,1,52,0,0.1218832,"\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","Int[Cos[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","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}","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 + a*B)*x + ((b*B + a*C)*ArcTanh[Sin[c + d*x]])/d + (a*A*Sin[c + d*x])/d + (b*C*Tan[c + d*x])/d","A",5,5,37,0.1351,1,"{4076, 4047, 8, 4045, 3770}"
866,1,69,0,0.1562199,"\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","Int[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{(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}","\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*b*B + a*(A + 2*C))*x)/2 + (b*C*ArcTanh[Sin[c + d*x]])/d + ((A*b + a*B)*Sin[c + d*x])/d + (a*A*Cos[c + d*x]*Sin[c + d*x])/(2*d)","A",5,5,39,0.1282,1,"{4074, 4047, 8, 4045, 3770}"
867,1,92,0,0.1795621,"\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","Int[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{\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}","\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,"((A*b + a*B + 2*b*C)*x)/2 + ((2*a*A + 3*b*B + 3*a*C)*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (a*A*Cos[c + d*x]^2*Sin[c + d*x])/(3*d)","A",5,5,39,0.1282,1,"{4074, 4047, 2637, 4045, 8}"
868,1,116,0,0.2146841,"\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","Int[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{\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}","\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,"((3*a*A + 4*b*B + 4*a*C)*x)/8 + ((A*b + a*B + b*C)*Sin[c + d*x])/d + ((3*a*A + 4*b*B + 4*a*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*A*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) - ((A*b + a*B)*Sin[c + d*x]^3)/(3*d)","A",7,6,39,0.1538,1,"{4074, 4047, 2635, 8, 4044, 3013}"
869,1,156,0,0.2346847,"\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","Int[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{\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}","-\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,"((3*A*b + 3*a*B + 4*b*C)*x)/8 + ((4*a*A + 5*b*B + 5*a*C)*Sin[c + d*x])/(5*d) + ((3*A*b + 3*a*B + 4*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + ((A*b + a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (a*A*Cos[c + d*x]^4*Sin[c + d*x])/(5*d) - ((4*a*A + 5*b*B + 5*a*C)*Sin[c + d*x]^3)/(15*d)","A",7,6,39,0.1538,1,"{4074, 4047, 2633, 4045, 2635, 8}"
870,1,281,0,0.5915904,"\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","Int[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{\tan (c+d x) \left(-4 a^2 b^2 (5 A+3 C)+5 a^3 b B-2 a^4 C-40 a b^3 B-4 b^4 (5 A+4 C)\right)}{30 b^2 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) \left(2 a^2 C-5 a b B+20 A b^2+16 b^2 C\right) (a+b \sec (c+d x))^2}{60 b^2 d}-\frac{\tan (c+d x) \sec (c+d x) \left(10 a^2 b B-4 a^3 C-2 a b^2 (20 A+13 C)-45 b^3 B\right)}{120 b d}+\frac{(5 b B-2 a C) \tan (c+d x) (a+b \sec (c+d x))^3}{20 b^2 d}+\frac{C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^3}{5 b d}","\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,"((8*a*A*b + 4*a^2*B + 3*b^2*B + 6*a*b*C)*ArcTanh[Sin[c + d*x]])/(8*d) - ((5*a^3*b*B - 40*a*b^3*B - 2*a^4*C - 4*a^2*b^2*(5*A + 3*C) - 4*b^4*(5*A + 4*C))*Tan[c + d*x])/(30*b^2*d) - ((10*a^2*b*B - 45*b^3*B - 4*a^3*C - 2*a*b^2*(20*A + 13*C))*Sec[c + d*x]*Tan[c + d*x])/(120*b*d) + ((20*A*b^2 - 5*a*b*B + 2*a^2*C + 16*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b^2*d) + ((5*b*B - 2*a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(5*b*d)","A",8,8,41,0.1951,1,"{4092, 4082, 4002, 3997, 3787, 3770, 3767, 8}"
871,1,200,0,0.3480077,"\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","Int[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{\tan (c+d x) \left(4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 b d}+\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{(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}","\frac{\tan (c+d x) \left(4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 b d}+\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{(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,"((8*a*b*B + 4*a^2*(2*A + C) + b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((4*a^2*b*B + 4*b^3*B - a^3*C + 4*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*b*d) + ((12*A*b^2 + 8*a*b*B - 2*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B - a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*b*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*b*d)","A",7,7,39,0.1795,1,"{4082, 4002, 3997, 3787, 3770, 3767, 8}"
872,1,134,0,0.1722161,"\int (a+b \sec (c+d x))^2 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"a^2*A*x + ((2*a^2*B + b^2*B + 2*a*b*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*A*b^2 + 6*a*b*B + 2*a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*d) + (b*(3*b*B + 2*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",6,5,33,0.1515,1,"{4056, 4048, 3770, 3767, 8}"
873,1,126,0,0.2026262,"\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","Int[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{\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}","\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,"a*(2*A*b + a*B)*x + ((2*A*b^2 + 4*a*b*B + 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/d - (b*(2*a*A - b*B - 2*a*C)*Tan[c + d*x])/d - (b^2*(2*A - C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",6,5,39,0.1282,1,"{4094, 4048, 3770, 3767, 8}"
874,1,118,0,0.3172194,"\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","Int[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{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}","\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*A*b^2 + 4*a*b*B + a^2*(A + 2*C))*x)/2 + (b*(b*B + 2*a*C)*ArcTanh[Sin[c + d*x]])/d + (a*(A*b + a*B)*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) - (b^2*(A - 2*C)*Tan[c + d*x])/(2*d)","A",6,6,41,0.1463,1,"{4094, 4076, 4047, 8, 4045, 3770}"
875,1,141,0,0.3650227,"\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","Int[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{\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}","\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,"((a^2*B + 2*b^2*B + 2*a*b*(A + 2*C))*x)/2 + (b^2*C*ArcTanh[Sin[c + d*x]])/d + ((2*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + (a*(2*A*b + 3*a*B)*Cos[c + d*x]*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d)","A",6,6,41,0.1463,1,"{4094, 4074, 4047, 8, 4045, 3770}"
876,1,175,0,0.4483595,"\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","Int[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{\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}","\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,"((8*a*b*B + 4*b^2*(A + 2*C) + a^2*(3*A + 4*C))*x)/8 + ((4*a*A*b + 2*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x])/(3*d) + ((2*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(A*b + 2*a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d)","A",6,6,41,0.1463,1,"{4094, 4074, 4047, 2637, 4045, 8}"
877,1,215,0,0.5135123,"\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","Int[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{\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}","-\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,"((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*x)/8 + ((10*a*b*B + a^2*(4*A + 5*C) + b^2*(4*A + 5*C))*Sin[c + d*x])/(5*d) + ((6*a*A*b + 3*a^2*B + 4*b^2*B + 8*a*b*C)*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (a*(2*A*b + 5*a*B)*Cos[c + d*x]^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) - ((2*A*b^2 + 10*a*b*B + a^2*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)","A",8,7,41,0.1707,1,"{4094, 4074, 4047, 2635, 8, 4044, 3013}"
878,1,381,0,0.8695322,"\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","Int[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{\tan (c+d x) \left(-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B+6 a^4 b B-2 a^5 C-24 a b^4 (5 A+4 C)-32 b^5 B\right)}{60 b^2 d}+\frac{\left(6 a^2 b (4 A+3 C)+8 a^3 B+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^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{\tan (c+d x) \left(6 a^2 b B-2 a^3 C-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(-12 a^2 b^2 (5 A+3 C)+12 a^3 b B-4 a^4 C-142 a b^3 B-15 b^4 (6 A+5 C)\right)}{240 b 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}","-\frac{\tan (c+d x) \left(-a^3 b^2 (30 A+17 C)-104 a^2 b^3 B+6 a^4 b B-2 a^5 C-24 a b^4 (5 A+4 C)-32 b^5 B\right)}{60 b^2 d}+\frac{\left(6 a^2 b (4 A+3 C)+8 a^3 B+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^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{\tan (c+d x) \left(6 a^2 b B-2 a^3 C-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(-12 a^2 b^2 (5 A+3 C)+12 a^3 b B-4 a^4 C-142 a b^3 B-15 b^4 (6 A+5 C)\right)}{240 b 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,"((8*a^3*B + 18*a*b^2*B + 6*a^2*b*(4*A + 3*C) + b^3*(6*A + 5*C))*ArcTanh[Sin[c + d*x]])/(16*d) - ((6*a^4*b*B - 104*a^2*b^3*B - 32*b^5*B - 2*a^5*C - 24*a*b^4*(5*A + 4*C) - a^3*b^2*(30*A + 17*C))*Tan[c + d*x])/(60*b^2*d) - ((12*a^3*b*B - 142*a*b^3*B - 4*a^4*C - 12*a^2*b^2*(5*A + 3*C) - 15*b^4*(6*A + 5*C))*Sec[c + d*x]*Tan[c + d*x])/(240*b*d) - ((6*a^2*b*B - 32*b^3*B - 2*a^3*C - 3*a*b^2*(10*A + 7*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b^2*d) + ((30*A*b^2 - 6*a*b*B + 2*a^2*C + 25*b^2*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b^2*d) + ((3*b*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(15*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(6*b*d)","A",9,8,41,0.1951,1,"{4092, 4082, 4002, 3997, 3787, 3770, 3767, 8}"
879,1,286,0,0.5872807,"\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","Int[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{\tan (c+d x) \left(4 a^2 b^2 (20 A+13 C)+15 a^3 b B-3 a^4 C+60 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 b d}+\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(30 a^2 b B-6 a^3 C+a b^2 (100 A+71 C)+45 b^3 B\right)}{120 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}","\frac{\tan (c+d x) \left(4 a^2 b^2 (20 A+13 C)+15 a^3 b B-3 a^4 C+60 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 b d}+\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(30 a^2 b B-6 a^3 C+a b^2 (100 A+71 C)+45 b^3 B\right)}{120 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,"((12*a^2*b*B + 3*b^3*B + 4*a^3*(2*A + C) + 3*a*b^2*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((15*a^3*b*B + 60*a*b^3*B - 3*a^4*C + 4*b^4*(5*A + 4*C) + 4*a^2*b^2*(20*A + 13*C))*Tan[c + d*x])/(30*b*d) + ((30*a^2*b*B + 45*b^3*B - 6*a^3*C + a*b^2*(100*A + 71*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((4*b^2*(5*A + 4*C) + 3*a*(5*b*B - a*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*b*d) + ((5*b*B - a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*b*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*b*d)","A",8,7,39,0.1795,1,"{4082, 4002, 3997, 3787, 3770, 3767, 8}"
880,1,207,0,0.3399173,"\int (a+b \sec (c+d x))^3 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(16 a^2 b B+3 a^3 C+6 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{\left(12 a^2 b (2 A+C)+8 a^3 B+12 a b^2 B+b^3 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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}+a^3 A x+\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}","\frac{\tan (c+d x) \left(16 a^2 b B+3 a^3 C+6 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{\left(12 a^2 b (2 A+C)+8 a^3 B+12 a b^2 B+b^3 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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}+a^3 A x+\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,"a^3*A*x + ((8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + ((16*a^2*b*B + 4*b^3*B + 3*a^3*C + 6*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) + (b*(12*A*b^2 + 20*a*b*B + 6*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + ((4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",7,5,33,0.1515,1,"{4056, 4048, 3770, 3767, 8}"
881,1,192,0,0.3712627,"\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","Int[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{b \tan (c+d x) \left(a^2 (-(6 A-8 C))+9 a b B+b^2 (3 A+2 C)\right)}{3 d}+\frac{\left(6 a^2 b B+2 a^3 C+3 a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 x (a B+3 A b)-\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}","\frac{b \tan (c+d x) \left(a^2 (-(6 A-8 C))+9 a b B+b^2 (3 A+2 C)\right)}{3 d}+\frac{\left(6 a^2 b B+2 a^3 C+3 a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 x (a B+3 A b)-\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,"a^2*(3*A*b + a*B)*x + ((6*a^2*b*B + b^3*B + 2*a^3*C + 3*a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + (A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d + (b*(9*a*b*B - a^2*(6*A - 8*C) + b^2*(3*A + 2*C))*Tan[c + d*x])/(3*d) - (b^2*(6*a*A - 3*b*B - 5*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(3*A - C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",7,6,39,0.1538,1,"{4094, 4056, 4048, 3770, 3767, 8}"
882,1,204,0,0.4556437,"\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","Int[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{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}","-\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,"(a*(6*A*b^2 + 6*a*b*B + a^2*(A + 2*C))*x)/2 + (b*(2*A*b^2 + 6*a*b*B + 6*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((3*A*b + 2*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(2*d) - (b*(9*a*A*b + 4*a^2*B - 2*b^2*B - 6*a*b*C)*Tan[c + d*x])/(2*d) - (b^2*(4*A*b + 2*a*B - b*C)*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",7,5,41,0.1220,1,"{4094, 4048, 3770, 3767, 8}"
883,1,196,0,0.6009098,"\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","Int[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 \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(3 a^2 b (A+2 C)+a^3 B+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}","\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(3 a^2 b (A+2 C)+a^3 B+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,"((2*A*b^3 + a^3*B + 6*a*b^2*B + 3*a^2*b*(A + 2*C))*x)/2 + (b^2*(b*B + 3*a*C)*ArcTanh[Sin[c + d*x]])/d + (a*(3*A*b^2 + 6*a*b*B + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*d) + ((A*b + a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(2*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) - (b^2*(5*A*b + 3*a*B - 6*b*C)*Tan[c + d*x])/(6*d)","A",7,6,41,0.1463,1,"{4094, 4076, 4047, 8, 4045, 3770}"
884,1,223,0,0.6566547,"\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","Int[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{\sin (c+d x) \left(6 a^2 b (2 A+3 C)+4 a^3 B+16 a b^2 B+3 A b^3\right)}{6 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{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}","\frac{\sin (c+d x) \left(6 a^2 b (2 A+3 C)+4 a^3 B+16 a b^2 B+3 A b^3\right)}{6 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{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*a^2*b*B + 8*b^3*B + 12*a*b^2*(A + 2*C) + a^3*(3*A + 4*C))*x)/8 + (b^3*C*ArcTanh[Sin[c + d*x]])/d + ((3*A*b^3 + 4*a^3*B + 16*a*b^2*B + 6*a^2*b*(2*A + 3*C))*Sin[c + d*x])/(6*d) + (a*(6*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(24*d) + ((3*A*b + 4*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(4*d)","A",7,6,41,0.1463,1,"{4094, 4074, 4047, 8, 4045, 3770}"
885,1,269,0,0.7511574,"\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","Int[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{\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{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) \cos (c+d x) \left(15 a^2 b (3 A+4 C)+15 a^3 B+50 a b^2 B+6 A b^3\right)}{40 d}+\frac{1}{8} x \left(3 a^2 b (3 A+4 C)+3 a^3 B+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}","\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{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) \cos (c+d x) \left(15 a^2 b (3 A+4 C)+15 a^3 B+50 a b^2 B+6 A b^3\right)}{40 d}+\frac{1}{8} x \left(3 a^2 b (3 A+4 C)+3 a^3 B+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,"((3*a^3*B + 12*a*b^2*B + 4*b^3*(A + 2*C) + 3*a^2*b*(3*A + 4*C))*x)/8 + ((30*a^2*b*B + 15*b^3*B + 15*a*b^2*(2*A + 3*C) + 2*a^3*(4*A + 5*C))*Sin[c + d*x])/(15*d) + ((6*A*b^3 + 15*a^3*B + 50*a*b^2*B + 15*a^2*b*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(40*d) + (a*(3*A*b^2 + 15*a*b*B + 2*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sin[c + d*x])/(30*d) + ((3*A*b + 5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d)","A",7,6,41,0.1463,1,"{4094, 4074, 4047, 2637, 4045, 8}"
886,1,320,0,0.8951963,"\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","Int[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{\sin ^3(c+d x) \left(3 a^2 b (4 A+5 C)+4 a^3 B+12 a b^2 B+A b^3\right)}{15 d}+\frac{\sin (c+d x) \left(9 a^2 b (4 A+5 C)+12 a^3 B+42 a b^2 B+b^3 (11 A+15 C)\right)}{15 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 (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}","-\frac{\sin ^3(c+d x) \left(3 a^2 b (4 A+5 C)+4 a^3 B+12 a b^2 B+A b^3\right)}{15 d}+\frac{\sin (c+d x) \left(9 a^2 b (4 A+5 C)+12 a^3 B+42 a b^2 B+b^3 (11 A+15 C)\right)}{15 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 (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,"((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*x)/16 + ((12*a^3*B + 42*a*b^2*B + 9*a^2*b*(4*A + 5*C) + b^3*(11*A + 15*C))*Sin[c + d*x])/(15*d) + ((18*a^2*b*B + 8*b^3*B + 6*a*b^2*(3*A + 4*C) + a^3*(5*A + 6*C))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(6*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*Sin[c + d*x])/(120*d) + ((A*b + 2*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(10*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) - ((A*b^3 + 4*a^3*B + 12*a*b^2*B + 3*a^2*b*(4*A + 5*C))*Sin[c + d*x]^3)/(15*d)","A",9,7,41,0.1707,1,"{4094, 4074, 4047, 2635, 8, 4044, 3013}"
887,1,491,0,1.2340427,"\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","Int[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{\tan (c+d x) \left(-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)-847 a^3 b^3 B+28 a^5 b B-8 a^6 C-896 a b^5 B-32 b^6 (7 A+6 C)\right)}{420 b^2 d}+\frac{\left(8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 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(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(28 a^2 b B-8 a^3 C-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{\tan (c+d x) \left(-12 a^2 b^2 (14 A+9 C)+28 a^3 b B-8 a^4 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(-48 a^3 b^2 (7 A+4 C)-1246 a^2 b^3 B+56 a^4 b B-16 a^5 C-4 a b^4 (406 A+333 C)-525 b^5 B\right)}{1680 b 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}","-\frac{\tan (c+d x) \left(-4 a^4 b^2 (42 A+23 C)-32 a^2 b^4 (49 A+39 C)-847 a^3 b^3 B+28 a^5 b B-8 a^6 C-896 a b^5 B-32 b^6 (7 A+6 C)\right)}{420 b^2 d}+\frac{\left(8 a^3 b (4 A+3 C)+36 a^2 b^2 B+8 a^4 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(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(28 a^2 b B-8 a^3 C-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{\tan (c+d x) \left(-12 a^2 b^2 (14 A+9 C)+28 a^3 b B-8 a^4 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(-48 a^3 b^2 (7 A+4 C)-1246 a^2 b^3 B+56 a^4 b B-16 a^5 C-4 a b^4 (406 A+333 C)-525 b^5 B\right)}{1680 b 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,"((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))*ArcTanh[Sin[c + d*x]])/(16*d) - ((28*a^5*b*B - 847*a^3*b^3*B - 896*a*b^5*B - 8*a^6*C - 32*b^6*(7*A + 6*C) - 4*a^4*b^2*(42*A + 23*C) - 32*a^2*b^4*(49*A + 39*C))*Tan[c + d*x])/(420*b^2*d) - ((56*a^4*b*B - 1246*a^2*b^3*B - 525*b^5*B - 16*a^5*C - 48*a^3*b^2*(7*A + 4*C) - 4*a*b^4*(406*A + 333*C))*Sec[c + d*x]*Tan[c + d*x])/(1680*b*d) - ((28*a^3*b*B - 371*a*b^3*B - 8*a^4*C - 32*b^4*(7*A + 6*C) - 12*a^2*b^2*(14*A + 9*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(840*b^2*d) - ((28*a^2*b*B - 175*b^3*B - 8*a^3*C - 4*a*b^2*(42*A + 31*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(840*b^2*d) + ((42*A*b^2 - 7*a*b*B + 2*a^2*C + 36*b^2*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(210*b^2*d) + ((7*b*B - 2*a*C)*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(42*b^2*d) + (C*Sec[c + d*x]*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(7*b*d)","A",10,8,41,0.1951,1,"{4092, 4082, 4002, 3997, 3787, 3770, 3767, 8}"
888,1,384,0,0.8757085,"\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","Int[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{\tan (c+d x) \left(a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+24 a^4 b B-4 a^5 C+32 a b^4 (5 A+4 C)+32 b^5 B\right)}{60 b d}+\frac{\left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+32 a^3 b B+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) \left(24 a^2 b B-4 a^3 C+a b^2 (70 A+53 C)+32 b^3 B\right) (a+b \sec (c+d x))^2}{120 b d}+\frac{\tan (c+d x) \sec (c+d x) \left(2 a^2 b^2 (130 A+89 C)+48 a^3 b B-8 a^4 C+232 a b^3 B+15 b^4 (6 A+5 C)\right)}{240 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}","\frac{\tan (c+d x) \left(a^3 b^2 (190 A+121 C)+224 a^2 b^3 B+24 a^4 b B-4 a^5 C+32 a b^4 (5 A+4 C)+32 b^5 B\right)}{60 b d}+\frac{\left(12 a^2 b^2 (4 A+3 C)+8 a^4 (2 A+C)+32 a^3 b B+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) \left(24 a^2 b B-4 a^3 C+a b^2 (70 A+53 C)+32 b^3 B\right) (a+b \sec (c+d x))^2}{120 b d}+\frac{\tan (c+d x) \sec (c+d x) \left(2 a^2 b^2 (130 A+89 C)+48 a^3 b B-8 a^4 C+232 a b^3 B+15 b^4 (6 A+5 C)\right)}{240 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,"((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))*ArcTanh[Sin[c + d*x]])/(16*d) + ((24*a^4*b*B + 224*a^2*b^3*B + 32*b^5*B - 4*a^5*C + 32*a*b^4*(5*A + 4*C) + a^3*b^2*(190*A + 121*C))*Tan[c + d*x])/(60*b*d) + ((48*a^3*b*B + 232*a*b^3*B - 8*a^4*C + 15*b^4*(6*A + 5*C) + 2*a^2*b^2*(130*A + 89*C))*Sec[c + d*x]*Tan[c + d*x])/(240*d) + ((24*a^2*b*B + 32*b^3*B - 4*a^3*C + a*b^2*(70*A + 53*C))*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(120*b*d) + ((5*b^2*(6*A + 5*C) + 4*a*(6*b*B - a*C))*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(120*b*d) + ((6*b*B - a*C)*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(30*b*d) + (C*(a + b*Sec[c + d*x])^5*Tan[c + d*x])/(6*b*d)","A",9,7,39,0.1795,1,"{4082, 4002, 3997, 3787, 3770, 3767, 8}"
889,1,290,0,0.542844,"\int (a+b \sec (c+d x))^4 \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\frac{\tan (c+d x) \left(2 a^2 b^2 (85 A+56 C)+95 a^3 b B+12 a^4 C+80 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 d}+\frac{\left(16 a^3 b (2 A+C)+24 a^2 b^2 B+8 a^4 B+4 a b^3 (4 A+3 C)+3 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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(130 a^2 b B+24 a^3 C+4 a b^2 (40 A+29 C)+45 b^3 B\right)}{120 d}+a^4 A x+\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}","\frac{\tan (c+d x) \left(2 a^2 b^2 (85 A+56 C)+95 a^3 b B+12 a^4 C+80 a b^3 B+4 b^4 (5 A+4 C)\right)}{30 d}+\frac{\left(16 a^3 b (2 A+C)+24 a^2 b^2 B+8 a^4 B+4 a b^3 (4 A+3 C)+3 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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(130 a^2 b B+24 a^3 C+4 a b^2 (40 A+29 C)+45 b^3 B\right)}{120 d}+a^4 A x+\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,"a^4*A*x + ((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))*ArcTanh[Sin[c + d*x]])/(8*d) + ((95*a^3*b*B + 80*a*b^3*B + 12*a^4*C + 4*b^4*(5*A + 4*C) + 2*a^2*b^2*(85*A + 56*C))*Tan[c + d*x])/(30*d) + (b*(130*a^2*b*B + 45*b^3*B + 24*a^3*C + 4*a*b^2*(40*A + 29*C))*Sec[c + d*x]*Tan[c + d*x])/(120*d) + ((20*A*b^2 + 35*a*b*B + 12*a^2*C + 16*b^2*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(60*d) + ((5*b*B + 4*a*C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(20*d) + (C*(a + b*Sec[c + d*x])^4*Tan[c + d*x])/(5*d)","A",8,5,33,0.1515,1,"{4056, 4048, 3770, 3767, 8}"
890,1,273,0,0.5826438,"\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","Int[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{b \tan (c+d x) \left(a^3 (-(12 A-19 C))+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{\left(24 a^2 b^2 (2 A+C)+32 a^3 b B+8 a^4 C+16 a b^3 B+b^4 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \tan (c+d x) \sec (c+d x) \left(a^2 (-(24 A-26 C))+32 a b B+3 b^2 (4 A+3 C)\right)}{24 d}+a^3 x (a B+4 A b)-\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}","\frac{b \tan (c+d x) \left(a^3 (-(12 A-19 C))+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right)}{6 d}+\frac{\left(24 a^2 b^2 (2 A+C)+32 a^3 b B+8 a^4 C+16 a b^3 B+b^4 (4 A+3 C)\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b^2 \tan (c+d x) \sec (c+d x) \left(a^2 (-(24 A-26 C))+32 a b B+3 b^2 (4 A+3 C)\right)}{24 d}+a^3 x (a B+4 A b)-\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,"a^3*(4*A*b + a*B)*x + ((32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4*A + 3*C))*ArcTanh[Sin[c + d*x]])/(8*d) + (A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/d + (b*(34*a^2*b*B + 4*b^3*B - a^3*(12*A - 19*C) + 8*a*b^2*(3*A + 2*C))*Tan[c + d*x])/(6*d) + (b^2*(32*a*b*B - a^2*(24*A - 26*C) + 3*b^2*(4*A + 3*C))*Sec[c + d*x]*Tan[c + d*x])/(24*d) - (b*(12*a*A - 4*b*B - 7*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) - (b*(4*A - C)*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",8,6,39,0.1538,1,"{4094, 4056, 4048, 3770, 3767, 8}"
891,1,274,0,0.6773952,"\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","Int[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{b \tan (c+d x) \left(a^2 b (39 A-34 C)+12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)\right)}{6 d}+\frac{b \left(12 a^2 b B+8 a^3 C+4 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) \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) (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}","-\frac{b \tan (c+d x) \left(a^2 b (39 A-34 C)+12 a^3 B-24 a b^2 B-2 b^3 (3 A+2 C)\right)}{6 d}+\frac{b \left(12 a^2 b B+8 a^3 C+4 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) \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) (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,"(a^2*(12*A*b^2 + 8*a*b*B + a^2*(A + 2*C))*x)/2 + (b*(12*a^2*b*B + b^3*B + 8*a^3*C + 4*a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*d) + ((2*A*b + a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/d + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(2*d) - (b*(12*a^3*B - 24*a*b^2*B + a^2*b*(39*A - 34*C) - 2*b^3*(3*A + 2*C))*Tan[c + d*x])/(6*d) - (b^2*(6*a^2*B - 3*b^2*B + 2*a*b*(9*A - 4*C))*Sec[c + d*x]*Tan[c + d*x])/(6*d) - (b*(15*A*b + 6*a*B - 2*b*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(6*d)","A",8,6,41,0.1463,1,"{4094, 4056, 4048, 3770, 3767, 8}"
892,1,303,0,0.8500536,"\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","Int[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{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{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{1}{2} a x \left(4 a^2 b (A+2 C)+a^3 B+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}","-\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{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{1}{2} a x \left(4 a^2 b (A+2 C)+a^3 B+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,"(a*(8*A*b^3 + a^3*B + 12*a*b^2*B + 4*a^2*b*(A + 2*C))*x)/2 + (b^2*(2*A*b^2 + 8*a*b*B + 12*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + ((12*A*b^2 + 15*a*b*B + a^2*(4*A + 6*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(6*d) + ((4*A*b + 3*a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(3*d) - (b*(39*a^2*b*B - 6*b^3*B + 4*a*b^2*(11*A - 6*C) + 4*a^3*(2*A + 3*C))*Tan[c + d*x])/(6*d) - (b^2*(18*a*b*B + 3*b^2*(6*A - C) + a^2*(4*A + 6*C))*Sec[c + d*x]*Tan[c + d*x])/(6*d)","A",8,5,41,0.1220,1,"{4094, 4048, 3770, 3767, 8}"
893,1,293,0,0.9724296,"\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","Int[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 \sin (c+d x) \left(a^2 b (23 A+36 C)+8 a^3 B+36 a b^2 B+12 A b^3\right)}{12 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{1}{8} x \left(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+16 a^3 b B+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}","\frac{a \sin (c+d x) \left(a^2 b (23 A+36 C)+8 a^3 B+36 a b^2 B+12 A b^3\right)}{12 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{1}{8} x \left(24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)+16 a^3 b B+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,"((8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B + 24*a^2*b^2*(A + 2*C) + a^4*(3*A + 4*C))*x)/8 + (b^3*(b*B + 4*a*C)*ArcTanh[Sin[c + d*x]])/d + (a*(12*A*b^3 + 8*a^3*B + 36*a*b^2*B + a^2*b*(23*A + 36*C))*Sin[c + d*x])/(12*d) + ((4*A*b^2 + 8*a*b*B + a^2*(3*A + 4*C))*Cos[c + d*x]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(8*d) + ((A*b + a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(4*d) - (b^2*(32*a*b*B + 2*b^2*(13*A - 12*C) + 3*a^2*(3*A + 4*C))*Tan[c + d*x])/(24*d)","A",8,6,41,0.1463,1,"{4094, 4076, 4047, 8, 4045, 3770}"
894,1,314,0,1.0487092,"\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","Int[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{\sin (c+d x) \left(2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right)}{30 d}+\frac{a \sin (c+d x) \cos (c+d x) \left(4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right)}{120 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{1}{8} x \left(4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 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}","\frac{\sin (c+d x) \left(2 a^2 b^2 (56 A+85 C)+4 a^4 (4 A+5 C)+80 a^3 b B+95 a b^3 B+12 A b^4\right)}{30 d}+\frac{a \sin (c+d x) \cos (c+d x) \left(4 a^2 b (29 A+40 C)+45 a^3 B+130 a b^2 B+24 A b^3\right)}{120 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{1}{8} x \left(4 a^3 b (3 A+4 C)+24 a^2 b^2 B+3 a^4 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,"((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*x)/8 + (b^4*C*ArcTanh[Sin[c + d*x]])/d + ((12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4*A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Sin[c + d*x])/(30*d) + (a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(29*A + 40*C))*Cos[c + d*x]*Sin[c + d*x])/(120*d) + ((12*A*b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(60*d) + ((4*A*b + 5*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(20*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(5*d)","A",8,6,41,0.1463,1,"{4094, 4074, 4047, 8, 4045, 3770}"
895,1,372,0,1.2083973,"\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","Int[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{\sin (c+d x) \left(8 a^3 b (4 A+5 C)+60 a^2 b^2 B+8 a^4 B+20 a b^3 (2 A+3 C)+15 b^4 B\right)}{15 d}+\frac{a \sin (c+d x) \cos ^2(c+d x) \left(a^2 b (39 A+50 C)+16 a^3 B+36 a b^2 B+4 A b^3\right)}{60 d}+\frac{\sin (c+d x) \cos (c+d x) \left(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+360 a^3 b B+336 a b^3 B+24 A b^4\right)}{240 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{1}{16} x \left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+24 a^3 b B+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}","\frac{\sin (c+d x) \left(8 a^3 b (4 A+5 C)+60 a^2 b^2 B+8 a^4 B+20 a b^3 (2 A+3 C)+15 b^4 B\right)}{15 d}+\frac{a \sin (c+d x) \cos ^2(c+d x) \left(a^2 b (39 A+50 C)+16 a^3 B+36 a b^2 B+4 A b^3\right)}{60 d}+\frac{\sin (c+d x) \cos (c+d x) \left(10 a^2 b^2 (49 A+66 C)+15 a^4 (5 A+6 C)+360 a^3 b B+336 a b^3 B+24 A b^4\right)}{240 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{1}{16} x \left(12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)+24 a^3 b B+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,"((24*a^3*b*B + 32*a*b^3*B + 8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*x)/16 + ((8*a^4*B + 60*a^2*b^2*B + 15*b^4*B + 20*a*b^3*(2*A + 3*C) + 8*a^3*b*(4*A + 5*C))*Sin[c + d*x])/(15*d) + ((24*A*b^4 + 360*a^3*b*B + 336*a*b^3*B + 15*a^4*(5*A + 6*C) + 10*a^2*b^2*(49*A + 66*C))*Cos[c + d*x]*Sin[c + d*x])/(240*d) + (a*(4*A*b^3 + 16*a^3*B + 36*a*b^2*B + a^2*b*(39*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/(60*d) + ((12*A*b^2 + 48*a*b*B + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(120*d) + ((2*A*b + 3*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(15*d) + (A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(6*d)","A",8,6,41,0.1463,1,"{4094, 4074, 4047, 2637, 4045, 8}"
896,1,438,0,1.3842658,"\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","Int[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{\sin ^3(c+d x) \left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right)}{105 d}+\frac{\sin (c+d x) \left(3 a^2 b^2 (162 A+203 C)+12 a^4 (6 A+7 C)+336 a^3 b B+371 a b^3 B+b^4 (74 A+105 C)\right)}{105 d}+\frac{a \sin (c+d x) \cos ^3(c+d x) \left(a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right)}{840 d}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)}{16 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{1}{16} x \left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 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}","-\frac{\sin ^3(c+d x) \left(3 a^2 b^2 (50 A+63 C)+4 a^4 (6 A+7 C)+112 a^3 b B+91 a b^3 B+4 A b^4\right)}{105 d}+\frac{\sin (c+d x) \left(3 a^2 b^2 (162 A+203 C)+12 a^4 (6 A+7 C)+336 a^3 b B+371 a b^3 B+b^4 (74 A+105 C)\right)}{105 d}+\frac{a \sin (c+d x) \cos ^3(c+d x) \left(a^2 (412 A b+504 b C)+175 a^3 B+336 a b^2 B+24 A b^3\right)}{840 d}+\frac{\sin (c+d x) \cos (c+d x) \left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 B+8 a b^3 (3 A+4 C)+8 b^4 B\right)}{16 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{1}{16} x \left(4 a^3 b (5 A+6 C)+36 a^2 b^2 B+5 a^4 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,"((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))*x)/16 + ((336*a^3*b*B + 371*a*b^3*B + 12*a^4*(6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^2*(162*A + 203*C))*Sin[c + d*x])/(105*d) + ((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))*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (a*(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412*A*b + 504*b*C))*Cos[c + d*x]^3*Sin[c + d*x])/(840*d) + ((4*A*b^2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(70*d) + ((4*A*b + 7*a*B)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(42*d) + (A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) - ((4*A*b^4 + 112*a^3*b*B + 91*a*b^3*B + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C))*Sin[c + d*x]^3)/(105*d)","A",10,7,41,0.1707,1,"{4094, 4074, 4047, 2635, 8, 4044, 3013}"
897,1,214,0,0.4760743,"\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","Int[(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{b^2 \left(34 a^2 b B-15 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 d}+\frac{b \left(8 a^2 b^2 C+32 a^3 b B-24 a^4 C+16 a b^3 B+3 b^4 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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}+a^4 x (b B-a C)+\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}","\frac{b^2 \left(34 a^2 b B-15 a^3 C+12 a b^2 C+4 b^3 B\right) \tan (c+d x)}{6 d}+\frac{b \left(8 a^2 b^2 C+32 a^3 b B-24 a^4 C+16 a b^3 B+3 b^4 C\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\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}+a^4 x (b B-a C)+\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,"a^4*(b*B - a*C)*x + (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]])/(8*d) + (b^2*(34*a^2*b*B + 4*b^3*B - 15*a^3*C + 12*a*b^2*C)*Tan[c + d*x])/(6*d) + (b^3*(32*a*b*B - 6*a^2*C + 9*b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(24*d) + (b^2*(4*b*B + 3*a*C)*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(12*d) + (b^2*C*(a + b*Sec[c + d*x])^3*Tan[c + d*x])/(4*d)","A",8,7,48,0.1458,1,"{4041, 3918, 4056, 4048, 3770, 3767, 8}"
898,1,149,0,0.308815,"\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","Int[(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{b^2 \left(a^2 (-C)+9 a b B+2 b^2 C\right) \tan (c+d x)}{3 d}+\frac{b \left(6 a^2 b B-4 a^3 C+2 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 x (b B-a C)+\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}","\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(6 a^2 b B-4 a^3 C+2 a b^2 C+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^3 x (b B-a C)+\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,"a^3*(b*B - a*C)*x + (b*(6*a^2*b*B + b^3*B - 4*a^3*C + 2*a*b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b^2*(9*a*b*B - a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*d) + (b^3*(3*b*B + 2*a*C)*Sec[c + d*x]*Tan[c + d*x])/(6*d) + (b^2*C*(a + b*Sec[c + d*x])^2*Tan[c + d*x])/(3*d)","A",7,6,48,0.1250,1,"{4041, 3918, 4048, 3770, 3767, 8}"
899,1,97,0,0.1687899,"\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","Int[(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 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}","\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,"a^2*(b*B - a*C)*x + (b*(4*a*b*B - 2*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*d) + (b^2*(2*b*B + a*C)*Tan[c + d*x])/(2*d) + (b^2*C*(a + b*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",6,5,46,0.1087,1,"{4041, 3918, 3770, 3767, 8}"
900,1,215,0,0.7408642,"\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","Int[(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{\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^2 b B-2 a^3 C-a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\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{(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}","\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^2 b B-2 a^3 C-a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\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{(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^2*b*B + b^3*B - 2*a^3*C - a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*b^4*d) + (2*a^2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^4*Sqrt[a + b]*d) + ((3*A*b^2 - 3*a*b*B + 3*a^2*C + 2*b^2*C)*Tan[c + d*x])/(3*b^3*d) + ((b*B - a*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*d) + (C*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*d)","A",8,8,41,0.1951,1,"{4102, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
901,1,153,0,0.4538707,"\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","Int[(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{\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}","\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,"((b^2*(2*A + C) - 2*a*(b*B - a*C))*ArcTanh[Sin[c + d*x]])/(2*b^3*d) - (2*a*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^3*Sqrt[a + b]*d) + ((b*B - a*C)*Tan[c + d*x])/(b^2*d) + (C*Sec[c + d*x]*Tan[c + d*x])/(2*b*d)","A",7,7,41,0.1707,1,"{4092, 4082, 3998, 3770, 3831, 2659, 208}"
902,1,106,0,0.2237639,"\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","Int[(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 \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}","\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,"((b*B - a*C)*ArcTanh[Sin[c + d*x]])/(b^2*d) + (2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*b^2*Sqrt[a + b]*d) + (C*Tan[c + d*x])/(b*d)","A",6,6,39,0.1538,1,"{4082, 3998, 3770, 3831, 2659, 208}"
903,1,94,0,0.172202,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{a+b \sec (c+d x)} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x]),x]","-\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}","-\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,"(A*x)/a + (C*ArcTanh[Sin[c + d*x]])/(b*d) - (2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*b*Sqrt[a + b]*d)","A",6,6,33,0.1818,1,"{4050, 3770, 3919, 3831, 2659, 208}"
904,1,98,0,0.2163965,"\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","Int[(Cos[c + d*x]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}","\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)*x)/a^2) + (2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*Sqrt[a - b]*Sqrt[a + b]*d) + (A*Sin[c + d*x])/(a*d)","A",5,5,39,0.1282,1,"{4104, 3919, 3831, 2659, 208}"
905,1,145,0,0.4522217,"\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","Int[(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 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{x \left(a^2 (A+2 C)-2 a b B+2 A b^2\right)}{2 a^3}-\frac{(A b-a B) \sin (c+d x)}{a^2 d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a 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{x \left(a^2 (A+2 C)-2 a b B+2 A b^2\right)}{2 a^3}-\frac{(A b-a B) \sin (c+d x)}{a^2 d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d}",1,"((2*A*b^2 - 2*a*b*B + a^2*(A + 2*C))*x)/(2*a^3) - (2*b*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*Sqrt[a - b]*Sqrt[a + b]*d) - ((A*b - a*B)*Sin[c + d*x])/(a^2*d) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d)","A",6,5,41,0.1220,1,"{4104, 3919, 3831, 2659, 208}"
906,1,205,0,0.7400606,"\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","Int[(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{\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{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{x \left(a^2 b (A+2 C)+a^3 (-B)-2 a b^2 B+2 A b^3\right)}{2 a^4}-\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a 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{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{x \left(a^2 b (A+2 C)+a^3 (-B)-2 a b^2 B+2 A b^3\right)}{2 a^4}-\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"-((2*A*b^3 - a^3*B - 2*a*b^2*B + a^2*b*(A + 2*C))*x)/(2*a^4) + (2*b^2*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*Sqrt[a - b]*Sqrt[a + b]*d) + ((3*A*b^2 - 3*a*b*B + a^2*(2*A + 3*C))*Sin[c + d*x])/(3*a^3*d) - ((A*b - a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*d) + (A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d)","A",7,5,41,0.1220,1,"{4104, 3919, 3831, 2659, 208}"
907,1,276,0,1.1038367,"\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","Int[(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{\sin (c+d x) \left(a^2 b (2 A+3 C)-2 a^3 B-3 a b^2 B+3 A b^3\right)}{3 a^4 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{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{x \left(4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)-4 a^3 b B-8 a b^3 B+8 A b^4\right)}{8 a^5}-\frac{(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}","-\frac{\sin (c+d x) \left(a^2 b (2 A+3 C)-2 a^3 B-3 a b^2 B+3 A b^3\right)}{3 a^4 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{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{x \left(4 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)-4 a^3 b B-8 a b^3 B+8 A b^4\right)}{8 a^5}-\frac{(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"((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))*x)/(8*a^5) - (2*b^3*(A*b^2 - a*(b*B - a*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*Sqrt[a - b]*Sqrt[a + b]*d) - ((3*A*b^3 - 2*a^3*B - 3*a*b^2*B + a^2*b*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*d) + ((4*A*b^2 - 4*a*b*B + a^2*(3*A + 4*C))*Cos[c + d*x]*Sin[c + d*x])/(8*a^3*d) - ((A*b - a*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*d) + (A*Cos[c + d*x]^3*Sin[c + d*x])/(4*a*d)","A",8,5,41,0.1220,1,"{4104, 3919, 3831, 2659, 208}"
908,1,407,0,1.7395234,"\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","Int[(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{\tan (c+d x) \left(-a^2 b^2 (6 A-7 C)+9 a^3 b B-12 a^4 C-6 a b^3 B+b^4 (3 A+2 C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\left(6 a^2 b B-8 a^3 C-2 a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{2 a^2 \left(2 a^2 A b^2-5 a^2 b^2 C-3 a^3 b B+4 a^4 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}}-\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(3 a^2 b B-4 a^3 C-2 a b^2 (A-C)-b^3 B\right)}{2 b^3 d \left(a^2-b^2\right)}","-\frac{\tan (c+d x) \left(-a^2 b^2 (6 A-7 C)+9 a^3 b B-12 a^4 C-6 a b^3 B+b^4 (3 A+2 C)\right)}{3 b^4 d \left(a^2-b^2\right)}+\frac{\left(6 a^2 b B-8 a^3 C-2 a b^2 (2 A+C)+b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{2 a^2 \left(2 a^2 A b^2-5 a^2 b^2 C-3 a^3 b B+4 a^4 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}}-\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(3 a^2 b B-4 a^3 C-2 a b^2 (A-C)-b^3 B\right)}{2 b^3 d \left(a^2-b^2\right)}",1,"((6*a^2*b*B + b^3*B - 8*a^3*C - 2*a*b^2*(2*A + C))*ArcTanh[Sin[c + d*x]])/(2*b^5*d) + (2*a^2*(2*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 4*a*b^3*B + 4*a^4*C - 5*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^5*(a + b)^(3/2)*d) - ((9*a^3*b*B - 6*a*b^3*B - a^2*b^2*(6*A - 7*C) - 12*a^4*C + b^4*(3*A + 2*C))*Tan[c + d*x])/(3*b^4*(a^2 - b^2)*d) + ((3*a^2*b*B - b^3*B - 2*a*b^2*(A - C) - 4*a^3*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 4*a^2*C - b^2*C)*Sec[c + d*x]^2*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,9,41,0.2195,1,"{4098, 4102, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
909,1,312,0,1.2333999,"\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","Int[(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{\tan (c+d x) \left(2 a^2 b B-3 a^3 C-a b^2 (A-2 C)-b^3 B\right)}{b^3 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{2 a \left(a^2 A b^2-4 a^2 b^2 C-2 a^3 b B+3 a^4 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}}-\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{\tan (c+d x) \left(2 a^2 b B-3 a^3 C-a b^2 (A-2 C)-b^3 B\right)}{b^3 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{2 a \left(a^2 A b^2-4 a^2 b^2 C-2 a^3 b B+3 a^4 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}}-\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)}",1,"((2*A*b^2 - 4*a*b*B + 6*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^4*d) - (2*a*(a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 4*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^4*(a + b)^(3/2)*d) + ((2*a^2*b*B - b^3*B - a*b^2*(A - 2*C) - 3*a^3*C)*Tan[c + d*x])/(b^3*(a^2 - b^2)*d) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,8,41,0.1951,1,"{4098, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
910,1,177,0,0.6474302,"\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","Int[(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 \left(3 a^2 b^2 C+a^3 b B-2 a^4 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{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{(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}","-\frac{2 \left(3 a^2 b^2 C+a^3 b B-2 a^4 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{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{(b B-2 a C) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{C \tan (c+d x)}{b^2 d}",1,"((b*B - 2*a*C)*ArcTanh[Sin[c + d*x]])/(b^3*d) - (2*(A*b^4 + a^3*b*B - 2*a*b^3*B - 2*a^4*C + 3*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^3*(a + b)^(3/2)*d) + (C*Tan[c + d*x])/(b^2*d) + (a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,7,41,0.1707,1,"{4090, 4082, 3998, 3770, 3831, 2659, 208}"
911,1,148,0,0.3037449,"\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","Int[(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 \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}","\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,"(C*ArcTanh[Sin[c + d*x]])/(b^2*d) + (2*(a*A*b^2 - b^3*B - a^3*C + 2*a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(3/2)*b^2*(a + b)^(3/2)*d) - ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,39,0.1538,1,"{4080, 3998, 3770, 3831, 2659, 208}"
912,1,138,0,0.2523923,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^2} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^2,x]","-\frac{2 \left(2 a^2 A b+a^2 b C+a^3 (-B)-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}}+\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(2 a^2 A b+a^2 b C+a^3 (-B)-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}}+\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}",1,"(A*x)/a^2 - (2*(2*a^2*A*b - A*b^3 - a^3*B + a^2*b*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",5,5,33,0.1515,1,"{4060, 3919, 3831, 2659, 208}"
913,1,202,0,0.6441869,"\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","Int[(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{\sin (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{2 \left(3 a^2 A b^2-2 a^3 b B+a^4 C+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}}+\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{x (2 A b-a B)}{a^3}","-\frac{\sin (c+d x) \left(a^2 (-(A-C))-a b B+2 A b^2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{2 \left(3 a^2 A b^2-2 a^3 b B+a^4 C+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}}+\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{x (2 A b-a B)}{a^3}",1,"-(((2*A*b - a*B)*x)/a^3) + (2*(3*a^2*A*b^2 - 2*A*b^4 - 2*a^3*b*B + a*b^3*B + a^4*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((2*A*b^2 - a*b*B - a^2*(A - C))*Sin[c + d*x])/(a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,39,0.1538,1,"{4100, 4104, 3919, 3831, 2659, 208}"
914,1,298,0,1.2351347,"\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","Int[(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{\sin (c+d x) \left(-a^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \cos (c+d x) \left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right)}{2 a^2 d \left(a^2-b^2\right)}-\frac{2 b \left(4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 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}}+\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^2 b (2 A-C)+a^3 B-2 a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \cos (c+d x) \left(a^2 (-(A-2 C))-2 a b B+3 A b^2\right)}{2 a^2 d \left(a^2-b^2\right)}-\frac{2 b \left(4 a^2 A b^2-a^2 b^2 C-3 a^3 b B+2 a^4 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}}+\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}",1,"((6*A*b^2 - 4*a*b*B + a^2*(A + 2*C))*x)/(2*a^4) - (2*b*(4*a^2*A*b^2 - 3*A*b^4 - 3*a^3*b*B + 2*a*b^3*B + 2*a^4*C - a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(3/2)*(a + b)^(3/2)*d) + ((3*A*b^3 + a^3*B - 2*a*b^2*B - a^2*b*(2*A - C))*Sin[c + d*x])/(a^3*(a^2 - b^2)*d) - ((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",7,6,41,0.1463,1,"{4100, 4104, 3919, 3831, 2659, 208}"
915,1,396,0,1.7588517,"\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","Int[(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{\sin (c+d x) \left(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+6 a^3 b B-9 a b^3 B+12 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \cos ^2(c+d x) \left(a^2 (-(A-3 C))-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 (c+d x) \left(-2 a^2 b (A-C)+a^3 B-3 a b^2 B+4 A b^3\right)}{2 a^3 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(5 a^2 A b^2-2 a^2 b^2 C-4 a^3 b B+3 a^4 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}}+\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{x \left(2 a^2 b (A+2 C)+a^3 (-B)-6 a b^2 B+8 A b^3\right)}{2 a^5}","-\frac{\sin (c+d x) \left(-a^2 b^2 (7 A-6 C)+a^4 (-(2 A+3 C))+6 a^3 b B-9 a b^3 B+12 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}-\frac{\sin (c+d x) \cos ^2(c+d x) \left(a^2 (-(A-3 C))-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 (c+d x) \left(-2 a^2 b (A-C)+a^3 B-3 a b^2 B+4 A b^3\right)}{2 a^3 d \left(a^2-b^2\right)}+\frac{2 b^2 \left(5 a^2 A b^2-2 a^2 b^2 C-4 a^3 b B+3 a^4 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}}+\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{x \left(2 a^2 b (A+2 C)+a^3 (-B)-6 a b^2 B+8 A b^3\right)}{2 a^5}",1,"-((8*A*b^3 - a^3*B - 6*a*b^2*B + 2*a^2*b*(A + 2*C))*x)/(2*a^5) + (2*b^2*(5*a^2*A*b^2 - 4*A*b^4 - 4*a^3*b*B + 3*a*b^3*B + 3*a^4*C - 2*a^2*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(3/2)*(a + b)^(3/2)*d) - ((12*A*b^4 + 6*a^3*b*B - 9*a*b^3*B - a^2*b^2*(7*A - 6*C) - a^4*(2*A + 3*C))*Sin[c + d*x])/(3*a^4*(a^2 - b^2)*d) + ((4*A*b^3 + a^3*B - 3*a*b^2*B - 2*a^2*b*(A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*d) - ((4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Cos[c + d*x]^2*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^2*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",8,6,41,0.1463,1,"{4100, 4104, 3919, 3831, 2659, 208}"
916,1,465,0,4.7450231,"\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","Int[(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{\tan (c+d x) \left(-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+6 a^4 b B-12 a^5 C+a b^4 (5 A-6 C)+2 b^5 B\right)}{2 b^4 d \left(a^2-b^2\right)^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{a \left(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+15 a^3 b^3 B-6 a^5 b B+12 a^6 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}}-\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{\tan (c+d x) \sec ^2(c+d x) \left(a \left(2 a^2 b B-4 a^3 C+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(-a^2 b^2 (A-10 C)+3 a^3 b B-6 a^4 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(-a^3 b^2 (2 A-21 C)-11 a^2 b^3 B+6 a^4 b B-12 a^5 C+a b^4 (5 A-6 C)+2 b^5 B\right)}{2 b^4 d \left(a^2-b^2\right)^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{a \left(a^4 b^2 (2 A-29 C)-5 a^2 b^4 (A-4 C)+15 a^3 b^3 B-6 a^5 b B+12 a^6 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}}-\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{\tan (c+d x) \sec ^2(c+d x) \left(a \left(2 a^2 b B-4 a^3 C+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(-a^2 b^2 (A-10 C)+3 a^3 b B-6 a^4 C-6 a b^3 B+b^4 (4 A-C)\right)}{2 b^3 d \left(a^2-b^2\right)^2}",1,"((2*A*b^2 - 6*a*b*B + 12*a^2*C + b^2*C)*ArcTanh[Sin[c + d*x]])/(2*b^5*d) - (a*(6*A*b^6 - 6*a^5*b*B + 15*a^3*b^3*B - 12*a*b^5*B + a^4*b^2*(2*A - 29*C) - 5*a^2*b^4*(A - 4*C) + 12*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^5*(a + b)^(5/2)*d) + ((6*a^4*b*B - 11*a^2*b^3*B + 2*b^5*B - a^3*b^2*(2*A - 21*C) + a*b^4*(5*A - 6*C) - 12*a^5*C)*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^2*d) - ((3*a^3*b*B - 6*a*b^3*B - a^2*b^2*(A - 10*C) + b^4*(4*A - C) - 6*a^4*C)*Sec[c + d*x]*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((3*A*b^4 + a*(2*a^2*b*B - 5*b^3*B - 4*a^3*C + 7*a*b^2*C))*Sec[c + d*x]^2*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",9,8,41,0.1951,1,"{4098, 4092, 4082, 3998, 3770, 3831, 2659, 208}"
917,1,323,0,2.986916,"\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","Int[(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{\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{\left(a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 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{\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{a \tan (c+d x) \left(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 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{(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}","\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{\left(a^2 b^4 (A+12 C)+5 a^3 b^3 B-15 a^4 b^2 C-2 a^5 b B+6 a^6 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{\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{a \tan (c+d x) \left(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 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{(b B-3 a C) \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"((b*B - 3*a*C)*ArcTanh[Sin[c + d*x]])/(b^4*d) + ((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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^4*(a + b)^(5/2)*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Tan[c + d*x])/(2*b^3*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - (a*(2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,8,41,0.1951,1,"{4098, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
918,1,242,0,0.9180528,"\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","Int[(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{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-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(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 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{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{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}","\frac{\left(a^2 b^3 B+5 a^3 b^2 C-2 a^5 C-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(a^2 b^2 (A+6 C)+a^3 b B-3 a^4 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{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{C \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"(C*ArcTanh[Sin[c + d*x]])/(b^3*d) + ((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))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*b^3*(a + b)^(5/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((2*A*b^4 + a^3*b*B - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 6*C))*Tan[c + d*x])/(2*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,41,0.1707,1,"{4090, 4080, 3998, 3770, 3831, 2659, 208}"
919,1,202,0,0.4213512,"\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","Int[(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{\left(a^2 (-(2 A+C))+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^2 b B+a^3 C-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))}-\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{\left(a^2 (-(2 A+C))+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^2 b B+a^3 C-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))}-\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}",1,"-(((3*a*b*B - a^2*(2*A + C) - b^2*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((a^2*b*B + 2*b^3*B + a^3*C - a*b^2*(3*A + 4*C))*Tan[c + d*x])/(2*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,6,39,0.1538,1,"{4080, 4003, 12, 3831, 2659, 208}"
920,1,229,0,0.7594505,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^3} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^3,x]","\frac{\left(5 a^2 A b^3-3 a^4 b (2 A+C)+a^3 b^2 B+2 a^5 B-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}}-\frac{\tan (c+d x) \left(-a^2 b^2 (5 A+2 C)+3 a^3 b B+a^4 (-C)+2 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\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{A x}{a^3}","\frac{\left(5 a^2 A b^3-3 a^4 b (2 A+C)+a^3 b^2 B+2 a^5 B-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}}-\frac{\tan (c+d x) \left(-a^2 b^2 (5 A+2 C)+3 a^3 b B+a^4 (-C)+2 A b^4\right)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\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{A x}{a^3}",1,"(A*x)/a^3 + ((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))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((2*A*b^4 + 3*a^3*b*B - a^4*C - a^2*b^2*(5*A + 2*C))*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",6,5,33,0.1515,1,"{4060, 3919, 3831, 2659, 208}"
921,1,330,0,3.1489457,"\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","Int[(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{\sin (c+d x) \left(11 a^2 A b^2+a^4 (-(2 A-3 C))-5 a^3 b B+2 a b^3 B-6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-5 a^3 b^3 B+6 a^5 b B-2 a^6 C+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}}-\frac{\sin (c+d x) \left(-a^2 b^2 (6 A+C)+4 a^3 b B-2 a^4 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{\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{x (3 A b-a B)}{a^4}","-\frac{\sin (c+d x) \left(11 a^2 A b^2+a^4 (-(2 A-3 C))-5 a^3 b B+2 a b^3 B-6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-a^4 b^2 (12 A+C)+15 a^2 A b^4-5 a^3 b^3 B+6 a^5 b B-2 a^6 C+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}}-\frac{\sin (c+d x) \left(-a^2 b^2 (6 A+C)+4 a^3 b B-2 a^4 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{\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{x (3 A b-a B)}{a^4}",1,"-(((3*A*b - a*B)*x)/a^4) - ((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*(12*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((11*a^2*A*b^2 - 6*A*b^4 - 5*a^3*b*B + 2*a*b^3*B - a^4*(2*A - 3*C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((3*A*b^4 + 4*a^3*b*B - a*b^3*B - 2*a^4*C - a^2*b^2*(6*A + C))*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,6,39,0.1538,1,"{4100, 4104, 3919, 3831, 2659, 208}"
922,1,453,0,4.8244772,"\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","Int[(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{\sin (c+d x) \left(-a^2 b^3 (21 A-2 C)+a^4 b (6 A-5 C)+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \cos (c+d x) \left(-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 a^3 b B-3 a b^3 B+6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+15 a^3 b^3 B-12 a^5 b B+6 a^6 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}}+\frac{\sin (c+d x) \cos (c+d x) \left(7 a^2 A b^2-5 a^3 b B+3 a^4 C+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) \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) \left(-a^2 b^3 (21 A-2 C)+a^4 b (6 A-5 C)+11 a^3 b^2 B-2 a^5 B-6 a b^4 B+12 A b^5\right)}{2 a^4 d \left(a^2-b^2\right)^2}+\frac{\sin (c+d x) \cos (c+d x) \left(-a^2 b^2 (10 A-C)+a^4 (A-4 C)+6 a^3 b B-3 a b^3 B+6 A b^4\right)}{2 a^3 d \left(a^2-b^2\right)^2}-\frac{b \left(5 a^4 b^2 (4 A-C)-a^2 b^4 (29 A-2 C)+15 a^3 b^3 B-12 a^5 b B+6 a^6 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}}+\frac{\sin (c+d x) \cos (c+d x) \left(7 a^2 A b^2-5 a^3 b B+3 a^4 C+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) \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}",1,"((12*A*b^2 - 6*a*b*B + a^2*(A + 2*C))*x)/(2*a^5) - (b*(12*A*b^6 - 12*a^5*b*B + 15*a^3*b^3*B - 6*a*b^5*B - a^2*b^4*(29*A - 2*C) + 5*a^4*b^2*(4*A - C) + 6*a^6*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(5/2)*(a + b)^(5/2)*d) - ((12*A*b^5 - 2*a^5*B + 11*a^3*b^2*B - 6*a*b^4*B + a^4*b*(6*A - 5*C) - a^2*b^3*(21*A - 2*C))*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^2*d) + ((6*A*b^4 + 6*a^3*b*B - 3*a*b^3*B + a^4*(A - 4*C) - a^2*b^2*(10*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((7*a^2*A*b^2 - 4*A*b^4 - 5*a^3*b*B + 2*a*b^3*B + 3*a^4*C)*Cos[c + d*x]*Sin[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",8,6,41,0.1463,1,"{4100, 4104, 3919, 3831, 2659, 208}"
923,1,470,0,9.9134103,"\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","Int[(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{\tan (c+d x) \left(23 a^2 b^2 C+3 a^3 b B-12 a^4 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{\left(a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 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{\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) \sec ^2(c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 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(3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 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{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}","-\frac{\tan (c+d x) \left(23 a^2 b^2 C+3 a^3 b B-12 a^4 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{\left(a^2 b^6 (3 A+20 C)-7 a^5 b^3 B+8 a^3 b^5 B+28 a^6 b^2 C-35 a^4 b^4 C+2 a^7 b B-8 a^8 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{\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) \sec ^2(c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 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(3 a^2 b^4 (A+4 C)+2 a^3 b^3 B-11 a^4 b^2 C-a^5 b B+4 a^6 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{(b B-4 a C) \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"((b*B - 4*a*C)*ArcTanh[Sin[c + d*x]])/(b^5*d) - ((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 + a^2*b^6*(3*A + 20*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^5*(a + b)^(7/2)*d) - ((5*A*b^4 + 3*a^3*b*B - 8*a*b^3*B - 12*a^4*C + 23*a^2*b^2*C - 6*b^4*C)*Tan[c + d*x])/(6*b^4*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^3*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Sec[c + d*x]^2*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (a*(2*A*b^6 - a^5*b*B + 2*a^3*b^3*B - 6*a*b^5*B + 4*a^6*C - 11*a^4*b^2*C + 3*a^2*b^4*(A + 4*C))*Tan[c + d*x])/(2*b^4*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,8,41,0.1951,1,"{4098, 4090, 4082, 3998, 3770, 3831, 2659, 208}"
924,1,358,0,2.5085397,"\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","Int[(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{\left(-a^3 b^4 (A-8 C)+3 a^2 b^5 B-7 a^5 b^2 C+2 a^7 C-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) \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{\tan (c+d x) \left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+a^3 b^3 B+9 a^6 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{a \tan (c+d x) \left(a^2 b^2 (3 A+8 C)-3 a^4 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{C \tanh ^{-1}(\sin (c+d x))}{b^4 d}","-\frac{\left(-a^3 b^4 (A-8 C)+3 a^2 b^5 B-7 a^5 b^2 C+2 a^7 C-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) \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{\tan (c+d x) \left(-a^4 b^2 (3 A+28 C)+2 a^2 b^4 (7 A+17 C)+a^3 b^3 B+9 a^6 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{a \tan (c+d x) \left(a^2 b^2 (3 A+8 C)-3 a^4 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{C \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(C*ArcTanh[Sin[c + d*x]])/(b^4*d) - ((3*a^2*b^5*B + 2*b^7*B - a^3*b^4*(A - 8*C) + 2*a^7*C - 7*a^5*b^2*C - 4*a*b^6*(A + 2*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*b^4*(a + b)^(7/2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - (a*(2*A*b^4 - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 8*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((4*A*b^6 + a^3*b^3*B - 16*a*b^5*B + 9*a^6*C + 2*a^2*b^4*(7*A + 17*C) - a^4*b^2*(3*A + 28*C))*Tan[c + d*x])/(6*b^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,8,41,0.1951,1,"{4098, 4090, 4080, 3998, 3770, 3831, 2659, 208}"
925,1,314,0,1.0392158,"\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","Int[(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{\left(-a^2 b (4 A+3 C)+a^3 B+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(a^3 b^2 (2 A-5 C)-10 a^2 b^3 B+a^4 b B+2 a^5 C+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))}+\frac{\tan (c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 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(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^2 b (4 A+3 C)+a^3 B+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(a^3 b^2 (2 A-5 C)-10 a^2 b^3 B+a^4 b B+2 a^5 C+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))}+\frac{\tan (c+d x) \left(a^2 b^2 (2 A+9 C)+a^3 b B-4 a^4 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(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}",1,"((a^3*B + 4*a*b^2*B - b^3*(A + 2*C) - a^2*b*(4*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d) + (a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((3*A*b^4 + a^3*b*B - 6*a*b^3*B - 4*a^4*C + a^2*b^2*(2*A + 9*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((a^4*b*B - 10*a^2*b^3*B - 6*b^5*B + a^3*b^2*(2*A - 5*C) + 2*a^5*C + a*b^4*(13*A + 18*C))*Tan[c + d*x])/(6*b^2*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,7,41,0.1707,1,"{4090, 4080, 4003, 12, 3831, 2659, 208}"
926,1,299,0,0.8558118,"\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","Int[(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(a^3 (-(2 A+C))+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^2 b^2 (11 A+10 C)+2 a^3 b B+a^4 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))}+\frac{\tan (c+d x) \left(2 a^2 b B+a^3 C-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 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(a^3 (-(2 A+C))+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^2 b^2 (11 A+10 C)+2 a^3 b B+a^4 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))}+\frac{\tan (c+d x) \left(2 a^2 b B+a^3 C-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 b^2-a (b B-a C)\right)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}",1,"-(((4*a^2*b*B + b^3*B - a^3*(2*A + C) - a*b^2*(3*A + 4*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/((a - b)^(7/2)*(a + b)^(7/2)*d)) - ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + ((2*a^2*b*B + 3*b^3*B + a^3*C - a*b^2*(5*A + 6*C))*Tan[c + d*x])/(6*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((2*a^3*b*B + 13*a*b^3*B + a^4*C - 2*b^4*(2*A + 3*C) - a^2*b^2*(11*A + 10*C))*Tan[c + d*x])/(6*b*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,6,39,0.1538,1,"{4080, 4003, 12, 3831, 2659, 208}"
927,1,336,0,2.1369532,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^4} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^4,x]","-\frac{\left(-a^4 b^3 (8 A-C)+7 a^2 A b^5+4 a^6 b (2 A+C)-3 a^5 b^2 B-2 a^7 B-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(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4+4 a^3 b^3 B+11 a^5 b B-2 a^6 C-6 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\tan (c+d x) \left(-a^2 b^2 (8 A+3 C)+5 a^3 b B-2 a^4 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{\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{A x}{a^4}","-\frac{\left(-a^4 b^3 (8 A-C)+7 a^2 A b^5+4 a^6 b (2 A+C)-3 a^5 b^2 B-2 a^7 B-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(-13 a^4 b^2 (2 A+C)+17 a^2 A b^4+4 a^3 b^3 B+11 a^5 b B-2 a^6 C-6 A b^6\right)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\tan (c+d x) \left(-a^2 b^2 (8 A+3 C)+5 a^3 b B-2 a^4 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{\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{A x}{a^4}",1,"(A*x)/a^4 - ((7*a^2*A*b^5 - 2*A*b^7 - 2*a^7*B - 3*a^5*b^2*B - a^4*b^3*(8*A - C) + 4*a^6*b*(2*A + C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((3*A*b^4 + 5*a^3*b*B - 2*a^4*C - a^2*b^2*(8*A + 3*C))*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((17*a^2*A*b^4 - 6*A*b^6 + 11*a^5*b*B + 4*a^3*b^3*B - 2*a^6*C - 13*a^4*b^2*(2*A + C))*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",7,5,33,0.1515,1,"{4060, 3919, 3831, 2659, 208}"
928,1,471,0,10.1005698,"\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","Int[(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{\sin (c+d x) \left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-8 a^5 b^3 B+7 a^3 b^5 B+8 a^7 b B-2 a^8 C-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}}-\frac{\sin (c+d x) \left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^3 b^3 B+6 a^5 b B-2 a^6 C+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{\sin (c+d x) \left(-a^2 b^2 (9 A+2 C)+6 a^3 b B-3 a^4 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 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 (4 A b-a B)}{a^5}","\frac{\sin (c+d x) \left(-a^4 b^2 (65 A+4 C)+68 a^2 A b^4+a^6 (6 A-11 C)-17 a^3 b^3 B+26 a^5 b B+6 a b^5 B-24 A b^6\right)}{6 a^4 d \left(a^2-b^2\right)^3}-\frac{\left(-a^6 b^2 (20 A+3 C)+35 a^4 A b^4-28 a^2 A b^6-8 a^5 b^3 B+7 a^3 b^5 B+8 a^7 b B-2 a^8 C-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}}-\frac{\sin (c+d x) \left(-3 a^4 b^2 (4 A+C)+11 a^2 A b^4-2 a^3 b^3 B+6 a^5 b B-2 a^6 C+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{\sin (c+d x) \left(-a^2 b^2 (9 A+2 C)+6 a^3 b B-3 a^4 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 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 (4 A b-a B)}{a^5}",1,"-(((4*A*b - a*B)*x)/a^5) - ((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 - a^6*b^2*(20*A + 3*C))*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^5*(a - b)^(7/2)*(a + b)^(7/2)*d) + ((68*a^2*A*b^4 - 24*A*b^6 + 26*a^5*b*B - 17*a^3*b^3*B + 6*a*b^5*B + a^6*(6*A - 11*C) - a^4*b^2*(65*A + 4*C))*Sin[c + d*x])/(6*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((4*A*b^4 + 6*a^3*b*B - a*b^3*B - 3*a^4*C - a^2*b^2*(9*A + 2*C))*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) - ((11*a^2*A*b^4 - 4*A*b^6 + 6*a^5*b*B - 2*a^3*b^3*B + a*b^5*B - 2*a^6*C - 3*a^4*b^2*(4*A + C))*Sin[c + d*x])/(2*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,6,39,0.1538,1,"{4100, 4104, 3919, 3831, 2659, 208}"
929,1,648,0,12.5141178,"\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","Int[(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{\sin (c+d x) \left(a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right)}{6 a^5 d \left(a^2-b^2\right)^3}-\frac{\sin (c+d x) \cos (c+d x) \left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 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}+\frac{\sin (c+d x) \cos (c+d x) \left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 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) \cos (c+d x) \left(-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 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(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) \left(a^4 b^3 (146 A-17 C)-a^2 b^5 (167 A-6 C)-a^6 (24 A b-26 b C)-65 a^5 b^2 B+68 a^3 b^4 B+6 a^7 B-24 a b^6 B+60 A b^7\right)}{6 a^5 d \left(a^2-b^2\right)^3}-\frac{\sin (c+d x) \cos (c+d x) \left(a^4 b^2 (23 A-2 C)-a^2 b^4 (27 A-C)+a^6 (-(A-6 C))+11 a^3 b^3 B-12 a^5 b B-4 a b^5 B+10 A b^6\right)}{2 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-8 a^6 b^2 (5 A-C)+7 a^4 b^4 (12 A-C)-a^2 b^6 (69 A-2 C)-35 a^5 b^3 B+28 a^3 b^5 B+20 a^7 b B-8 a^8 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}+\frac{\sin (c+d x) \cos (c+d x) \left(a^4 b^2 (48 A+C)-a^2 b^4 (53 A-2 C)+20 a^3 b^3 B-27 a^5 b B+12 a^6 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) \cos (c+d x) \left(-a^2 b^2 (10 A+C)+7 a^3 b B-4 a^4 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(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}",1,"((20*A*b^2 - 8*a*b*B + a^2*(A + 2*C))*x)/(2*a^6) + (b*(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 - a^2*b^6*(69*A - 2*C) - 8*a^6*b^2*(5*A - C) + 7*a^4*b^4*(12*A - C) - 8*a^8*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^6*Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)^3*d) + ((60*A*b^7 + 6*a^7*B - 65*a^5*b^2*B + 68*a^3*b^4*B - 24*a*b^6*B + a^4*b^3*(146*A - 17*C) - a^2*b^5*(167*A - 6*C) - a^6*(24*A*b - 26*b*C))*Sin[c + d*x])/(6*a^5*(a^2 - b^2)^3*d) - ((10*A*b^6 - 12*a^5*b*B + 11*a^3*b^3*B - 4*a*b^5*B - a^6*(A - 6*C) + a^4*b^2*(23*A - 2*C) - a^2*b^4*(27*A - C))*Cos[c + d*x]*Sin[c + d*x])/(2*a^4*(a^2 - b^2)^3*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) - ((5*A*b^4 + 7*a^3*b*B - 2*a*b^3*B - 4*a^4*C - a^2*b^2*(10*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + ((20*A*b^6 - 27*a^5*b*B + 20*a^3*b^3*B - 8*a*b^5*B - a^2*b^4*(53*A - 2*C) + 12*a^6*C + a^4*b^2*(48*A + C))*Cos[c + d*x]*Sin[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",9,6,41,0.1463,1,"{4100, 4104, 3919, 3831, 2659, 208}"
930,1,24,0,0.0232022,"\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","Int[(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]","x (b B-a C)+\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 - a*C)*x + (b*C*ArcTanh[Sin[c + d*x]])/d","A",3,2,48,0.04167,1,"{24, 3770}"
931,1,75,0,0.1509273,"\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","Int[(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{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}}","\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)*x)/a - (2*b*(b*B - 2*a*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a*Sqrt[a - b]*Sqrt[a + b]*d)","A",5,5,48,0.1042,1,"{24, 3919, 3831, 2659, 208}"
932,1,140,0,0.3896992,"\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","Int[(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{2 b \left(2 a^2 b B-3 a^3 C+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}}+\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(2 a^2 b B-3 a^3 C+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}}+\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}",1,"((b*B - a*C)*x)/a^2 - (2*b*(2*a^2*b*B - b^3*B - 3*a^3*C + a*b^2*C)*ArcTanh[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^2*(a - b)^(3/2)*(a + b)^(3/2)*d) + (b^2*(b*B - 2*a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",6,6,48,0.1250,1,"{24, 3923, 3919, 3831, 2659, 208}"
933,1,231,0,1.1000682,"\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","Int[(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{b \left(-5 a^2 b^3 B+4 a^3 b^2 C+6 a^4 b B-8 a^5 C-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}}+\frac{b^2 \left(5 a^2 b B-8 a^3 C+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^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{x (b B-a C)}{a^3}","-\frac{b \left(-5 a^2 b^3 B+4 a^3 b^2 C+6 a^4 b B-8 a^5 C-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}}+\frac{b^2 \left(5 a^2 b B-8 a^3 C+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^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{x (b B-a C)}{a^3}",1,"((b*B - a*C)*x)/a^3 - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^3*(a - b)^(5/2)*(a + b)^(5/2)*d) + (b^2*(b*B - 2*a*C)*Tan[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + (b^2*(5*a^2*b*B - 2*b^3*B - 8*a^3*C + 2*a*b^2*C)*Tan[c + d*x])/(2*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",7,7,48,0.1458,1,"{24, 3923, 4060, 3919, 3831, 2659, 208}"
934,1,336,0,4.6712673,"\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","Int[(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 \left(-8 a^4 b^3 B+7 a^2 b^5 B+5 a^5 b^2 C-7 a^3 b^4 C+8 a^6 b B-10 a^7 C+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}}+\frac{b^2 \left(-17 a^2 b^3 B+13 a^3 b^2 C+26 a^4 b B-37 a^5 C-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^2 \left(8 a^2 b B-13 a^3 C+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 (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{x (b B-a C)}{a^4}","-\frac{b \left(-8 a^4 b^3 B+7 a^2 b^5 B+5 a^5 b^2 C-7 a^3 b^4 C+8 a^6 b B-10 a^7 C+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}}+\frac{b^2 \left(-17 a^2 b^3 B+13 a^3 b^2 C+26 a^4 b B-37 a^5 C-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^2 \left(8 a^2 b B-13 a^3 C+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 (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{x (b B-a C)}{a^4}",1,"((b*B - a*C)*x)/a^4 - (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[(Sqrt[a - b]*Tan[(c + d*x)/2])/Sqrt[a + b]])/(a^4*(a - b)^(7/2)*(a + b)^(7/2)*d) + (b^2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^3) + (b^2*(8*a^2*b*B - 3*b^3*B - 13*a^3*C + 3*a*b^2*C)*Tan[c + d*x])/(6*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^2) + (b^2*(26*a^4*b*B - 17*a^2*b^3*B + 6*b^5*B - 37*a^5*C + 13*a^3*b^2*C - 6*a*b^4*C)*Tan[c + d*x])/(6*a^3*(a^2 - b^2)^3*d*(a + b*Sec[c + d*x]))","A",8,7,48,0.1458,1,"{24, 3923, 4060, 3919, 3831, 2659, 208}"
935,1,517,0,1.5567954,"\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","Int[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{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(12 a^2 b B-8 a^3 C-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(12 a^2 b (2 B-C)-16 a^3 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(-6 a^2 b^2 (7 A+4 C)+24 a^3 b B-16 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}","\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(12 a^2 b B-8 a^3 C-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(12 a^2 b (2 B-C)-16 a^3 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(-6 a^2 b^2 (7 A+4 C)+24 a^3 b B-16 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,"(-2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^5*d) - (2*(a - b)*Sqrt[a + b]*(12*a^2*b*(2*B - C) - 16*a^3*C - 6*a*b^2*(7*A - 3*B + 6*C) - 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) - (2*(12*a^2*b*B - 75*b^3*B - 8*a^3*C - a*b^2*(21*A + 13*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^3*d) + (2*(63*A*b^2 + 9*a*b*B - 6*a^2*C + 49*b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B + a*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(63*b*d) + (2*C*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(9*d)","A",7,7,43,0.1628,1,"{4096, 4102, 4092, 4082, 4005, 3832, 4004}"
936,1,413,0,0.9203816,"\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","Int[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]","\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(14 a^2 b B-8 a^3 C-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}","\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(14 a^2 b B-8 a^3 C-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,"(2*(a - b)*Sqrt[a + b]*(14*a^2*b*B - 63*b^3*B - 8*a^3*C - a*b^2*(35*A + 19*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) - (2*(a - b)*Sqrt[a + b]*(35*A*b^2 - b^2*(63*B - 25*C) + 8*a^2*C - a*(14*b*B - 6*b*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(35*A*b^2 - 14*a*b*B + 8*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^2*d) + (2*(7*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(7*b*d)","A",6,6,43,0.1395,1,"{4092, 4082, 4002, 4005, 3832, 4004}"
937,1,324,0,0.5920765,"\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","Int[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{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}","\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,"(-2*(a - b)*Sqrt[a + b]*(3*b^2*(5*A + 3*C) + a*(5*b*B - 2*a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(a - b)*Sqrt[a + b]*(15*A*b - 5*b*B + 2*a*C + 9*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*(5*b*B - 2*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*b*d)","A",5,5,41,0.1220,1,"{4082, 4002, 4005, 3832, 4004}"
938,1,366,0,0.4062828,"\int \sqrt{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(-2*(a - b)*Sqrt[a + b]*(3*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*Sqrt[a + b]*(3*A*b + (a - b)*(3*B - C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",6,6,35,0.1714,1,"{4056, 4058, 3921, 3784, 3832, 4004}"
939,1,362,0,0.4117065,"\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","Int[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{\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}","\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,"((a - b)*Sqrt[a + b]*(A - 2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) + (Sqrt[a + b]*(A*b + 2*b*B + 2*a*C - 2*b*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (Sqrt[a + b]*(A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",6,6,41,0.1463,1,"{4094, 4058, 3921, 3784, 3832, 4004}"
940,1,435,0,0.7732076,"\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","Int[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{\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}","\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 - b)*Sqrt[a + b]*(A*b + 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*b*d) + (Sqrt[a + b]*(A*b + 2*a*(A + 2*B + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + (Sqrt[a + b]*(A*b^2 - 4*a*b*B - 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) + ((A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",7,7,43,0.1628,1,"{4094, 4104, 4058, 3921, 3784, 3832, 4004}"
941,1,538,0,1.2309382,"\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","Int[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{\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(4 a^2 b (A+2 C)+8 a^3 B-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}","-\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(4 a^2 b (A+2 C)+8 a^3 B-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,"-((a - b)*Sqrt[a + b]*(3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*b*d) - (Sqrt[a + b]*(3*A*b^2 - 2*a*b*(A + 3*B) - 4*a^2*(4*A + 3*B + 6*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a^2*d) - (Sqrt[a + b]*(A*b^3 + 8*a^3*B - 2*a*b^2*B + 4*a^2*b*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^3*d) - ((3*A*b^2 - 6*a*b*B - 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a^2*d) + ((A*b + 6*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*a*d) + (A*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",8,7,43,0.1628,1,"{4094, 4104, 4058, 3921, 3784, 3832, 4004}"
942,1,628,0,2.6274327,"\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","Int[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{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(33 a^2 b B-18 a^3 C+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(-3 a^2 b^2 (33 A+19 C)+44 a^3 b B-24 a^4 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(-6 a^2 b^2 (33 A-11 B+24 C)+4 a^3 b (22 B-9 C)-48 a^4 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(-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+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}","\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(33 a^2 b B-18 a^3 C+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(-3 a^2 b^2 (33 A+19 C)+44 a^3 b B-24 a^4 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(-6 a^2 b^2 (33 A-11 B+24 C)+4 a^3 b (22 B-9 C)-48 a^4 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(-18 a^3 b^2 (11 A+6 C)+363 a^2 b^3 B+88 a^4 b B-48 a^5 C+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,"(-2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^5*d) - (2*(a - b)*Sqrt[a + b]*(4*a^3*b*(22*B - 9*C) - 48*a^4*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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) - (2*(44*a^3*b*B - 968*a*b^3*B - 24*a^4*C - 75*b^4*(11*A + 9*C) - 3*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^3*d) + (2*(33*a^2*b*B + 539*b^3*B - 18*a^3*C + 6*a*b^2*(132*A + 101*C))*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 + 110*a*b*B + 3*a^2*C + 81*b^2*C)*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(693*b*d) + (2*(11*b*B + 3*a*C)*Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(99*d) + (2*C*Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(11*d)","A",8,7,43,0.1628,1,"{4096, 4102, 4092, 4082, 4005, 3832, 4004}"
943,1,505,0,1.3209257,"\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","Int[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]","\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(18 a^2 b B-8 a^3 C-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(6 a^2 b (3 B-C)-8 a^3 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(-3 a^2 b^2 (21 A+11 C)+18 a^3 b B-8 a^4 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}","\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(18 a^2 b B-8 a^3 C-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(6 a^2 b (3 B-C)-8 a^3 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(-3 a^2 b^2 (21 A+11 C)+18 a^3 b B-8 a^4 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,"(2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^4*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2*b*(3*B - C) - 8*a^3*C - 3*a*b^2*(21*A - 57*B + 13*C) + 3*b^3*(63*A - 25*B + 49*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) - (2*(18*a^2*b*B - 75*b^3*B - 8*a^3*C - 3*a*b^2*(21*A + 13*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b^2*d) + (2*(63*A*b^2 - 18*a*b*B + 8*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b^2*d) + (2*(9*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(9*b*d)","A",7,6,43,0.1395,1,"{4092, 4082, 4002, 4005, 3832, 4004}"
944,1,406,0,0.8302249,"\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","Int[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{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(21 a^2 b B-6 a^3 C+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}","\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(21 a^2 b B-6 a^3 C+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,"(-2*(a - b)*Sqrt[a + b]*(21*a^2*b*B + 63*b^3*B - 6*a^3*C + 2*a*b^2*(70*A + 41*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^3*d) + (2*(a - b)*Sqrt[a + b]*(6*a^2*C + 3*a*b*(35*A - 7*B + 19*C) - b^2*(35*A - 63*B + 25*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*(35*A*b^2 + 21*a*b*B - 6*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b*d) + (2*(7*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*b*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*b*d)","A",6,5,41,0.1220,1,"{4082, 4002, 4005, 3832, 4004}"
945,1,443,0,0.6607181,"\int (a+b \sec (c+d x))^{3/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(-2*(a - b)*Sqrt[a + b]*(15*A*b^2 + 20*a*b*B + 3*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^2*d) + (2*Sqrt[a + b]*(3*a^2*(5*B - C) + 2*a*b*(15*A - 10*B + 6*C) - b^2*(15*A - 5*B + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (2*a*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(5*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",7,6,35,0.1714,1,"{4056, 4058, 3921, 3784, 3832, 4004}"
946,1,426,0,0.6644327,"\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","Int[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]","\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}","\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,"((a - b)*Sqrt[a + b]*(3*a*A - 6*b*B - 8*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) + (Sqrt[a + b]*(a*b*(3*A + 12*B - 8*C) + 6*a^2*C + 2*b^2*(3*A - 3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b*d) - (Sqrt[a + b]*(3*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/d - (b*(3*A - 2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,41,0.1707,1,"{4094, 4056, 4058, 3921, 3784, 3832, 4004}"
947,1,442,0,0.8105192,"\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","Int[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{\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}","-\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,"((a - b)*Sqrt[a + b]*(5*A*b + 4*a*B - 8*b*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*b*d) + (Sqrt[a + b]*(b*(5*A + 8*B - 8*C) + 2*a*(A + 2*B + 8*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) - (Sqrt[a + b]*(3*A*b^2 + 12*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a*d) + ((3*A*b + 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",7,6,43,0.1395,1,"{4094, 4058, 3921, 3784, 3832, 4004}"
948,1,540,0,1.3133732,"\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","Int[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]","\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(-12 a^2 b (A+2 C)-8 a^3 B-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}","\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(-12 a^2 b (A+2 C)-8 a^3 B-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,"((a - b)*Sqrt[a + b]*(3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*b*d) + (Sqrt[a + b]*(3*A*b^2 + 4*a^2*(4*A + 3*B + 6*C) + 2*a*b*(7*A + 15*B + 24*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*a*d) + (Sqrt[a + b]*(A*b^3 - 8*a^3*B - 6*a*b^2*B - 12*a^2*b*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a^2*d) + ((3*A*b^2 + 30*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*a*d) + ((A*b + 2*a*B)*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",8,7,43,0.1628,1,"{4094, 4104, 4058, 3921, 3784, 3832, 4004}"
949,1,650,0,1.9062874,"\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","Int[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{\sin (c+d x) \left(-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right) \sqrt{a+b \sec (c+d x)}}{192 a^2 d}+\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{\sqrt{a+b} \cot (c+d x) \left(-4 a^2 b (39 A+28 B+60 C)-8 a^3 (9 A+16 B+12 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(-12 a^2 b (13 A+20 C)-128 a^3 B-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(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-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}","-\frac{\sin (c+d x) \left(-12 a^2 b (13 A+20 C)-128 a^3 B-24 a b^2 B+9 A b^3\right) \sqrt{a+b \sec (c+d x)}}{192 a^2 d}+\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{\sqrt{a+b} \cot (c+d x) \left(-4 a^2 b (39 A+28 B+60 C)-8 a^3 (9 A+16 B+12 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(-12 a^2 b (13 A+20 C)-128 a^3 B-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(24 a^2 b^2 (A+2 C)+16 a^4 (3 A+4 C)+96 a^3 b B-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,"-((a - b)*Sqrt[a + b]*(9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^2*b*d) - (Sqrt[a + b]*(9*A*b^3 - 6*a*b^2*(A + 4*B) - 8*a^3*(9*A + 16*B + 12*C) - 4*a^2*b*(39*A + 28*B + 60*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a^2*d) - (Sqrt[a + b]*(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))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^3*d) - ((9*A*b^3 - 128*a^3*B - 24*a*b^2*B - 12*a^2*b*(13*A + 20*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a^2*d) + ((3*A*b^2 + 56*a*b*B + 12*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*a*d) + ((3*A*b + 8*a*B)*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",9,7,43,0.1628,1,"{4094, 4104, 4058, 3921, 3784, 3832, 4004}"
950,1,610,0,2.1095869,"\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","Int[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{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(110 a^2 b B-40 a^3 C-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(-15 a^2 b^2 (33 A+19 C)+110 a^3 b B-40 a^4 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(-15 a^2 b^2 (33 A-121 B+19 C)+10 a^3 b (11 B-3 C)-40 a^4 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(-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B+110 a^4 b B-40 a^5 C-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}","\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(110 a^2 b B-40 a^3 C-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(-15 a^2 b^2 (33 A+19 C)+110 a^3 b B-40 a^4 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(-15 a^2 b^2 (33 A-121 B+19 C)+10 a^3 b (11 B-3 C)-40 a^4 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(-15 a^3 b^2 (33 A+17 C)-3069 a^2 b^3 B+110 a^4 b B-40 a^5 C-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,"(2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^4*d) + (2*(a - b)*Sqrt[a + b]*(10*a^3*b*(11*B - 3*C) - 40*a^4*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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3465*b^3*d) - (2*(110*a^3*b*B - 1254*a*b^3*B - 40*a^4*C - 75*b^4*(11*A + 9*C) - 15*a^2*b^2*(33*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3465*b^2*d) - (2*(110*a^2*b*B - 539*b^3*B - 40*a^3*C - 5*a*b^2*(99*A + 67*C))*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(3465*b^2*d) + (2*(99*A*b^2 - 22*a*b*B + 8*a^2*C + 81*b^2*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(693*b^2*d) + (2*(11*b*B - 4*a*C)*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(99*b^2*d) + (2*C*Sec[c + d*x]*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(11*b*d)","A",8,6,43,0.1395,1,"{4092, 4082, 4002, 4005, 3832, 4004}"
951,1,502,0,1.2559448,"\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","Int[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{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(45 a^2 b B-10 a^3 C+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(15 a^2 b (21 A-3 B+11 C)+10 a^3 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(3 a^2 b^2 (161 A+93 C)+45 a^3 b B-10 a^4 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}","\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(45 a^2 b B-10 a^3 C+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(15 a^2 b (21 A-3 B+11 C)+10 a^3 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(3 a^2 b^2 (161 A+93 C)+45 a^3 b B-10 a^4 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,"(-2*(a - b)*Sqrt[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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^3*d) + (2*(a - b)*Sqrt[a + 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(315*b^2*d) + (2*(45*a^2*b*B + 75*b^3*B - 10*a^3*C + 6*a*b^2*(28*A + 19*C))*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(315*b*d) + (2*(63*A*b^2 + 45*a*b*B - 10*a^2*C + 49*b^2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(315*b*d) + (2*(9*b*B - 2*a*C)*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(63*b*d) + (2*C*(a + b*Sec[c + d*x])^(7/2)*Tan[c + d*x])/(9*b*d)","A",7,5,41,0.1220,1,"{4082, 4002, 4005, 3832, 4004}"
952,1,521,0,0.9714559,"\int (a+b \sec (c+d x))^{5/2} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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 \sqrt{a+b} \cot (c+d x) \left(a^2 b (315 A-161 B+135 C)+15 a^3 (7 B-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(161 a^2 b B+15 a^3 C+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 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 (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}","\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 \sqrt{a+b} \cot (c+d x) \left(a^2 b (315 A-161 B+135 C)+15 a^3 (7 B-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(161 a^2 b B+15 a^3 C+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 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 (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,"(-2*(a - b)*Sqrt[a + b]*(161*a^2*b*B + 63*b^3*B + 15*a^3*C + 5*a*b^2*(49*A + 29*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^2*d) + (2*Sqrt[a + b]*(15*a^3*(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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b*d) - (2*a^2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*(35*A*b^2 + 56*a*b*B + 15*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*d) + (2*(7*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(35*d) + (2*C*(a + b*Sec[c + d*x])^(5/2)*Tan[c + d*x])/(7*d)","A",8,6,35,0.1714,1,"{4056, 4058, 3921, 3784, 3832, 4004}"
953,1,505,0,0.9651778,"\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","Int[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{\sqrt{a+b} \cot (c+d x) \left(a^2 b (15 A+90 B-46 C)+30 a^3 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{(a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (-(15 A-46 C))+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{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}","\frac{\sqrt{a+b} \cot (c+d x) \left(a^2 b (15 A+90 B-46 C)+30 a^3 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{(a-b) \sqrt{a+b} \cot (c+d x) \left(a^2 (-(15 A-46 C))+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{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,"-((a - b)*Sqrt[a + b]*(70*a*b*B - a^2*(15*A - 46*C) + 6*b^2*(5*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) + (Sqrt[a + b]*(a^2*b*(15*A + 90*B - 46*C) + 30*a^3*C - 2*b^3*(15*A - 5*B + 9*C) + 2*a*b^2*(45*A - 35*B + 17*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b*d) - (a*Sqrt[a + b]*(5*A*b + 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/d - (b*(15*a*A - 10*b*B - 16*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) - (b*(5*A - 2*C)*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",8,7,41,0.1707,1,"{4094, 4056, 4058, 3921, 3784, 3832, 4004}"
954,1,507,0,1.0613753,"\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","Int[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]","\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}","\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,"((a - b)*Sqrt[a + b]*(12*a^2*B - 24*b^2*B + a*b*(27*A - 56*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*b*d) + (Sqrt[a + b]*(a*b*(27*A + 72*B - 56*C) + 8*b^2*(3*A - 3*B + C) + 6*a^2*(A + 2*(B + 6*C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(12*d) - (Sqrt[a + b]*(15*A*b^2 + 20*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*d) + ((5*A*b + 4*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d) + (A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(2*d) - (b*(21*A*b + 12*a*B - 8*b*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(12*d)","A",8,7,43,0.1628,1,"{4094, 4056, 4058, 3921, 3784, 3832, 4004}"
955,1,549,0,1.2786652,"\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","Int[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]","\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(20 a^2 b (A+2 C)+8 a^3 B+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}","\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(20 a^2 b (A+2 C)+8 a^3 B+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,"((a - b)*Sqrt[a + b]*(54*a*b*B + 3*b^2*(11*A - 16*C) + 8*a^2*(2*A + 3*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*b*d) + (Sqrt[a + b]*(3*b^2*(11*A + 16*(B - C)) + 4*a^2*(4*A + 3*B + 6*C) + 2*a*b*(13*A + 27*B + 72*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(24*d) - (Sqrt[a + b]*(5*A*b^3 + 8*a^3*B + 30*a*b^2*B + 20*a^2*b*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(8*a*d) + ((15*A*b^2 + 42*a*b*B + 8*a^2*(2*A + 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + ((5*A*b + 6*a*B)*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (A*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",8,6,43,0.1395,1,"{4094, 4058, 3921, 3784, 3832, 4004}"
956,1,652,0,2.01955,"\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","Int[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]","\frac{\sin (c+d x) \left(4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{192 a d}+\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{\sqrt{a+b} \cot (c+d x) \left(4 a^2 b (71 A+52 B+108 C)+8 a^3 (9 A+16 B+12 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(4 a^2 b (71 A+108 C)+128 a^3 B+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(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)-160 a^3 b B-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}","\frac{\sin (c+d x) \left(4 a^2 b (71 A+108 C)+128 a^3 B+264 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{192 a d}+\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{\sqrt{a+b} \cot (c+d x) \left(4 a^2 b (71 A+52 B+108 C)+8 a^3 (9 A+16 B+12 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(4 a^2 b (71 A+108 C)+128 a^3 B+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(-120 a^2 b^2 (A+2 C)-16 a^4 (3 A+4 C)-160 a^3 b B-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,"((a - b)*Sqrt[a + b]*(15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*b*d) + (Sqrt[a + b]*(15*A*b^3 + 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(192*a*d) + (Sqrt[a + b]*(5*A*b^4 - 160*a^3*b*B - 40*a*b^3*B - 120*a^2*b^2*(A + 2*C) - 16*a^4*(3*A + 4*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(64*a^2*d) + ((15*A*b^3 + 128*a^3*B + 264*a*b^2*B + 4*a^2*b*(71*A + 108*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*a*d) + ((5*A*b^2 + 24*a*b*B + 4*a^2*(3*A + 4*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + ((5*A*b + 8*a*B)*Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (A*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",9,7,43,0.1628,1,"{4094, 4104, 4058, 3921, 3784, 3832, 4004}"
957,1,774,0,3.2032549,"\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","Int[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{\sin (c+d x) \left(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right) \sqrt{a+b \sec (c+d x)}}{1920 a^2 d}+\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(4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{960 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(-4 a^2 b^2 (423 A+295 B+660 C)-8 a^3 b (193 A+355 B+260 C)-16 a^4 (64 A+45 B+80 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(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-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(40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-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}","-\frac{\sin (c+d x) \left(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-150 a b^3 B+45 A b^4\right) \sqrt{a+b \sec (c+d x)}}{1920 a^2 d}+\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(4 a^2 b (193 A+260 C)+360 a^3 B+590 a b^2 B+15 A b^3\right) \sqrt{a+b \sec (c+d x)}}{960 a d}-\frac{\sqrt{a+b} \cot (c+d x) \left(-4 a^2 b^2 (423 A+295 B+660 C)-8 a^3 b (193 A+355 B+260 C)-16 a^4 (64 A+45 B+80 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(-12 a^2 b^2 (141 A+220 C)-256 a^4 (4 A+5 C)-2840 a^3 b B-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(40 a^2 b^3 (A+2 C)+80 a^4 b (3 A+4 C)+240 a^3 b^2 B+96 a^5 B-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,"-((a - b)*Sqrt[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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*b*d) - (Sqrt[a + b]*(45*A*b^4 - 30*a*b^3*(A + 5*B) - 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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(1920*a^2*d) - (Sqrt[a + b]*(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))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(128*a^3*d) - ((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))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(1920*a^2*d) + ((15*A*b^3 + 360*a^3*B + 590*a*b^2*B + 4*a^2*b*(193*A + 260*C))*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(960*a*d) + ((15*A*b^2 + 110*a*b*B + 16*a^2*(4*A + 5*C))*Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(240*d) + ((A*b + 2*a*B)*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d) + (A*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",10,7,43,0.1628,1,"{4094, 4104, 4058, 3921, 3784, 3832, 4004}"
958,1,429,0,1.0572961,"\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","Int[(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]","\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 \sqrt{a+b} \cot (c+d x) \left(-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)\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 (a-b) \sqrt{a+b} \cot (c+d x) \left(56 a^2 b B-48 a^3 C-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 (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}","\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 \sqrt{a+b} \cot (c+d x) \left(-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)\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 (a-b) \sqrt{a+b} \cot (c+d x) \left(56 a^2 b B-48 a^3 C-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 (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,"(-2*(a - b)*Sqrt[a + b]*(56*a^2*b*B + 63*b^3*B - 48*a^3*C - 2*a*b^2*(35*A + 22*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^5*d) + (2*Sqrt[a + b]*(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))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(105*b^4*d) + (2*(35*A*b^2 - 28*a*b*B + 24*a^2*C + 25*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(105*b^3*d) + (2*(7*b*B - 6*a*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(35*b^2*d) + (2*C*Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(7*b*d)","A",6,6,43,0.1395,1,"{4102, 4092, 4082, 4005, 3832, 4004}"
959,1,342,0,0.6640791,"\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","Int[(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]","-\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 (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 (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}","-\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 (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 (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,"(-2*(a - b)*Sqrt[a + b]*(15*A*b^2 - 10*a*b*B + 8*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*d) - (2*Sqrt[a + b]*(15*A*b^2 - b^2*(5*B - 9*C) + 8*a^2*C - 2*a*b*(5*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^3*d) + (2*(5*b*B - 4*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^2*d) + (2*C*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b*d)","A",5,5,43,0.1163,1,"{4092, 4082, 4005, 3832, 4004}"
960,1,267,0,0.3645792,"\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","Int[(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 \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}","\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,"(-2*(a - b)*Sqrt[a + b]*(3*b*B - 2*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*d) + (2*Sqrt[a + b]*(3*A*b - b*(3*B - C) + 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*d) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b*d)","A",4,4,41,0.09756,1,"{4082, 4005, 3832, 4004}"
961,1,317,0,0.2489377,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[a + b*Sec[c + d*x]],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 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}","-\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,"(-2*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*d) + (2*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",5,5,35,0.1429,1,"{4058, 3921, 3784, 3832, 4004}"
962,1,358,0,0.4243979,"\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","Int[(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{\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}","\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,"(A*(a - b)*Sqrt[a + b]*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*(A*b + 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*d) + (Sqrt[a + b]*(A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(a*d)","A",6,6,41,0.1463,1,"{4104, 4058, 3921, 3784, 3832, 4004}"
963,1,439,0,0.749037,"\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","Int[(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{\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{(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{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a 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{(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{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"-((a - b)*Sqrt[a + b]*(3*A*b - 4*a*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*b*d) - (Sqrt[a + b]*(3*A*b - 2*a*(A + 2*B))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^2*d) - (Sqrt[a + b]*(3*A*b^2 - 4*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*d) - ((3*A*b - 4*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*d) + (A*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*a*d)","A",7,6,43,0.1395,1,"{4104, 4058, 3921, 3784, 3832, 4004}"
964,1,510,0,1.3451894,"\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","Int[(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{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(20 a^2 b B-24 a^3 C-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(a^2 b (40 B-36 C)-48 a^3 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(-6 a^2 b^2 (5 A-4 C)+40 a^3 b B-48 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}}","-\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(20 a^2 b B-24 a^3 C-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(a^2 b (40 B-36 C)-48 a^3 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(-6 a^2 b^2 (5 A-4 C)+40 a^3 b B-48 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,"(2*(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))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^5*Sqrt[a + b]*d) + (2*(a^2*b*(40*B - 36*C) - 48*a^3*C - 6*a*b^2*(5*A - 5*B + 2*C) - b^3*(15*A - 5*B + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*b^4*Sqrt[a + b]*d) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(20*a^2*b*B - 5*b^3*B - 3*a*b^2*(5*A - 3*C) - 24*a^3*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*b^3*(a^2 - b^2)*d) + (2*(5*A*b^2 - 5*a*b*B + 6*a^2*C - b^2*C)*Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(5*b^2*(a^2 - b^2)*d)","A",6,6,43,0.1395,1,"{4098, 4092, 4082, 4005, 3832, 4004}"
965,1,352,0,0.8012512,"\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","Int[(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]","\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(6 a^2 b B-8 a^3 C-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}","\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(6 a^2 b B-8 a^3 C-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,"(-2*(6*a^2*b*B - 3*b^3*B - a*b^2*(3*A - 5*C) - 8*a^3*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*d) + (2*(3*A*b^2 - (2*a + b)*(b*(3*B - C) - 4*a*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*d) + (2*a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^2*d)","A",5,5,43,0.1163,1,"{4090, 4082, 4005, 3832, 4004}"
966,1,293,0,0.4640374,"\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","Int[(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{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}}","-\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,"(-2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^3*Sqrt[a + b]*d) + (2*(A*b + b*(B - C) - 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b^2*Sqrt[a + b]*d) - (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",4,4,41,0.09756,1,"{4080, 4005, 3832, 4004}"
967,1,395,0,0.4887388,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(3/2),x]","\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}}","\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,"(2*(A*b^2 - a*(b*B - a*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b^2*Sqrt[a + b]*d) - (2*(A*b - a*(B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*b*Sqrt[a + b]*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",6,6,35,0.1714,1,"{4060, 4058, 3921, 3784, 3832, 4004}"
968,1,451,0,0.7507073,"\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","Int[(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 \tan (c+d x) \left(a^2 (-(A-2 C))-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(a^2 (-(A-2 C))-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{\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{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}","-\frac{b \tan (c+d x) \left(a^2 (-(A-2 C))-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(a^2 (-(A-2 C))-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{\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{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"-(((3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d)) + ((3*A*b^2 + a*b*(A - 2*B) + 2*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*b*Sqrt[a + b]*d) + (Sqrt[a + b]*(3*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (A*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d*x]]) - (b*(3*A*b^2 - 2*a*b*B - a^2*(A - 2*C))*Tan[c + d*x])/(a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,41,0.1707,1,"{4104, 4060, 4058, 3921, 3784, 3832, 4004}"
969,1,552,0,1.2464368,"\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","Int[(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]","\frac{b \tan (c+d x) \left(-a^2 (7 A b-8 b C)+4 a^3 B-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(-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{\cot (c+d x) \left(-a^2 (7 A b-8 b C)+4 a^3 B-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{\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{(5 A b-4 a B) \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}","\frac{b \tan (c+d x) \left(-a^2 (7 A b-8 b C)+4 a^3 B-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(-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{\cot (c+d x) \left(-a^2 (7 A b-8 b C)+4 a^3 B-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{\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{(5 A b-4 a B) \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"((15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*b*Sqrt[a + b]*d) - ((15*A*b^2 + a*b*(5*A - 12*B) - 2*a^2*(A + 2*B - 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^3*Sqrt[a + b]*d) - (Sqrt[a + b]*(15*A*b^2 - 12*a*b*B + 4*a^2*(A + 2*C))*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(4*a^4*d) - ((5*A*b - 4*a*B)*Sin[c + d*x])/(4*a^2*d*Sqrt[a + b*Sec[c + d*x]]) + (A*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(15*A*b^3 + 4*a^3*B - 12*a*b^2*B - a^2*(7*A*b - 8*b*C))*Tan[c + d*x])/(4*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4104, 4060, 4058, 3921, 3784, 3832, 4004}"
970,1,549,0,1.850342,"\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","Int[(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{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(3 a^2 b B-6 a^3 C+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(-2 a^2 b^2 (A-3 B-8 C)+a^3 b (8 B-12 C)-16 a^4 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(-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+8 a^4 b B-16 a^5 C+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)}","-\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(3 a^2 b B-6 a^3 C+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(-2 a^2 b^2 (A-3 B-8 C)+a^3 b (8 B-12 C)-16 a^4 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(-2 a^3 b^2 (A-14 C)-15 a^2 b^3 B+8 a^4 b B-16 a^5 C+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,"(-2*(8*a^4*b*B - 15*a^2*b^3*B + 3*b^5*B - 2*a^3*b^2*(A - 14*C) + 2*a*b^4*(3*A - 4*C) - 16*a^5*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^5*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(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) - 16*a^4*C + b^4*(3*A - 3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^2*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*a*(4*A*b^4 + a*(3*a^2*b*B - 7*b^3*B - 6*a^3*C + 10*a*b^2*C))*Tan[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 - a*b*B + 2*a^2*C - b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*b^3*(a^2 - b^2)*d)","A",6,6,43,0.1395,1,"{4098, 4090, 4082, 4005, 3832, 4004}"
971,1,449,0,1.0437343,"\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","Int[(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]","\frac{2 \tan (c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 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 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(2 a^2 b (B-3 C)-8 a^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 \cot (c+d x) \left(a^2 b^2 (A+15 C)+2 a^3 b B-8 a^4 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)}","\frac{2 \tan (c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 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 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(2 a^2 b (B-3 C)-8 a^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 \cot (c+d x) \left(a^2 b^2 (A+15 C)+2 a^3 b B-8 a^4 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,"(2*(2*a^3*b*B - 6*a*b^3*B + 3*b^4*(A - C) - 8*a^4*C + a^2*b^2*(A + 15*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^4*Sqrt[a + b]*(a^2 - b^2)*d) + (2*(2*a^2*b*(B - 3*C) - 3*b^3*(A + B - C) - 8*a^3*C + a*b^2*(A + 3*B + 9*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^3*Sqrt[a + b]*(a^2 - b^2)*d) + (2*a*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Tan[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,43,0.1163,1,"{4090, 4080, 4005, 3832, 4004}"
972,1,416,0,0.7939272,"\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","Int[(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]","\frac{2 \tan (c+d x) \left(a^2 b B+2 a^3 C-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 \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 \cot (c+d x) \left(-a^2 b B-2 a^3 C+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}}","\frac{2 \tan (c+d x) \left(a^2 b B+2 a^3 C-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 \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 \cot (c+d x) \left(-a^2 b B-2 a^3 C+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,"(-2*(4*a*A*b^2 - a^2*b*B - 3*b^3*B - 2*a^3*C + 6*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*(a - b)*b^3*(a + b)^(3/2)*d) + (2*(2*a^2*C + a*b*(3*A + B + 3*C) - b^2*(A + 3*(B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*b^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(a^2*b*B + 3*b^3*B + 2*a^3*C - 2*a*b^2*(2*A + 3*C))*Tan[c + d*x])/(3*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",5,5,41,0.1220,1,"{4080, 4003, 4005, 3832, 4004}"
973,1,541,0,0.9103641,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(a + b*Sec[c + d*x])^(5/2),x]","-\frac{2 \tan (c+d x) \left(-a^2 b^2 (7 A+3 C)+4 a^3 b B+a^4 (-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 \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^2 b (6 A+B+3 C)+a^3 (3 B+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 \cot (c+d x) \left(7 a^2 A b^2+3 a^2 b^2 C-4 a^3 b B+a^4 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}}-\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^2 b^2 (7 A+3 C)+4 a^3 b B+a^4 (-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 \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^2 b (6 A+B+3 C)+a^3 (3 B+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 \cot (c+d x) \left(7 a^2 A b^2+3 a^2 b^2 C-4 a^3 b B+a^4 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}}-\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}",1,"(2*(7*a^2*A*b^2 - 3*A*b^4 - 4*a^3*b*B + a^4*C + 3*a^2*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*b^2*(a + b)^(3/2)*d) + (2*(a*A*b^2 + 3*A*b^3 + a^3*(3*B + C) - a^2*b*(6*A + B + 3*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*b*Sqrt[a + b]*(a^2 - b^2)*d) - (2*A*Sqrt[a + b]*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*(A*b^2 - a*(b*B - a*C))*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(3*A*b^4 + 4*a^3*b*B - a^4*C - a^2*b^2*(7*A + 3*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",7,6,35,0.1714,1,"{4060, 4058, 3921, 3784, 3832, 4004}"
974,1,618,0,1.4812209,"\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","Int[(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]","-\frac{b \tan (c+d x) \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+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{b \tan (c+d x) \left(a^2 (-(3 A-2 C))-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{\cot (c+d x) \left(-a^2 b^2 (21 A+2 B)-a^3 b (3 A-2 (6 B+C))-6 a^4 C+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(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+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{\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{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}","-\frac{b \tan (c+d x) \left(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+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{b \tan (c+d x) \left(a^2 (-(3 A-2 C))-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{\cot (c+d x) \left(-a^2 b^2 (21 A+2 B)-a^3 b (3 A-2 (6 B+C))-6 a^4 C+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(26 a^2 A b^2+a^4 (-(3 A-8 C))-14 a^3 b B+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{\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{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"-((26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*(a - b)*b*(a + b)^(3/2)*d) - ((15*A*b^4 + a*b^3*(5*A - 6*B) - a^2*b^2*(21*A + 2*B) - 6*a^4*C - a^3*b*(3*A - 2*(6*B + C)))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^3*b*Sqrt[a + b]*(a^2 - b^2)*d) + (Sqrt[a + b]*(5*A*b - 2*a*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^4*d) + (A*Sin[c + d*x])/(a*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(5*A*b^2 - 2*a*b*B - a^2*(3*A - 2*C))*Tan[c + d*x])/(3*a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (b*(26*a^2*A*b^2 - 15*A*b^4 - 14*a^3*b*B + 6*a*b^3*B - a^4*(3*A - 8*C))*Tan[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",8,7,41,0.1707,1,"{4104, 4060, 4058, 3921, 3784, 3832, 4004}"
975,1,448,0,0.8928829,"\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","Int[(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]","-\frac{2 \sqrt{a+b} \left(-3 a^2 b (15 B+4 C)+30 a^3 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 (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 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}","-\frac{2 \sqrt{a+b} \left(-3 a^2 b (15 B+4 C)+30 a^3 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 (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 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,"(-2*(a - b)*Sqrt[a + b]*(35*a*b*B - 12*a^2*C + 9*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) - (2*Sqrt[a + b]*(a*b^2*(35*B - 12*C) - b^3*(5*B - 9*C) + 30*a^3*C - 3*a^2*b*(15*B + 4*C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(15*d) - (2*a^2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b^2*(5*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(15*d) + (2*b^2*C*(a + b*Sec[c + d*x])^(3/2)*Tan[c + d*x])/(5*d)","A",8,8,50,0.1600,1,"{4041, 3918, 4056, 4058, 3921, 3784, 3832, 4004}"
976,1,382,0,0.6234828,"\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","Int[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{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}","-\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)*Sqrt[a + b]*(3*b*B + a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*Sqrt[a + b]*(b^2*(3*B - C) - a*b*(6*B - C) + 3*a^2*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*d) - (2*a*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b^2*C*Sqrt[a + b*Sec[c + d*x]]*Tan[c + d*x])/(3*d)","A",7,7,50,0.1400,1,"{4041, 3918, 4058, 3921, 3784, 3832, 4004}"
977,1,316,0,0.4103862,"\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","Int[(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 \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}","\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*(a - b)*Sqrt[a + b]*C*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d + (2*b*Sqrt[a + b]*(B - C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d","A",6,6,50,0.1200,1,"{4041, 3916, 3784, 4005, 3832, 4004}"
978,1,212,0,0.1641725,"\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","Int[(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{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}","\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,"(2*Sqrt[a + b]*C*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d)","A",4,4,50,0.08000,1,"{24, 3921, 3784, 3832}"
979,1,379,0,0.5370114,"\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","Int[(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]","\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}}","\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,"(2*(b*B - 2*a*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*(b*B - 2*a*C)*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*Sqrt[a + b]*d) - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^2*d) + (2*b^2*(b*B - 2*a*C)*Tan[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",7,7,50,0.1400,1,"{24, 3923, 4058, 3921, 3784, 3832, 4004}"
980,1,519,0,0.9998868,"\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","Int[(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{2 b^2 \left(7 a^2 b B-11 a^3 C+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 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 \left(-2 a^2 b (3 B+C)+9 a^3 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(7 a^2 b B-11 a^3 C+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}}-\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 \left(7 a^2 b B-11 a^3 C+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 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 \left(-2 a^2 b (3 B+C)+9 a^3 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(7 a^2 b B-11 a^3 C+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}}-\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}",1,"(2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*(a - b)*(a + b)^(3/2)*d) + (2*(3*b^3*B + a*b^2*(B - 3*C) + 9*a^3*C - 2*a^2*b*(3*B + C))*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(3*a^2*Sqrt[a + b]*(a^2 - b^2)*d) - (2*Sqrt[a + b]*(b*B - a*C)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a^3*d) + (2*b^2*(b*B - 2*a*C)*Tan[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*b^2*(7*a^2*b*B - 3*b^3*B - 11*a^3*C + 3*a*b^2*C)*Tan[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,50,0.1600,1,"{24, 3923, 4060, 4058, 3921, 3784, 3832, 4004}"
981,1,266,0,0.300583,"\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","Int[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{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}","\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*(9*A*b + 9*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(7*a*A + 5*b*B + 5*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(9*A*b + 9*a*B + 7*b*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*(b*B + a*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d) + (2*b*C*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(9*d)","A",10,7,41,0.1707,1,"{4076, 4047, 3768, 3771, 2639, 4046, 2641}"
982,1,230,0,0.2760809,"\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","Int[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{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}","\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*(5*a*A + 3*b*B + 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(5*a*A + 3*b*B + 3*a*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(b*B + a*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d) + (2*b*C*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(7*d)","A",9,7,41,0.1707,1,"{4076, 4047, 3768, 3771, 2641, 4046, 2639}"
983,1,192,0,0.2346588,"\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","Int[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{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}","\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*(5*A*b + 5*a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(b*B + a*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(5*A*b + 5*a*B + 3*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(b*B + a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*b*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*d)","A",8,7,41,0.1707,1,"{4076, 4047, 3768, 3771, 2639, 4046, 2641}"
984,1,152,0,0.223363,"\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","Int[((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{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}","\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,"(-2*(b*B - a*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*A*b + 3*a*B + b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(b*B + a*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d + (2*b*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d)","A",7,6,41,0.1463,1,"{4076, 4047, 3771, 2641, 4046, 2639}"
985,1,146,0,0.2265891,"\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","Int[((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{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}","\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,"(2*(A*b + a*B - b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (2*(3*b*B + a*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*b*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/d","A",7,6,41,0.1463,1,"{4074, 4047, 3771, 2641, 4046, 2639}"
986,1,156,0,0.2265585,"\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","Int[((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{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)}","\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,"(2*(3*a*A + 5*b*B + 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(A*b + a*B + 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*A*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",7,6,41,0.1463,1,"{4074, 4047, 3771, 2639, 4045, 2641}"
987,1,194,0,0.2525957,"\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","Int[((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{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)}","\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,"(2*(3*A*b + 3*a*B + 5*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(5*a*A + 7*b*B + 7*a*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",8,7,41,0.1707,1,"{4074, 4047, 3769, 3771, 2641, 4045, 2639}"
988,1,230,0,0.2805176,"\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","Int[((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{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)}","\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,"(2*(7*a*A + 9*b*B + 9*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*A*b + 5*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*A*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(7*a*A + 9*b*B + 9*a*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(5*A*b + 5*a*B + 7*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]])","A",9,7,41,0.1707,1,"{4074, 4047, 3769, 3771, 2639, 4045, 2641}"
989,1,266,0,0.3080119,"\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","Int[((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{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)}","\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,"(2*(7*A*b + 7*a*B + 9*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (10*(9*a*A + 11*b*B + 11*a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*A*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2)) + (2*(A*b + a*B)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(9*a*A + 11*b*B + 11*a*C)*Sin[c + d*x])/(77*d*Sec[c + d*x]^(5/2)) + (2*(7*A*b + 7*a*B + 9*b*C)*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (10*(9*a*A + 11*b*B + 11*a*C)*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]])","A",10,7,41,0.1707,1,"{4074, 4047, 3769, 3771, 2641, 4045, 2639}"
990,1,343,0,0.5876214,"\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","Int[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{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}","\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*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(18*a*b*B + 3*a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(14*a*A*b + 7*a^2*B + 5*b^2*B + 10*a*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*(9*A*b^2 + 18*a*b*B + 4*a^2*C + 7*b^2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(45*d) + (2*b*(9*b*B + 4*a*C)*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(63*d) + (2*C*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d)","A",10,8,43,0.1860,1,"{4096, 4076, 4047, 3768, 3771, 2641, 4046, 2639}"
991,1,289,0,0.529411,"\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","Int[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{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}","\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,"(-2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(21*d) + (2*b*(7*b*B + 4*a*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(35*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d)","A",9,8,43,0.1860,1,"{4096, 4076, 4047, 3768, 3771, 2639, 4046, 2641}"
992,1,241,0,0.5127399,"\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","Int[((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{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}","\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,"(-2*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*(5*A*b^2 + 10*a*b*B + 4*a^2*C + 3*b^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*b*(5*b*B + 4*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d)","A",8,7,43,0.1628,1,"{4096, 4076, 4047, 3771, 2641, 4046, 2639}"
993,1,224,0,0.4997575,"\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","Int[((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 \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}","\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^2*B - b^2*B + 2*a*b*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/d + (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]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(3*b*B - 2*a*(A - 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*d) - (2*b^2*(A - C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*d) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",8,7,43,0.1628,1,"{4094, 4076, 4047, 3771, 2641, 4046, 2639}"
994,1,225,0,0.5177835,"\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","Int[((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 \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}","\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*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) + (2*a*(4*A*b + 5*a*B)*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(A - 5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",8,7,43,0.1628,1,"{4094, 4074, 4047, 3771, 2641, 4046, 2639}"
995,1,242,0,0.5217542,"\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","Int[((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{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)}","\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,"(2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(4*A*b + 7*a*B)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",8,7,43,0.1628,1,"{4094, 4074, 4047, 3771, 2639, 4045, 2641}"
996,1,290,0,0.5520368,"\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","Int[((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{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)}","\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,"(2*(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]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(4*A*b + 9*a*B)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Sin[c + d*x])/(45*d*Sec[c + d*x]^(3/2)) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sin[c + d*x])/(21*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",9,8,43,0.1860,1,"{4094, 4074, 4047, 3769, 3771, 2641, 4045, 2639}"
997,1,397,0,0.8665761,"\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","Int[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 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(54 a^2 b B+8 a^3 C+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(9 a^2 b (5 A+3 C)+15 a^3 B+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(9 a^2 b (5 A+3 C)+15 a^3 B+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}","\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(54 a^2 b B+8 a^3 C+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(9 a^2 b (5 A+3 C)+15 a^3 B+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(9 a^2 b (5 A+3 C)+15 a^3 B+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*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(63*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(315*d) + (2*(3*b*B + 2*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d)","A",10,8,43,0.1860,1,"{4096, 4076, 4047, 3768, 3771, 2639, 4046, 2641}"
998,1,334,0,0.7859187,"\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","Int[((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{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(98 a^2 b B+24 a^3 C+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^2 b (3 A+C)+21 a^3 B+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}","\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(98 a^2 b B+24 a^3 C+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^2 b (3 A+C)+21 a^3 B+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,"(-2*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(98*a^2*b*B + 21*b^3*B + 24*a^3*C + 21*a*b^2*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*b*(35*A*b^2 + 63*a*b*B + 24*a^2*C + 25*b^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(7*b*B + 6*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(35*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7*d)","A",9,7,43,0.1628,1,"{4096, 4076, 4047, 3771, 2641, 4046, 2639}"
999,1,319,0,0.8299167,"\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","Int[((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 b \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (-(10 A-42 C))+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(15 a^2 b (A-C)+5 a^3 B-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)}}","\frac{2 b \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 (-(10 A-42 C))+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(15 a^2 b (A-C)+5 a^3 B-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*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(3*d) + (2*b*(45*a*b*B - a^2*(10*A - 42*C) + 3*b^2*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(5*a*A - 5*b*B - 9*a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) - (2*b*(5*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",9,8,43,0.1860,1,"{4094, 4096, 4076, 4047, 3771, 2641, 4046, 2639}"
1000,1,313,0,0.8335262,"\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","Int[((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{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(3 a^2 b (A+3 C)+a^3 B+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)}","-\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(3 a^2 b (A+3 C)+a^3 B+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,"(2*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b*(10*a^2*B - 15*b^2*B + 3*a*b*(7*A - 15*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(9*A*b + 5*a*B - 5*b*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(6*A*b + 5*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",9,7,43,0.1628,1,"{4094, 4076, 4047, 3771, 2641, 4046, 2639}"
1001,1,317,0,0.8318749,"\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","Int[((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{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^2 b (3 A+5 C)+3 a^3 B+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)}","\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^2 b (3 A+5 C)+3 a^3 B+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,"(2*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(24*A*b^2 + 63*a*b*B + 5*a^2*(5*A + 7*C))*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(11*A*b + 7*a*B - 35*b*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*(6*A*b + 7*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",9,7,43,0.1628,1,"{4094, 4074, 4047, 3771, 2641, 4046, 2639}"
1002,1,336,0,0.8560307,"\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","Int[((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{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(9 a^2 b (5 A+7 C)+15 a^3 B+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(3 a^2 b (5 A+7 C)+5 a^3 B+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)}","\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(9 a^2 b (5 A+7 C)+15 a^3 B+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(3 a^2 b (5 A+7 C)+5 a^3 B+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,"(2*(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]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sin[c + d*x])/(63*d*Sqrt[Sec[c + d*x]]) + (2*(2*A*b + 3*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",9,7,43,0.1628,1,"{4094, 4074, 4047, 3771, 2639, 4045, 2641}"
1003,1,401,0,0.9147223,"\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","Int[((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{2 \sin (c+d x) \left(33 a^2 b (7 A+9 C)+77 a^3 B+242 a b^2 B+24 A b^3\right)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\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(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(3 a^2 b (7 A+9 C)+7 a^3 B+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)}","\frac{2 \sin (c+d x) \left(33 a^2 b (7 A+9 C)+77 a^3 B+242 a b^2 B+24 A b^3\right)}{495 d \sec ^{\frac{3}{2}}(c+d x)}+\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(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(3 a^2 b (7 A+9 C)+7 a^3 B+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*(7*a^3*B + 27*a*b^2*B + 3*b^3*(3*A + 5*C) + 3*a^2*b*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*(24*A*b^2 + 143*a*b*B + 9*a^2*(9*A + 11*C))*Sin[c + d*x])/(693*d*Sec[c + d*x]^(5/2)) + (2*(24*A*b^3 + 77*a^3*B + 242*a*b^2*B + 33*a^2*b*(7*A + 9*C))*Sin[c + d*x])/(495*d*Sec[c + d*x]^(3/2)) + (2*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(6*A*b + 11*a*B)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",10,8,43,0.1860,1,"{4094, 4074, 4047, 3769, 3771, 2641, 4045, 2639}"
1004,1,515,0,1.3055047,"\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","Int[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 \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(1353 a^2 b B+192 a^3 C+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(9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 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(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 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(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+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(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 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}","\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(1353 a^2 b B+192 a^3 C+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(9 a^2 b^2 (143 A+101 C)+682 a^3 b B+64 a^4 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(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 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(66 a^2 b^2 (7 A+5 C)+77 a^4 (3 A+C)+308 a^3 b B+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(12 a^3 b (5 A+3 C)+54 a^2 b^2 B+15 a^4 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*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*(15*a^4*B + 54*a^2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2*b^2*(143*A + 101*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(693*d) + (2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 673*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3465*d) + (2*(33*A*b^2 + 55*a*b*B + 16*a^2*C + 27*b^2*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(11*b*B + 8*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d)","A",11,8,43,0.1860,1,"{4096, 4076, 4047, 3768, 3771, 2639, 4046, 2641}"
1005,1,441,0,1.2392134,"\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","Int[((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 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(261 a^2 b B+64 a^3 C+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(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 \sin (c+d x) \sqrt{\sec (c+d x)} \left(7 a^2 b^2 (261 A+155 C)+1098 a^3 b B+192 a^4 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(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 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(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+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}","\frac{2 b \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(261 a^2 b B+64 a^3 C+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(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 \sin (c+d x) \sqrt{\sec (c+d x)} \left(7 a^2 b^2 (261 A+155 C)+1098 a^3 b B+192 a^4 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(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 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(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+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*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*(1098*a^3*b*B + 756*a*b^3*B + 192*a^4*C + 21*b^4*(9*A + 7*C) + 7*a^2*b^2*(261*A + 155*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*b*(261*a^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(315*d) + (2*(9*b*B + 8*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(63*d) + (2*C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(9*d)","A",10,7,43,0.1628,1,"{4096, 4076, 4047, 3771, 2641, 4046, 2639}"
1006,1,419,0,1.2256467,"\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","Int[((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 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (-(35 A-87 C))+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(a^3 (-(70 A-366 C))+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(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+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(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 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)}}","\frac{2 b^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 (-(35 A-87 C))+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(a^3 (-(70 A-366 C))+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(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+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(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 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*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*b*(609*a^2*b*B + 63*b^3*B - a^3*(70*A - 366*C) + 84*a*b^2*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*b^2*(98*a*b*B - a^2*(35*A - 87*C) + 5*b^2*(7*A + 5*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) - (2*b*(35*a*A - 21*b*B - 39*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(105*d) - (2*b*(7*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(21*d) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",10,8,43,0.1860,1,"{4094, 4096, 4076, 4047, 3771, 2641, 4046, 2639}"
1007,1,409,0,1.2468449,"\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","Int[((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 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(a^2 b (31 A-87 C)+10 a^3 B-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(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 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(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-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)}","-\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(a^2 b (31 A-87 C)+10 a^3 B-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(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 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(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-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*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*d) - (2*b*(10*a^3*B - 60*a*b^2*B + a^2*b*(31*A - 87*C) - 3*b^3*(5*A + 3*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d) - (2*b^2*(5*a^2*B - 5*b^2*B + 14*a*b*(A - C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) - (2*b*(11*A*b + 5*a*B - 3*b*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(15*d) + (2*(8*A*b + 5*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",10,8,43,0.1860,1,"{4094, 4096, 4076, 4047, 3771, 2641, 4046, 2639}"
1008,1,429,0,1.3093629,"\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","Int[((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{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(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+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(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 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)}","-\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(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+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(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 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,"(2*(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]*Sqrt[Sec[c + d*x]])/(5*d) + (2*(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]*Sqrt[Sec[c + d*x]])/(21*d) - (2*b*(217*a^2*b*B - 105*b^3*B + 12*a*b^2*(19*A - 35*C) + 10*a^3*(5*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(105*d) - (2*b^2*(98*a*b*B + b^2*(87*A - 35*C) + 5*a^2*(5*A + 7*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*(48*A*b^2 + 77*a*b*B + 5*a^2*(5*A + 7*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*(8*A*b + 7*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",10,7,43,0.1628,1,"{4094, 4076, 4047, 3771, 2641, 4046, 2639}"
1009,1,426,0,1.3130309,"\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","Int[((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{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(a^2 (202 A b+294 b C)+75 a^3 B+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(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 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(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+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)}","\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(a^2 (202 A b+294 b C)+75 a^3 B+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(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 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(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+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*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*d) + (2*a*(64*A*b^3 + 75*a^3*B + 261*a*b^2*B + a^2*(202*A*b + 294*b*C))*Sin[c + d*x])/(315*d*Sqrt[Sec[c + d*x]]) - (2*b^2*(162*a*b*B + 3*b^2*(41*A - 105*C) + 7*a^2*(7*A + 9*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(48*A*b^2 + 117*a*b*B + 7*a^2*(7*A + 9*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b + 9*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",10,7,43,0.1628,1,"{4094, 4074, 4047, 3771, 2641, 4046, 2639}"
1010,1,444,0,1.3158137,"\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","Int[((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{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(2 a^2 b (673 A+891 C)+539 a^3 B+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(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+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(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+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(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 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)}","\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(2 a^2 b (673 A+891 C)+539 a^3 B+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(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+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(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+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(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 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*(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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(15*d) + (2*(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))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(3/2)) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sin[c + d*x])/(693*d*Sqrt[Sec[c + d*x]]) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*(8*A*b + 11*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",10,7,43,0.1628,1,"{4094, 4074, 4047, 3771, 2639, 4045, 2641}"
1011,1,516,0,1.4005783,"\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","Int[((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{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 \sin (c+d x) \left(11 a^2 b^2 (491 A+637 C)+77 a^4 (11 A+13 C)+4004 a^3 b B+3458 a b^3 B+192 A b^4\right)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(a^2 (2518 A b+3146 b C)+1053 a^3 B+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(20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 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(20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 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(78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)+364 a^3 b B+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)}","\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 \sin (c+d x) \left(11 a^2 b^2 (491 A+637 C)+77 a^4 (11 A+13 C)+4004 a^3 b B+3458 a b^3 B+192 A b^4\right)}{6435 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a \sin (c+d x) \left(a^2 (2518 A b+3146 b C)+1053 a^3 B+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(20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 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(20 a^3 b (9 A+11 C)+330 a^2 b^2 B+45 a^4 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(78 a^2 b^2 (7 A+9 C)+a^4 (77 A+91 C)+364 a^3 b B+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*(364*a^3*b*B + 468*a*b^3*B + 39*b^4*(3*A + 5*C) + 78*a^2*b^2*(7*A + 9*C) + a^4*(77*A + 91*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(195*d) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b^3*(5*A + 7*C) + 20*a^3*b*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(231*d) + (2*a*(192*A*b^3 + 1053*a^3*B + 2171*a*b^2*B + a^2*(2518*A*b + 3146*b*C))*Sin[c + d*x])/(9009*d*Sec[c + d*x]^(5/2)) + (2*(192*A*b^4 + 4004*a^3*b*B + 3458*a*b^3*B + 77*a^4*(11*A + 13*C) + 11*a^2*b^2*(491*A + 637*C))*Sin[c + d*x])/(6435*d*Sec[c + d*x]^(3/2)) + (2*(45*a^4*B + 330*a^2*b^2*B + 77*b^4*B + 44*a*b^3*(5*A + 7*C) + 20*a^3*b*(9*A + 11*C))*Sin[c + d*x])/(231*d*Sqrt[Sec[c + d*x]]) + (2*(48*A*b^2 + 221*a*b*B + 11*a^2*(11*A + 13*C))*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(1287*d*Sec[c + d*x]^(7/2)) + (2*(8*A*b + 13*a*B)*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(143*d*Sec[c + d*x]^(9/2)) + (2*A*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(13*d*Sec[c + d*x]^(11/2))","A",11,8,43,0.1860,1,"{4094, 4074, 4047, 3769, 3771, 2641, 4045, 2639}"
1012,1,296,0,1.1102062,"\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","Int[(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]","\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}","\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,"(-2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*b^3*d) + (2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^2*d) - (2*a*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a + b)*d) + (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(5*b^3*d) + (2*(b*B - a*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*d) + (2*C*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(5*b*d)","A",11,8,43,0.1860,1,"{4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1013,1,218,0,0.7732513,"\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","Int[(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]","\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}","\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,"(-2*(b*B - a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*d) + (2*C*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a + b)*d) + (2*(b*B - a*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*d) + (2*C*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*d)","A",10,8,43,0.1860,1,"{4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1014,1,178,0,0.477231,"\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","Int[(Sqrt[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 \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}","-\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,"(-2*C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b*d) + (2*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a + b)*d) + (2*C*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",9,8,43,0.1860,1,"{4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1015,1,157,0,0.2884978,"\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","Int[(A + 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{\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}","\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,"(2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a + b)*d)","A",8,7,43,0.1628,1,"{4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1016,1,207,0,0.530612,"\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","Int[(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]","\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 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 A \sin (c+d x)}{3 a 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 (A+3 C)-3 a b B+3 A b^2\right)}{3 a^3 d}-\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 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}",1,"(-2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*d) + (2*(3*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^3*d) - (2*b*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a + b)*d) + (2*A*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",9,8,43,0.1860,1,"{4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1017,1,269,0,0.8554607,"\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","Int[(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]","-\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 (A+3 C)+a^3 (-B)-3 a b^2 B+3 A b^3\right)}{3 a^4 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)-5 a b B+5 A b^2\right)}{5 a^3 d}+\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 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(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 b (A+3 C)+a^3 (-B)-3 a b^2 B+3 A b^3\right)}{3 a^4 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)-5 a b B+5 A b^2\right)}{5 a^3 d}+\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 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^3*d) - (2*(3*A*b^3 - a^3*B - 3*a*b^2*B + a^2*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^4*d) + (2*b^2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^4*(a + b)*d) + (2*A*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(3*a^2*d*Sqrt[Sec[c + d*x]])","A",10,8,43,0.1860,1,"{4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1018,1,342,0,1.2220824,"\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","Int[(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]","\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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)-7 a^3 b B-21 a b^3 B+21 A b^4\right)}{21 a^5 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 (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right)}{5 a^4 d}-\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 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(7 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)-7 a^3 b B-21 a b^3 B+21 A b^4\right)}{21 a^5 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 (3 A+5 C)-3 a^3 B-5 a b^2 B+5 A b^3\right)}{5 a^4 d}-\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 A \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}",1,"(-2*(5*A*b^3 - 3*a^3*B - 5*a*b^2*B + a^2*b*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^4*d) + (2*(21*A*b^4 - 7*a^3*b*B - 21*a*b^3*B + 7*a^2*b^2*(A + 3*C) + a^4*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(21*a^5*d) - (2*b^3*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^5*(a + b)*d) + (2*A*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (2*(A*b - a*B)*Sin[c + d*x])/(5*a^2*d*Sec[c + d*x]^(3/2)) + (2*(7*A*b^2 - 7*a*b*B + a^2*(5*A + 7*C))*Sin[c + d*x])/(21*a^3*d*Sqrt[Sec[c + d*x]])","A",11,8,43,0.1860,1,"{4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1019,1,447,0,1.370409,"\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","Int[(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{\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{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 b B-5 a^3 C-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)} 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{\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 B-5 a^3 C-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(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 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}","-\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{\sin (c+d x) \sqrt{\sec (c+d x)} \left(3 a^2 b B-5 a^3 C-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)} 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{\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 B-5 a^3 C-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(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 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,"-(((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^3*(a^2 - b^2)*d)) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/((a - b)*b^3*(a + b)^2*d) + ((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^3*(a^2 - b^2)*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",11,9,43,0.2093,1,"{4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1020,1,363,0,0.959012,"\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","Int[(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{\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(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 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}","-\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(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 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,"-(((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(b^2*(a^2 - b^2)*d)) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*(a - b)*b^2*(a + b)^2*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",10,9,43,0.2093,1,"{4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1021,1,299,0,0.6726263,"\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","Int[(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{\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(a^2 (-(2 A+C))+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(-3 a^2 b^2 (A+C)+a^3 b B+a^4 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}","-\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(a^2 (-(2 A+C))+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(-3 a^2 b^2 (A+C)+a^3 b B+a^4 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,"((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a*b*(a^2 - b^2)*d) - ((A*b^2 + a*b*B - a^2*(2*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a - b)*b*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,43,0.1860,1,"{4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1022,1,317,0,0.678562,"\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","Int[(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{\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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A+C)+2 a^3 B-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)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-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)} \left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-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}","\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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A+C)+2 a^3 B-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)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-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)} \left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-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,"-(((3*A*b^2 - a*b*B - a^2*(2*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^2*(a^2 - b^2)*d)) + ((3*A*b^3 + 2*a^3*B - a*b^2*B - a^2*b*(4*A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a - b)*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(a + b*Sec[c + d*x]))","A",9,8,43,0.1860,1,"{4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1023,1,406,0,1.0300906,"\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","Int[(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{\sin (c+d x) \left(a^2 (-(2 A-3 C))-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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (16 A-3 C)-2 a^4 (A+3 C)+12 a^3 b B-9 a b^3 B+15 A b^4\right)}{3 a^4 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^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 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}","-\frac{\sin (c+d x) \left(a^2 (-(2 A-3 C))-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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (16 A-3 C)-2 a^4 (A+3 C)+12 a^3 b B-9 a b^3 B+15 A b^4\right)}{3 a^4 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^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 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,"((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^3*(a^2 - b^2)*d) - ((15*A*b^4 + 12*a^3*b*B - 9*a*b^3*B - a^2*b^2*(16*A - 3*C) - 2*a^4*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d) + (b*(5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^4*(a - b)*(a + b)^2*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))","A",10,9,43,0.2093,1,"{4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1024,1,507,0,1.5159791,"\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","Int[(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{\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(a^2 (-(2 A-5 C))-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(-a^2 (4 A b-3 b C)+2 a^3 B-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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)+16 a^3 b^2 B+2 a^5 B-15 a b^4 B+21 A b^5\right)}{3 a^5 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 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+20 a^3 b B-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(-3 a^2 b^2 (3 A-C)+7 a^3 b B-5 a^4 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{\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(a^2 (-(2 A-5 C))-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(-a^2 (4 A b-3 b C)+2 a^3 B-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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (20 A-9 C)-4 a^4 b (A+3 C)+16 a^3 b^2 B+2 a^5 B-15 a b^4 B+21 A b^5\right)}{3 a^5 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 b^2 (8 A-5 C)-2 a^4 (3 A+5 C)+20 a^3 b B-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(-3 a^2 b^2 (3 A-C)+7 a^3 b B-5 a^4 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}",1,"-((35*A*b^4 + 20*a^3*b*B - 25*a*b^3*B - 3*a^2*b^2*(8*A - 5*C) - 2*a^4*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(5*a^4*(a^2 - b^2)*d) + ((21*A*b^5 + 2*a^5*B + 16*a^3*b^2*B - 15*a*b^4*B - a^2*b^3*(20*A - 9*C) - 4*a^4*b*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(3*a^5*(a^2 - b^2)*d) - (b^2*(7*A*b^4 + 7*a^3*b*B - 5*a*b^3*B - 3*a^2*b^2*(3*A - C) - 5*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(a^5*(a - b)*(a + b)^2*d) - ((7*A*b^2 - 5*a*b*B - a^2*(2*A - 5*C))*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) + ((7*A*b^3 + 2*a^3*B - 5*a*b^2*B - a^2*(4*A*b - 3*b*C))*Sin[c + d*x])/(3*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x]))","A",11,9,43,0.2093,1,"{4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1025,1,667,0,2.2479338,"\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","Int[(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{\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(a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 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(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 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(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 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)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+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(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 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}","-\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(a^2 b^2 (A+13 C)+3 a^3 b B-7 a^4 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(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 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(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+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)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (3 A-61 C)+15 a^3 b B-35 a^4 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)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^3 b^2 (3 A-65 C)-29 a^2 b^3 B+15 a^4 b B-35 a^5 C+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(a^4 b^2 (3 A-86 C)-3 a^2 b^4 (2 A-21 C)+38 a^3 b^3 B-15 a^5 b B+35 a^6 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,"-((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^4*(a^2 - b^2)^2*d) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*b^3*(a^2 - b^2)^2*d) + ((15*A*b^6 - 15*a^5*b*B + 38*a^3*b^3*B - 35*a*b^5*B + a^4*b^2*(3*A - 86*C) - 3*a^2*b^4*(2*A - 21*C) + 35*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*(a - b)^2*b^4*(a + b)^3*d) + ((15*a^4*b*B - 29*a^2*b^3*B + 8*b^5*B - a^3*b^2*(3*A - 65*C) + 3*a*b^4*(3*A - 8*C) - 35*a^5*C)*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^4*(a^2 - b^2)^2*d) - ((15*a^3*b*B - 33*a*b^3*B - a^2*b^2*(3*A - 61*C) + b^4*(21*A - 8*C) - 35*a^4*C)*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(12*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((5*A*b^4 + 3*a^3*b*B - 9*a*b^3*B - 7*a^4*C + a^2*b^2*(A + 13*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",12,9,43,0.2093,1,"{4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1026,1,556,0,1.6922467,"\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","Int[(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{\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(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 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}","-\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(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 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,"((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*b^3*(a^2 - b^2)^2*d) + ((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))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((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))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",11,9,43,0.2093,1,"{4098, 4102, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1027,1,469,0,1.1797456,"\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","Int[(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{\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(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 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(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 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(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 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^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 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}","-\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(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 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(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 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(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 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^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 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,"-((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a*b^2*(a^2 - b^2)^2*d) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) - ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[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))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,8,43,0.1860,1,"{4098, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1028,1,478,0,1.2076903,"\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","Int[(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{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 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{\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{\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 (5 A-3 C)+a^4 (8 A+3 C)-7 a^3 b B+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^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-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(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-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}","\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 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{\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{\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 (5 A-3 C)+a^4 (8 A+3 C)-7 a^3 b B+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^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-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(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-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,"-((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^2*b*(a^2 - b^2)^2*d) + ((3*A*b^4 - 7*a^3*b*B + a*b^3*B - a^2*b^2*(5*A - 3*C) + a^4*(8*A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a - b)^2*b*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,9,43,0.2093,1,"{4098, 4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1029,1,486,0,1.2228625,"\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","Int[(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{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 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{\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{\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^3 (33 A+C)+a^4 b (24 A+7 C)+5 a^3 b^2 B-8 a^5 B-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)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-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)} \left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 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}","-\frac{\sin (c+d x) \sqrt{\sec (c+d x)} \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 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{\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{\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^3 (33 A+C)+a^4 b (24 A+7 C)+5 a^3 b^2 B-8 a^5 B-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)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-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)} \left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 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,"((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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^3*(a^2 - b^2)^2*d) - ((15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B - a^2*b^3*(33*A + C) + a^4*b*(24*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^2) - ((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))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x]))","A",10,8,43,0.1860,1,"{4100, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1030,1,598,0,1.8051972,"\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","Int[(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{\sin (c+d x) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-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(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 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{\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{\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^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)+99 a^3 b^3 B-72 a^5 b B-45 a b^5 B+105 A b^6\right)}{12 a^5 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^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 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}","\frac{\sin (c+d x) \left(-a^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-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(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 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{\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{\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^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)+99 a^3 b^3 B-72 a^5 b B-45 a b^5 B+105 A b^6\right)}{12 a^5 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^2 b^3 (65 A-3 C)+3 a^4 b (8 A-3 C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 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,"-((35*A*b^5 - 8*a^5*B + 29*a^3*b^2*B - 15*a*b^4*B + 3*a^4*b*(8*A - 3*C) - a^2*b^3*(65*A - 3*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^4*(a^2 - b^2)^2*d) + ((105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(12*a^5*(a^2 - b^2)^2*d) - (b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*Sqrt[Cos[c + d*x]]*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]) + ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2) - ((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))*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x]))","A",11,9,43,0.2093,1,"{4100, 4104, 4106, 3849, 2805, 3787, 3771, 2639, 2641}"
1031,1,447,0,1.6247503,"\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","Int[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{\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(2 a^2 b B+a^3 (-C)-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}","\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(2 a^2 b B+a^3 (-C)-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,"((24*A*b^2 + 18*a*b*B - a^2*C + 16*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d) + ((6*b*B + a*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*b*d) + (C*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",14,13,45,0.2889,1,"{4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1032,1,346,0,1.1858888,"\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","Int[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{\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}","\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,"((8*a*A + 4*b*B + 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B + a*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*b*B + a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d)","A",13,13,45,0.2889,1,"{4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1033,1,258,0,0.8322067,"\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","Int[(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{(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}","\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,"((2*a*B + b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B + a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A - C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",12,12,45,0.2667,1,"{4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1034,1,277,0,0.9036502,"\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","Int[(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]","-\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)}}","-\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,"(-2*(A*b^2 - a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",12,12,45,0.2667,1,"{4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1035,1,273,0,0.8066913,"\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","Int[(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]","-\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)}","-\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,"(-2*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2)) + (2*(A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d*Sqrt[Sec[c + d*x]])","A",9,9,45,0.2000,1,"{4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1036,1,360,0,1.173141,"\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","Int[(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]","-\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(a^2 b (19 A+35 C)+63 a^3 B-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)}","-\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(a^2 b (19 A+35 C)+63 a^3 B-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,"(2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2)) + (2*(A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d*Sqrt[Sec[c + d*x]])","A",10,9,45,0.2000,1,"{4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1037,1,457,0,1.6436238,"\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","Int[(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]","-\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(a^2 b (13 A+21 C)+75 a^3 B-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(6 a^2 b (6 A+7 C)-75 a^3 B-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(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-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)}","-\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(a^2 b (13 A+21 C)+75 a^3 B-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(6 a^2 b (6 A+7 C)-75 a^3 B-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(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-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,"(-2*(a^2 - b^2)*(16*A*b^3 - 75*a^3*B - 24*a*b^2*B + 6*a^2*b*(6*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^4*d*Sqrt[a + b*Sec[c + d*x]]) - (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))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2)) + (2*(A*b + 9*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*a*d*Sec[c + d*x]^(5/2)) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Sec[c + d*x]^(3/2)) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^3*d*Sqrt[Sec[c + d*x]])","A",11,9,45,0.2000,1,"{4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1038,1,551,0,2.1621462,"\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","Int[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{\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(24 a^2 b B-9 a^3 C+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(136 a^2 b B-3 a^3 C+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(24 a^2 b B-9 a^3 C+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(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 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}","\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(24 a^2 b B-9 a^3 C+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(136 a^2 b B-3 a^3 C+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(24 a^2 b B-9 a^3 C+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(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 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,"((136*a^2*b*B + 128*b^3*B - 3*a^3*C + 12*a*b^2*(28*A + 19*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(192*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(64*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*b*d) + ((8*b*B + 3*a*C)*Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d)","A",15,13,45,0.2889,1,"{4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1039,1,446,0,1.6544809,"\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","Int[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{\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(6 a^2 b B+a^3 (-C)+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}","\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(6 a^2 b B+a^3 (-C)+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,"((42*a*b*B + 8*b^2*(3*A + 2*C) + a^2*(48*A + 17*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d) + ((2*b*B + a*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",14,13,45,0.2889,1,"{4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1040,1,353,0,1.1818973,"\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","Int[((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{\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}","\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,"((8*a^2*B + 4*b^2*B + a*b*(8*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*a*A - 4*b*B - 5*a*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*b*B + 3*a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d) + (C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d)","A",13,12,45,0.2667,1,"{4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1041,1,340,0,1.2313607,"\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","Int[((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{\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)}}","\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,"((6*a*b*B - b^2*(2*A - 3*C) + 2*a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*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[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b + 6*a*B - 3*b*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(2*A - 3*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",13,13,45,0.2889,1,"{4094, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1042,1,356,0,1.2545591,"\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","Int[((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]","-\frac{2 \sqrt{\sec (c+d x)} \left(-3 a^2 b (A+5 C)-5 a^3 B+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 \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 (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)}}","-\frac{2 \sqrt{\sec (c+d x)} \left(-3 a^2 b (A+5 C)-5 a^3 B+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 \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 (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,"(-2*(3*A*b^3 - 5*a^3*B + 5*a*b^2*B - 3*a^2*b*(A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(3*A*b + 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",13,12,45,0.2667,1,"{4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1043,1,359,0,1.2359395,"\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","Int[((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]","\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(-2 a^2 b (41 A+70 C)-63 a^3 B-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)}","\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(-2 a^2 b (41 A+70 C)-63 a^3 B-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,"(2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(3*A*b + 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",10,9,45,0.2000,1,"{4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1044,1,455,0,1.7090928,"\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","Int[((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]","\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(-2 a^2 b (44 A+63 C)-75 a^3 B-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(a^2 (39 A b+63 b C)+75 a^3 B-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(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-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)}","\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(-2 a^2 b (44 A+63 C)-75 a^3 B-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(a^2 (39 A b+63 b C)+75 a^3 B-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(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-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,"(2*(a^2 - b^2)*(8*A*b^3 + 75*a^3*B - 18*a*b^2*B + a^2*(39*A*b + 63*b*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (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))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b + 3*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d*Sec[c + d*x]^(5/2)) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",11,9,45,0.2000,1,"{4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1045,1,550,0,2.1894957,"\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","Int[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{\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(264 a^2 b B+15 a^3 C+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(264 a^2 b B+15 a^3 C+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(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 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}","\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(264 a^2 b B+15 a^3 C+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(264 a^2 b B+15 a^3 C+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(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 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,"((472*a^2*b*B + 128*b^3*B + 4*a*b^2*(132*A + 89*C) + a^3*(384*A + 133*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(192*d*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(64*b*d*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d) + ((8*b*B + 5*a*C)*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d) + (C*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d)","A",15,13,45,0.2889,1,"{4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1046,1,453,0,1.681357,"\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","Int[((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{\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{\sqrt{\sec (c+d x)} \left(a^2 b (96 A+59 C)+48 a^3 B+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{\left(a^2 (-(48 A-33 C))+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(30 a^2 b B+5 a^3 C+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}","\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{\sqrt{\sec (c+d x)} \left(a^2 b (96 A+59 C)+48 a^3 B+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{\left(a^2 (-(48 A-33 C))+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(30 a^2 b B+5 a^3 C+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,"((48*a^3*B + 66*a*b^2*B + 8*b^3*(3*A + 2*C) + a^2*b*(96*A + 59*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(24*d*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(8*d*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d) + ((6*b*B + 5*a*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d) + (C*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",14,12,45,0.2667,1,"{4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1047,1,427,0,1.6622926,"\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","Int[((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{\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{\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{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)}}","\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{\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{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,"((48*a^2*b*B + 12*b^3*B + 8*a^3*(A + 3*C) + a*b^2*(16*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(12*d*Sqrt[a + b*Sec[c + d*x]]) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*d*Sqrt[a + b*Sec[c + d*x]]) + ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(12*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(8*a*A - 12*b*B - 21*a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d) - (b*(4*A - 3*C)*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]])","A",14,13,45,0.2889,1,"{4094, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1048,1,419,0,1.6451763,"\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","Int[((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{\sqrt{\sec (c+d x)} \left(4 a^2 b (4 A+15 C)+10 a^3 B+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{\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{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)}}","\frac{\sqrt{\sec (c+d x)} \left(4 a^2 b (4 A+15 C)+10 a^3 B+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{\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{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,"((10*a^3*B + 20*a*b^2*B - b^3*(16*A - 15*C) + 4*a^2*b*(4*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*d*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*b*B + 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (b*(16*A*b + 10*a*B - 15*b*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(A*b + a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Sqrt[Sec[c + d*x]]) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Sec[c + d*x]^(3/2))","A",14,13,45,0.2889,1,"{4094, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1049,1,441,0,1.6708251,"\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","Int[((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]","\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 \sqrt{\sec (c+d x)} \left(10 a^2 b^2 (A-7 C)-5 a^4 (5 A+7 C)-56 a^3 b B+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 \left(5 a^2 b (29 A+49 C)+63 a^3 B+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 (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)}}","\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 \sqrt{\sec (c+d x)} \left(10 a^2 b^2 (A-7 C)-5 a^4 (5 A+7 C)-56 a^3 b B+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 \left(5 a^2 b (29 A+49 C)+63 a^3 B+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 (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,"(-2*(15*A*b^4 - 56*a^3*b*B + 56*a*b^3*B + 10*a^2*b^2*(A - 7*C) - 5*a^4*(5*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b + 7*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d*Sec[c + d*x]^(3/2)) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d*Sec[c + d*x]^(5/2))","A",14,12,45,0.2667,1,"{4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1050,1,452,0,1.7458712,"\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","Int[((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]","\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(a^2 b (163 A+231 C)+75 a^3 B+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(-6 a^2 b (19 A+28 C)-75 a^3 B-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(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-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)}","\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(a^2 b (163 A+231 C)+75 a^3 B+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(-6 a^2 b (19 A+28 C)-75 a^3 B-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(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-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,"(-2*(a^2 - b^2)*(10*A*b^3 - 75*a^3*B - 45*a*b^2*B - 6*a^2*b*(19*A + 28*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(315*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (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))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d*Sec[c + d*x]^(3/2)) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b + 9*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(63*d*Sec[c + d*x]^(5/2)) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d*Sec[c + d*x]^(7/2))","A",11,9,45,0.2000,1,"{4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1051,1,565,0,2.298664,"\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","Int[((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]","\frac{2 \sin (c+d x) \left(5 a^2 b (229 A+297 C)+539 a^3 B+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(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(-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-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(15 a^2 b^2 (19 A+33 C)+75 a^4 (9 A+11 C)+1254 a^3 b B-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(15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-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)}","\frac{2 \sin (c+d x) \left(5 a^2 b (229 A+297 C)+539 a^3 B+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(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(-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-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(15 a^2 b^2 (19 A+33 C)+75 a^4 (9 A+11 C)+1254 a^3 b B-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(15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-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,"(2*(a^2 - b^2)*(40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B + 75*a^4*(9*A + 11*C) + 15*a^2*b^2*(19*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3465*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (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))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(231*d*Sec[c + d*x]^(5/2)) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d*Sec[c + d*x]^(3/2)) - (2*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d*Sqrt[Sec[c + d*x]]) + (2*(5*A*b + 11*a*B)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d*Sec[c + d*x]^(7/2)) + (2*A*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d*Sec[c + d*x]^(9/2))","A",12,9,45,0.2000,1,"{4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1052,1,350,0,1.1448486,"\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","Int[(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{\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}","\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,"((4*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(4*b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B - 3*a*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + ((4*b*B - 3*a*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d) + (C*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d)","A",13,12,45,0.2667,1,"{4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1053,1,260,0,0.8397858,"\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","Int[(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 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}}}","\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*A + C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) - (C*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (C*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d)","A",12,12,45,0.2667,1,"{4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1054,1,219,0,0.6181266,"\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","Int[(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]","-\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)}}","-\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,"(-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + (2*A*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",11,11,45,0.2444,1,"{4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1055,1,216,0,0.5210096,"\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","Int[(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{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)}}","\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,"(2*(2*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d*Sqrt[Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1056,1,291,0,0.8171014,"\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","Int[(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]","-\frac{2 \sqrt{\sec (c+d x)} \left(a^2 b (7 A+15 C)-5 a^3 B-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 \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 (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 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}","-\frac{2 \sqrt{\sec (c+d x)} \left(a^2 b (7 A+15 C)-5 a^3 B-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 \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 (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 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"(-2*(8*A*b^3 - 5*a^3*B - 10*a*b^2*B + a^2*b*(7*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d*Sec[c + d*x]^(3/2)) - (2*(4*A*b - 5*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d*Sqrt[Sec[c + d*x]])","A",9,8,45,0.1778,1,"{4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1057,1,380,0,1.1777492,"\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","Int[(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]","\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 \sqrt{\sec (c+d x)} \left(2 a^2 b^2 (16 A+35 C)+5 a^4 (5 A+7 C)-49 a^3 b B-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 \left(a^2 (44 A b+70 b C)-63 a^3 B-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 (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 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 a d \sec ^{\frac{5}{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 \sqrt{\sec (c+d x)} \left(2 a^2 b^2 (16 A+35 C)+5 a^4 (5 A+7 C)-49 a^3 b B-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 \left(a^2 (44 A b+70 b C)-63 a^3 B-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 (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 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 a d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*(48*A*b^4 - 49*a^3*b*B - 56*a*b^3*B + 5*a^4*(5*A + 7*C) + 2*a^2*b^2*(16*A + 35*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(105*a^4*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*A*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*a*d*Sec[c + d*x]^(5/2)) - (2*(6*A*b - 7*a*B)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a^2*d*Sec[c + d*x]^(3/2)) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^3*d*Sqrt[Sec[c + d*x]])","A",10,8,45,0.1778,1,"{4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1058,1,253,0,1.0789548,"\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","Int[(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{(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}}}","\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,"((2*a*A + b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(d*Sqrt[a + b*Sec[c + d*x]]) - (B*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (B*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/d","A",13,13,54,0.2407,1,"{4072, 4031, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1059,1,393,0,1.3542968,"\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","Int[(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{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)}}","-\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,"(C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*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[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d)","A",13,13,45,0.2889,1,"{4098, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1060,1,311,0,0.9770167,"\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","Int[(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]","-\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)}}","-\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,"(2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",12,12,45,0.2667,1,"{4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1061,1,249,0,0.6054766,"\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","Int[(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]","\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(a^2 (-(A-C))-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)}}","\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(a^2 (-(A-C))-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,"(-2*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(a^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^2 - a*b*B - a^2*(A - C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1062,1,350,0,0.9432835,"\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","Int[(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]","-\frac{2 \sin (c+d x) \left(a^2 (-(A-3 C))-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(-a^2 (5 A b-3 b C)+3 a^3 B-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}}}","-\frac{2 \sin (c+d x) \left(a^2 (-(A-3 C))-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(-a^2 (5 A b-3 b C)+3 a^3 B-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,"(2*(8*A*b^2 - 6*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",9,9,45,0.2000,1,"{4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1063,1,461,0,1.3726766,"\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","Int[(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]","-\frac{2 \sin (c+d x) \left(a^2 (-(A-5 C))-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(-a^2 (9 A b-15 b C)+5 a^3 B-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(6 a^2 b (2 A+5 C)-5 a^3 B-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(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-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}}}","-\frac{2 \sin (c+d x) \left(a^2 (-(A-5 C))-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(-a^2 (9 A b-15 b C)+5 a^3 B-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(6 a^2 b (2 A+5 C)-5 a^3 B-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(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-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,"(-2*(48*A*b^3 - 5*a^3*B - 40*a*b^2*B + 6*a^2*b*(2*A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^4*d*Sqrt[a + b*Sec[c + d*x]]) - (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))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]])","A",10,9,45,0.2000,1,"{4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1064,1,563,0,1.9470442,"\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","Int[(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{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{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 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(26 a^2 b^2 C+6 a^3 b B-15 a^4 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{\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{\left(26 a^2 b^2 C+6 a^3 b B-15 a^4 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)}}","-\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{2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 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(26 a^2 b^2 C+6 a^3 b B-15 a^4 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{\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{\left(26 a^2 b^2 C+6 a^3 b B-15 a^4 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,"((2*A*b^2 - 2*a*b*B + 5*a^2*C - 3*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*b^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^3*d*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(5/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d)","A",14,13,45,0.2889,1,"{4098, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1065,1,447,0,1.4490144,"\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","Int[(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]","-\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 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 b^2 (3 A+7 C)-3 a^4 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 \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 \left(a^2 b^2 (3 A+7 C)-3 a^4 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)}}","-\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 \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^2 b^2 (3 A+7 C)-3 a^4 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 \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 \left(a^2 b^2 (3 A+7 C)-3 a^4 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,"(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a*b*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(b^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",13,12,45,0.2667,1,"{4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1066,1,378,0,1.0374481,"\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","Int[(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]","\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-5 a^2 b^2 (A+C)+2 a^3 b B+a^4 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)}}-\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(a^2 (-(3 A+C))+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(-2 a^2 b (3 A+2 C)+3 a^3 B+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(-5 a^2 b^2 (A+C)+2 a^3 b B+a^4 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)}}-\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(a^2 (-(3 A+C))+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(-2 a^2 b (3 A+2 C)+3 a^3 B+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}}}",1,"(-2*(2*A*b^2 + a*b*B - a^2*(3*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) - (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 + 2*a^3*b*B + 2*a*b^3*B + a^4*C - 5*a^2*b^2*(A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*b*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,9,45,0.2000,1,"{4098, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1067,1,401,0,1.034339,"\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","Int[(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]","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 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 \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(-a^2 b (9 A+C)+3 a^3 B-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 \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-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}}}","-\frac{2 \sin (c+d x) \sqrt{\sec (c+d x)} \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 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 \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(-a^2 b (9 A+C)+3 a^3 B-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 \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-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,"(2*(8*A*b^3 + 3*a^3*B - 2*a*b^2*B - a^2*b*(9*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (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))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]])","A",9,8,45,0.1778,1,"{4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1068,1,521,0,1.6024842,"\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","Int[(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]","\frac{2 \sin (c+d x) \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-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(10 a^2 A b^2-7 a^3 b B+4 a^4 C+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 \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 \sqrt{\sec (c+d x)} \left(-2 a^2 b^2 (8 A-C)+a^4 (-(A+3 C))+9 a^3 b B-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(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-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}}}","\frac{2 \sin (c+d x) \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-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(10 a^2 A b^2-7 a^3 b B+4 a^4 C+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 \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 \sqrt{\sec (c+d x)} \left(-2 a^2 b^2 (8 A-C)+a^4 (-(A+3 C))+9 a^3 b B-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(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-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,"(-2*(16*A*b^4 + 9*a^3*b*B - 8*a*b^3*B - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(3*a^4*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (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))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",10,9,45,0.2000,1,"{4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1069,1,663,0,2.2268465,"\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","Int[(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]","\frac{2 \sin (c+d x) \left(-a^2 b^2 (71 A-15 C)+a^4 (3 A-35 C)+50 a^3 b B-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(-2 a^2 b^2 (6 A-C)+9 a^3 b B-6 a^4 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(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(-2 a^2 b^3 (49 A-10 C)+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-5 a^5 B-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(-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)+80 a^3 b^2 B+5 a^5 B-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(5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)+140 a^3 b^3 B-40 a^5 b B-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}}}","\frac{2 \sin (c+d x) \left(-a^2 b^2 (71 A-15 C)+a^4 (3 A-35 C)+50 a^3 b B-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(-2 a^2 b^2 (6 A-C)+9 a^3 b B-6 a^4 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(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(-2 a^2 b^3 (49 A-10 C)+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-5 a^5 B-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(-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)+80 a^3 b^2 B+5 a^5 B-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(5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)+140 a^3 b^3 B-40 a^5 b B-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,"(2*(128*A*b^5 + 5*a^5*B + 80*a^3*b^2*B - 80*a*b^4*B - 4*a^2*b^3*(29*A - 10*C) - a^4*b*(17*A + 45*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[Sec[c + d*x]])/(15*a^5*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (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) - 4*a^2*b^4*(53*A - 10*C) + 3*a^6*(3*A + 5*C))*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*(8*A*b^4 + 9*a^3*b*B - 5*a*b^3*B - 2*a^2*b^2*(6*A - C) - 6*a^4*C)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + (2*(48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B + a^4*(3*A - 35*C) - a^2*b^2*(71*A - 15*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d*Sec[c + d*x]^(3/2)) - (2*(64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B + 2*a^4*b*(7*A - 20*C) - 2*a^2*b^3*(49*A - 10*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d*Sqrt[Sec[c + d*x]])","A",11,9,45,0.2000,1,"{4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1070,0,0,0,0.3228525,"\int (a+b \sec (c+d x))^{2/3} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(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,"(Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -5/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(2/3), x]","A",0,0,0,0,-1,"{}"
1071,0,0,0,0.3045656,"\int \sqrt[3]{a+b \sec (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[(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,"(Sqrt[2]*(a + b)*C*AppellF1[1/2, 1/2, -4/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(1/3), x]","A",0,0,0,0,-1,"{}"
1072,0,0,0,0.303332,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Int[(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,"(Sqrt[2]*C*AppellF1[1/2, 1/2, -2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(2/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(1/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(-1/3), x]","A",0,0,0,0,-1,"{}"
1073,0,0,0,0.3116736,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Int[(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,"(Sqrt[2]*C*AppellF1[1/2, 1/2, -1/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^(1/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^(1/3)) + (Sqrt[2]*(b*B - a*C)*AppellF1[1/2, 1/2, 2/3, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*((a + b*Sec[c + d*x])/(a + b))^(2/3)*Tan[c + d*x])/(b*d*Sqrt[1 + Sec[c + d*x]]*(a + b*Sec[c + d*x])^(2/3)) + A*Defer[Int][(a + b*Sec[c + d*x])^(-2/3), x]","A",0,0,0,0,-1,"{}"
1074,0,0,0,0.2400096,"\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","Int[(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,"(Sqrt[2]*b*(a + b)*C*AppellF1[1/2, 1/2, -1 - m, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^m*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))^m) + (b*B - a*C)*Defer[Int][(a + b*Sec[c + d*x])^(1 + m), x]","A",0,0,0,0,-1,"{}"
1075,1,80,0,0.0790408,"\int \cos ^{\frac{9}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(9/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",4,4,23,0.1739,1,"{4066, 3014, 2635, 2639}"
1076,1,80,0,0.0743782,"\int \cos ^{\frac{7}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[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)}{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{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])/(21*d) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",4,4,23,0.1739,1,"{4066, 3014, 2635, 2641}"
1077,1,50,0,0.0592337,"\int \cos ^{\frac{5}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[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)}{5 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(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])/(5*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",3,3,23,0.1304,1,"{4066, 3014, 2639}"
1078,1,48,0,0.0600691,"\int \cos ^{\frac{3}{2}}(c+d x) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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 (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,"(2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",3,3,23,0.1304,1,"{4066, 3014, 2641}"
1079,1,44,0,0.0618465,"\int \sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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)}}","\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,"(2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",3,3,23,0.1304,1,"{4066, 3012, 2639}"
1080,1,48,0,0.0603084,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(A + 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)}{3 d}+\frac{2 C \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}","\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])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",3,3,23,0.1304,1,"{4066, 3012, 2641}"
1081,1,80,0,0.0747611,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2),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)}","-\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)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",4,4,23,0.1739,1,"{4066, 3012, 2636, 2639}"
1082,1,80,0,0.0755043,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + C*Sec[c + d*x]^2)/Cos[c + d*x]^(5/2),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)}","\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)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2))","A",4,4,23,0.1739,1,"{4066, 3012, 2636, 2641}"
1083,1,165,0,0.2414356,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*a*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",8,7,33,0.2121,1,"{4114, 3034, 3023, 2748, 2635, 2641, 2639}"
1084,1,134,0,0.2185758,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*a*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,7,33,0.2121,1,"{4114, 3034, 3023, 2748, 2639, 2635, 2641}"
1085,1,101,0,0.2031626,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*a*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,33,0.1818,1,"{4114, 3034, 3023, 2748, 2641, 2639}"
1086,1,95,0,0.2038751,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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)}}","\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,"(2*a*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,33,0.1818,1,"{4114, 3032, 3023, 2748, 2641, 2639}"
1087,1,95,0,0.2073928,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2),x]","\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)}}","\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,"(2*a*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,33,0.1818,1,"{4114, 3032, 3021, 2748, 2641, 2639}"
1088,1,132,0,0.222512,"\int \frac{(a+a \sec (c+d x)) \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\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)}","\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,"(-2*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,7,33,0.2121,1,"{4114, 3032, 3021, 2748, 2636, 2639, 2641}"
1089,1,165,0,0.2446989,"\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","Int[((a + a*Sec[c + d*x])*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\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)}","\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,"(-2*a*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,33,0.2121,1,"{4114, 3032, 3021, 2748, 2636, 2641, 2639}"
1090,1,230,0,0.5181567,"\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","Int[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(4*a^2*(7*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^2*(25*A + 33*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^2*(25*A + 33*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(7*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(89*A + 99*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (8*A*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)","A",10,9,35,0.2571,1,"{4114, 3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
1091,1,197,0,0.4859662,"\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","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(16*a^2*(2*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (8*A*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)","A",9,9,35,0.2571,1,"{4114, 3046, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1092,1,164,0,0.4660333,"\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","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(4*a^2*(3*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(3*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (8*A*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",8,8,35,0.2286,1,"{4114, 3046, 2976, 2968, 3023, 2748, 2641, 2639}"
1093,1,158,0,0.4641379,"\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","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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)}}","\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,"(16*a^2*A*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*A - 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 5*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",8,8,35,0.2286,1,"{4114, 3044, 2976, 2968, 3023, 2748, 2641, 2639}"
1094,1,154,0,0.4676166,"\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","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (8*a^2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(A - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",8,8,35,0.2286,1,"{4114, 3044, 2975, 2968, 3023, 2748, 2641, 2639}"
1095,1,156,0,0.4823273,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(-16*a^2*C*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(15*A + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",8,8,35,0.2286,1,"{4114, 3044, 2975, 2968, 3021, 2748, 2641, 2639}"
1096,1,197,0,0.5124745,"\int \frac{(a+a \sec (c+d x))^2 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\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)}","\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,"(-4*a^2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^2*(7*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 33*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))","A",9,9,35,0.2571,1,"{4114, 3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1097,1,230,0,0.5417322,"\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","Int[((a + a*Sec[c + d*x])^2*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\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)}","\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,"(-16*a^2*(3*A + 2*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(21*A + 19*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (16*a^2*(3*A + 2*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (8*C*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))","A",10,9,35,0.2571,1,"{4114, 3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
1098,1,279,0,0.685408,"\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","Int[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(4*a^3*(175*A + 221*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (4*a^3*(95*A + 121*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(95*A + 121*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(175*A + 221*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(585*d) + (40*a^3*(118*A + 143*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(13*d) + (12*A*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(143*a*d) + (2*(145*A + 143*C)*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(1287*d)","A",11,9,35,0.2571,1,"{4114, 3046, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
1099,1,246,0,0.6452048,"\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","Int[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(105*A + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(105*A + 143*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (8*a^3*(35*A + 44*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(385*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (4*A*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d) + (2*(35*A + 33*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d)","A",10,9,35,0.2571,1,"{4114, 3046, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1100,1,213,0,0.6225286,"\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","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(4*a^3*(17*A + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(16*A + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (4*A*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(73*A + 63*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)","A",9,8,35,0.2286,1,"{4114, 3046, 2976, 2968, 3023, 2748, 2641, 2639}"
1101,1,215,0,0.6280099,"\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","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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)}}","\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,"(4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(41*A - 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 7*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) + (2*(11*A - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",9,8,35,0.2286,1,"{4114, 3044, 2976, 2968, 3023, 2748, 2641, 2639}"
1102,1,211,0,0.6287904,"\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","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(4*a^3*(9*A - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (8*a^3*(3*A - 10*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + (2*(3*A - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)","A",9,9,35,0.2571,1,"{4114, 3044, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
1103,1,213,0,0.63446,"\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","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(4*a^3*(5*A - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(5*A + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(3/2)) + (2*(5*A + 11*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",9,8,35,0.2286,1,"{4114, 3044, 2975, 2968, 3023, 2748, 2641, 2639}"
1104,1,213,0,0.6515243,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2),x]","\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)}","\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,"(-4*a^3*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(70*A + 53*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (12*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 7*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",9,8,35,0.2286,1,"{4114, 3044, 2975, 2968, 3021, 2748, 2641, 2639}"
1105,1,246,0,0.6781749,"\int \frac{(a+a \sec (c+d x))^3 \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\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)}","\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,"(-4*a^3*(27*A + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (8*a^3*(21*A + 16*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(27*A + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(63*A + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))","A",10,9,35,0.2571,1,"{4114, 3044, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1106,1,279,0,0.7130535,"\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","Int[((a + a*Sec[c + d*x])^3*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\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)}","\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,"(-4*a^3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(143*A + 105*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^3*(44*A + 35*C)*Sin[c + d*x])/(385*d*Cos[c + d*x]^(5/2)) + (4*a^3*(143*A + 105*C)*Sin[c + d*x])/(231*d*Cos[c + d*x]^(3/2)) + (4*a^3*(7*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (4*C*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(33*a*d*Cos[c + d*x]^(9/2)) + (2*(33*A + 35*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2))","A",11,9,35,0.2571,1,"{4114, 3044, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
1107,1,192,0,0.2874306,"\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","Int[(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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,"(-3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(9*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(9*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((7*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((9*A + 7*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",8,6,35,0.1714,1,"{4114, 3042, 2748, 2635, 2639, 2641}"
1108,1,159,0,0.2646704,"\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","Int[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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}","-\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,"(3*(7*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((7*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,6,35,0.1714,1,"{4114, 3042, 2748, 2635, 2641, 2639}"
1109,1,122,0,0.2406371,"\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","Int[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","\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}","\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*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((5*A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,35,0.1714,1,"{4114, 3042, 2748, 2639, 2635, 2641}"
1110,1,84,0,0.2308355,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{a+a \sec (c+d x)} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x]),x]","-\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)}","-\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*A + C)*EllipticE[(c + d*x)/2, 2])/(a*d) - ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,35,0.1429,1,"{4114, 3042, 2748, 2641, 2639}"
1111,1,112,0,0.2437819,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])),x]","\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)}","\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,"-(((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((A - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((A + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))","A",6,6,35,0.1714,1,"{4114, 3042, 2748, 2636, 2639, 2641}"
1112,1,150,0,0.2627722,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\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)}}","\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,"((A + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((A + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))","A",7,6,35,0.1714,1,"{4114, 3042, 2748, 2636, 2641, 2639}"
1113,1,192,0,0.273037,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","-\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)}}","-\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*(5*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A + 7*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((3*A + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(5*A + 7*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))","A",8,6,35,0.1714,1,"{4114, 3042, 2748, 2636, 2639, 2641}"
1114,1,196,0,0.4046562,"\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","Int[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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,"(4*(14*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(3*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (4*(14*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((3*A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",8,7,35,0.2000,1,"{4114, 3042, 2977, 2748, 2635, 2641, 2639}"
1115,1,161,0,0.3812117,"\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","Int[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","\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}","\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*A + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + (2*(5*A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(5*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((7*A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,35,0.2000,1,"{4114, 3042, 2977, 2748, 2639, 2635, 2641}"
1116,1,130,0,0.3489907,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^2} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^2,x]","-\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}","-\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*A*EllipticE[(c + d*x)/2, 2])/(a^2*d) - ((5*A - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((5*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,35,0.1714,1,"{4114, 3042, 2977, 2748, 2641, 2639}"
1117,1,125,0,0.3582316,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","\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}","\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,"-(((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + (2*(A + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,35,0.1714,1,"{4114, 3042, 2978, 2748, 2641, 2639}"
1118,1,151,0,0.3762756,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\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}","\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*C*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((A - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (4*C*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((A - 5*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)","A",7,7,35,0.2000,1,"{4114, 3042, 2978, 2748, 2636, 2639, 2641}"
1119,1,189,0,0.4144052,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","\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}","\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,"((A + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(A + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + (2*(A + 5*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((A + 7*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A + 7*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)","A",8,7,35,0.2000,1,"{4114, 3042, 2978, 2748, 2636, 2641, 2639}"
1120,1,250,0,0.5881117,"\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","Int[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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,"(7*(33*A + 7*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((63*A + 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((63*A + 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(33*A + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(6*A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((63*A + 13*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",9,7,35,0.2000,1,"{4114, 3042, 2977, 2748, 2635, 2641, 2639}"
1121,1,209,0,0.5422443,"\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","Int[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","\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}","\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*A + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((11*A + C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((11*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a^3*d) - ((A + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,35,0.2000,1,"{4114, 3042, 2977, 2748, 2639, 2635, 2641}"
1122,1,186,0,0.5406633,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{(a+a \sec (c+d x))^3} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^3,x]","-\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}","-\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*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(4*A - C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,35,0.1714,1,"{4114, 3042, 2977, 2748, 2641, 2639}"
1123,1,184,0,0.5310156,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","\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}","\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*A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - (2*(3*A - 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((9*A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",7,7,35,0.2000,1,"{4114, 3042, 2977, 2978, 2748, 2641, 2639}"
1124,1,180,0,0.5231939,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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,"-((A - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + (2*(2*A - 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,35,0.1714,1,"{4114, 3042, 2978, 2748, 2641, 2639}"
1125,1,209,0,0.5654634,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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,"((A - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - 49*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) + (2*(A - 4*C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) + ((A - 13*C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))","A",8,7,35,0.2000,1,"{4114, 3042, 2978, 2748, 2636, 2639, 2641}"
1126,1,242,0,0.5667259,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3),x]","\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}","\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*A + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 11*C)*EllipticF[(c + d*x)/2, 2])/(2*a^3*d) + ((A + 11*C)*Sin[c + d*x])/(2*a^3*d*Cos[c + d*x]^(3/2)) - ((9*A + 119*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) - (2*C*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((9*A + 119*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))","A",9,7,35,0.2000,1,"{4114, 3042, 2978, 2748, 2636, 2641, 2639}"
1127,1,213,0,0.5704342,"\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","Int[Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(16*a*(16*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)","A",6,5,37,0.1351,1,"{4265, 4087, 4015, 3805, 3804}"
1128,1,168,0,0.4928025,"\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","Int[Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}}","\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,"(4*a*(24*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",5,5,37,0.1351,1,"{4265, 4087, 4015, 3805, 3804}"
1129,1,122,0,0.4228754,"\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","Int[Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*a*(7*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",4,4,37,0.1081,1,"{4265, 4087, 4013, 3804}"
1130,1,136,0,0.3990669,"\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","Int[Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,37,0.1351,1,"{4265, 4087, 4015, 3801, 215}"
1131,1,135,0,0.4094909,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2),x]","\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}","\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[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,37,0.1351,1,"{4265, 4089, 4015, 3801, 215}"
1132,1,144,0,0.4064219,"\int \frac{\sqrt{a+a \sec (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{\cos (c+d x)}} \, dx","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],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}}","\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,"(Sqrt[a]*(8*A + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*C*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))","A",5,5,37,0.1351,1,"{4265, 4089, 4016, 3801, 215}"
1133,1,189,0,0.4846954,"\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","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),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}}","\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,"(Sqrt[a]*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*C*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*A + 5*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))","A",6,6,37,0.1622,1,"{4265, 4089, 4016, 3803, 3801, 215}"
1134,1,234,0,0.5725573,"\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","Int[(Sqrt[a + a*Sec[c + d*x]]*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),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}}","\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,"(Sqrt[a]*(48*A + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*C*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 35*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))","A",7,6,37,0.1622,1,"{4265, 4089, 4016, 3803, 3801, 215}"
1135,1,266,0,0.8066455,"\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","Int[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(16*a^2*(112*A + 143*C)*Sin[c + d*x])/(1155*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(112*A + 143*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(112*A + 143*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(385*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(28*A + 33*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(231*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(33*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)","A",7,6,37,0.1622,1,"{4265, 4087, 4017, 4015, 3805, 3804}"
1136,1,219,0,0.7229111,"\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","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(4*a^2*(136*A + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 189*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 63*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)","A",6,6,37,0.1622,1,"{4265, 4087, 4017, 4015, 3805, 3804}"
1137,1,169,0,0.5445181,"\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","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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}","\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,"(8*a^2*(19*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 35*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (6*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)","A",5,5,37,0.1351,1,"{4265, 4087, 4013, 3809, 3804}"
1138,1,183,0,0.5961671,"\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","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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 \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}","\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^{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 \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,"(2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(4*A + 5*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,37,0.1622,1,"{4265, 4087, 4017, 4015, 3801, 215}"
1139,1,189,0,0.6002523,"\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","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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{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 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}","\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{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 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,"(3*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*(8*A - 3*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,37,0.1622,1,"{4265, 4087, 4018, 4015, 3801, 215}"
1140,1,191,0,0.6132353,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2),x]","\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{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{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)}}","\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{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{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^(3/2)*(8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(8*A - 5*C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])","A",6,6,37,0.1622,1,"{4265, 4089, 4018, 4015, 3801, 215}"
1141,1,191,0,0.6197895,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\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^{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 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)}","\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^{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 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^(3/2)*(24*A + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 19*C)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,37,0.1622,1,"{4265, 4089, 4018, 4016, 3801, 215}"
1142,1,238,0,0.731376,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\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^{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 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)}","\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^{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 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^(3/2)*(112*A + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(16*A + 13*C)*Sin[c + d*x])/(32*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(112*A + 75*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))","A",7,7,37,0.1892,1,"{4265, 4089, 4018, 4016, 3803, 3801, 215}"
1143,1,285,0,0.8079927,"\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","Int[((a + a*Sec[c + d*x])^(3/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\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{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{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)}","\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{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{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^(3/2)*(176*A + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 67*C)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 133*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (3*a*C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))","A",8,7,37,0.1892,1,"{4265, 4089, 4018, 4016, 3803, 3801, 215}"
1144,1,313,0,1.0157554,"\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","Int[Cos[c + d*x]^(13/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^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 \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}","\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^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 \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,"(16*a^3*(8368*A + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 10439*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 10439*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2224*A + 2717*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 143*C)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d) + (10*a*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*A*Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)","A",8,6,37,0.1622,1,"{4265, 4087, 4017, 4015, 3805, 3804}"
1145,1,266,0,0.9371031,"\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","Int[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^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 \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}","\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^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 \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,"(4*a^3*(568*A + 759*C)*Sin[c + d*x])/(693*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(568*A + 759*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(232*A + 297*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 33*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d) + (10*a*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d)","A",7,6,37,0.1622,1,"{4265, 4087, 4017, 4015, 3805, 3804}"
1146,1,216,0,0.6319345,"\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","Int[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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{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{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}","\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{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{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,"(64*a^3*(13*A + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 21*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(13*A + 21*C)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (10*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d)","A",6,5,37,0.1351,1,"{4265, 4087, 4013, 3809, 3804}"
1147,1,230,0,0.7810663,"\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","Int[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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 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}","\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^{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 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,"(2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(32*A + 49*C)*Sin[c + d*x])/(21*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(8*A + 7*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",7,6,37,0.1622,1,"{4265, 4087, 4017, 4015, 3801, 215}"
1148,1,230,0,0.7982736,"\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","Int[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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{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{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}","\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{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{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,"(5*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*(64*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A - 15*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,37,0.1892,1,"{4265, 4087, 4017, 4018, 4015, 3801, 215}"
1149,1,244,0,0.8128846,"\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","Int[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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 (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}","\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^{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 (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^(5/2)*(8*A + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^3*(56*A - 27*C)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (a*(4*A - 3*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",7,6,37,0.1622,1,"{4265, 4087, 4018, 4015, 3801, 215}"
1150,1,238,0,0.7927752,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} \left(A+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2),x]","\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^{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{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)}}","\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^{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{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,"(5*a^(5/2)*(8*A + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(24*A - 49*C)*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (5*a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",7,6,37,0.1622,1,"{4265, 4089, 4018, 4015, 3801, 215}"
1151,1,238,0,0.8342653,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[Cos[c + d*x]],x]","\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{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{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)}","\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{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{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^(5/2)*(304*A + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 299*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 17*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d*Cos[c + d*x]^(3/2)) + (5*a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2))","A",7,6,37,0.1622,1,"{4265, 4089, 4018, 4016, 3801, 215}"
1152,1,285,0,0.9328562,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(3/2),x]","\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^{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 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)}","\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^{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 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^(5/2)*(400*A + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 787*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(400*A + 283*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 79*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",8,7,37,0.1892,1,"{4265, 4089, 4018, 4016, 3803, 3801, 215}"
1153,1,332,0,1.0323533,"\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","Int[((a + a*Sec[c + d*x])^(5/2)*(A + C*Sec[c + d*x]^2))/Cos[c + d*x]^(5/2),x]","\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^{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 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)}","\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^{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 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^(5/2)*(1304*A + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(136*A + 109*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(768*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1015*C)*Sin[c + d*x])/(512*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 23*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(96*d*Cos[c + d*x]^(7/2)) + (a*C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(7/2))","A",9,7,37,0.1892,1,"{4265, 4089, 4018, 4016, 3803, 3801, 215}"
1154,1,244,0,0.8199358,"\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","Int[(Cos[c + d*x]^(7/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"(Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,37,0.1622,1,"{4265, 4087, 4022, 4013, 3808, 206}"
1155,1,201,0,0.6252515,"\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","Int[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"-((Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4265, 4087, 4022, 4013, 3808, 206}"
1156,1,156,0,0.4522821,"\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","Int[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","\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}}","\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,"(Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*A*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,37,0.1351,1,"{4265, 4087, 4013, 3808, 206}"
1157,1,175,0,0.485774,"\int \frac{\sqrt{\cos (c+d x)} \left(A+C \sec ^2(c+d x)\right)}{\sqrt{a+a \sec (c+d x)}} \, dx","Int[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/Sqrt[a + a*Sec[c + d*x]],x]","-\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}","-\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*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4265, 4087, 4023, 3808, 206, 3801, 215}"
1158,1,173,0,0.4842626,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\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}","\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,"-((C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4265, 4089, 4023, 3808, 206, 3801, 215}"
1159,1,223,0,0.6784294,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\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}}","-\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,"((8*A + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - (C*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,8,37,0.2162,1,"{4265, 4089, 4021, 4023, 3808, 206, 3801, 215}"
1160,1,266,0,0.8480731,"\int \frac{A+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\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}}","\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,"-((8*A + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) - (C*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((8*A + 7*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,8,37,0.2162,1,"{4265, 4089, 4021, 4023, 3808, 206, 3801, 215}"
1161,1,268,0,0.8652772,"\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","Int[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((15*A + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((49*A + 25*C)*Sin[c + d*x])/(10*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((13*A + 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((9*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,37,0.1622,1,"{4265, 4085, 4022, 4013, 3808, 206}"
1162,1,221,0,0.6680095,"\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","Int[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","\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}}","\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,"((11*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((19*A + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((7*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4265, 4085, 4022, 4013, 3808, 206}"
1163,1,172,0,0.4909609,"\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","Int[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(3/2),x]","-\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}}","-\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,"-((7*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((5*A + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",5,5,37,0.1351,1,"{4265, 4085, 4013, 3808, 206}"
1164,1,185,0,0.5281129,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\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}}","\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,"(2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",7,7,37,0.1892,1,"{4265, 4085, 4023, 3808, 206, 3801, 215}"
1165,1,228,0,0.7035997,"\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","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\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}}","\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,"(-3*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A + 3*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,8,37,0.2162,1,"{4265, 4085, 4021, 4023, 3808, 206, 3801, 215}"
1166,1,285,0,0.9105848,"\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","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\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}}","-\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,"((8*A + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A + 13*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A + 2*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - ((2*A + 7*C)*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,8,37,0.2162,1,"{4265, 4085, 4021, 4023, 3808, 206, 3801, 215}"
1167,1,315,0,1.0703935,"\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","Int[(Cos[c + d*x]^(5/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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{(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{(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}}","\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{(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{(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,"-((283*A + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((21*A + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((2671*A + 735*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((787*A + 195*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((157*A + 45*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,37,0.1892,1,"{4265, 4085, 4020, 4022, 4013, 3808, 206}"
1168,1,266,0,0.8592934,"\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","Int[(Cos[c + d*x]^(3/2)*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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{(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{(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}}","\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{(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{(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,"((163*A + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((17*A + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((299*A + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (5*(19*A + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,37,0.1892,1,"{4265, 4085, 4020, 4022, 4013, 3808, 206}"
1169,1,219,0,0.6878858,"\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","Int[(Sqrt[Cos[c + d*x]]*(A + C*Sec[c + d*x]^2))/(a + a*Sec[c + d*x])^(5/2),x]","\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{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{(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}}","\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{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{(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,"(-5*(15*A - C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((49*A + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",6,6,37,0.1622,1,"{4265, 4085, 4020, 4013, 3808, 206}"
1170,1,174,0,0.5075336,"\int \frac{A+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Int[(A + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\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}}","\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,"((19*A + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - 7*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",5,5,37,0.1351,1,"{4265, 4085, 4012, 3808, 206}"
1171,1,232,0,0.7111807,"\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","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\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}}","\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*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((5*A - 11*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",8,8,37,0.2162,1,"{4265, 4085, 4019, 4023, 3808, 206, 3801, 215}"
1172,1,277,0,0.9191933,"\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","Int[(A + C*Sec[c + d*x]^2)/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\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{(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{(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}}","\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{(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{(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,"(-5*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) + ((A - 15*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((3*A + 35*C)*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,9,37,0.2432,1,"{4265, 4085, 4019, 4021, 4023, 3808, 206, 3801, 215}"
1173,1,111,0,0.094935,"\int \cos ^{\frac{9}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(9/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(6*C*EllipticE[(c + d*x)/2, 2])/(5*d) + (10*B*EllipticF[(c + d*x)/2, 2])/(21*d) + (10*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*C*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*B*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,5,30,0.1667,1,"{4064, 2748, 2635, 2639, 2641}"
1174,1,87,0,0.0833361,"\int \cos ^{\frac{7}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(7/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),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) \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}","\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,"(6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*C*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,5,30,0.1667,1,"{4064, 2748, 2635, 2641, 2639}"
1175,1,61,0,0.0714616,"\int \cos ^{\frac{5}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(5/2)*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),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) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 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)}{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*C*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,30,0.1667,1,"{4064, 2748, 2639, 2635, 2641}"
1176,1,35,0,0.0613801,"\int \cos ^{\frac{3}{2}}(c+d x) \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[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",4,4,30,0.1333,1,"{4064, 2748, 2641, 2639}"
1177,1,57,0,0.0695075,"\int \sqrt{\cos (c+d x)} \left(B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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)}}","\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])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,30,0.1667,1,"{4064, 2748, 2636, 2639, 2641}"
1178,1,83,0,0.0789359,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Sqrt[Cos[c + d*x]],x]","-\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)}","-\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,"(-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*C*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,5,30,0.1667,1,"{4064, 2748, 2636, 2641, 2639}"
1179,1,111,0,0.0936062,"\int \frac{B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2),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)}}","\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,"(-6*C*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (6*C*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,5,30,0.1667,1,"{4064, 2748, 2636, 2639, 2641}"
1180,1,123,0,0.1296528,"\int \cos ^{\frac{7}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x] + 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)}{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}","\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,"(6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*A + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*A + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*B*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,6,31,0.1935,1,"{4064, 3023, 2748, 2635, 2641, 2639}"
1181,1,93,0,0.1161454,"\int \cos ^{\frac{5}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x] + 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)}{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}","\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*A + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*B*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,31,0.1935,1,"{4064, 3023, 2748, 2639, 2635, 2641}"
1182,1,65,0,0.1038656,"\int \cos ^{\frac{3}{2}}(c+d x) \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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}","\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,"(2*B*EllipticE[(c + d*x)/2, 2])/d + (2*(A + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,31,0.1613,1,"{4064, 3023, 2748, 2641, 2639}"
1183,1,61,0,0.1067505,"\int \sqrt{\cos (c+d x)} \left(A+B \sec (c+d x)+C \sec ^2(c+d x)\right) \, dx","Int[Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]","\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)}}","\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,"(2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (2*B*EllipticF[(c + d*x)/2, 2])/d + (2*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,31,0.1613,1,"{4064, 3021, 2748, 2641, 2639}"
1184,1,87,0,0.1209716,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sqrt{\cos (c+d x)}} \, dx","Int[(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)}{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)}","\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,"(-2*B*EllipticE[(c + d*x)/2, 2])/d + (2*(3*A + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*B*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,31,0.1935,1,"{4064, 3021, 2748, 2636, 2639, 2641}"
1185,1,123,0,0.1342583,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(3/2),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)}","-\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,"(-2*(5*A + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*B*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*B*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(5*A + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,6,31,0.1935,1,"{4064, 3021, 2748, 2636, 2641, 2639}"
1186,1,147,0,0.1522706,"\int \frac{A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/Cos[c + d*x]^(5/2),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)}","\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,"(-6*B*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*A + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*B*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(7*A + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (6*B*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,6,31,0.1935,1,"{4064, 3021, 2748, 2636, 2639, 2641}"
1187,1,175,0,0.3076652,"\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","Int[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]","\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}","\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,"(2*a*(7*A + 9*(B + C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*a*(5*(A + B) + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(5*(A + B) + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(7*A + 9*(B + C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(A + B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",8,7,41,0.1707,1,"{4112, 3033, 3023, 2748, 2635, 2641, 2639}"
1188,1,142,0,0.2794901,"\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","Int[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]","\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}","\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,"(2*a*(3*(A + B) + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(5*A + 7*(B + C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(5*A + 7*(B + C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(A + B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,7,41,0.1707,1,"{4112, 3033, 3023, 2748, 2639, 2635, 2641}"
1189,1,106,0,0.2442987,"\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","Int[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]","\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}","\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,"(2*a*(3*A + 5*(B + C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(A + B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,41,0.1463,1,"{4112, 3033, 3023, 2748, 2641, 2639}"
1190,1,98,0,0.2395491,"\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","Int[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]","\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)}}","\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,"(2*a*(A + B - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(A + 3*(B + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,41,0.1463,1,"{4112, 3031, 3023, 2748, 2641, 2639}"
1191,1,103,0,0.2486447,"\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","Int[Sqrt[Cos[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 (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)}","\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,"(2*a*(A - B - C)*EllipticE[(c + d*x)/2, 2])/d + (2*a*(3*A + 3*B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(B + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,41,0.1463,1,"{4112, 3031, 3021, 2748, 2641, 2639}"
1192,1,141,0,0.2668086,"\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","Int[((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{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)}","\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,"(-2*a*(5*A + 5*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(3*A + B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 5*B + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,7,41,0.1707,1,"{4112, 3031, 3021, 2748, 2636, 2639, 2641}"
1193,1,177,0,0.30244,"\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","Int[((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{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)}","\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,"(-2*a*(5*A + 3*(B + C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*a*(7*A + 7*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*a*(B + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*a*(7*A + 7*B + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*a*(5*A + 3*(B + C))*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,41,0.1707,1,"{4112, 3031, 3021, 2748, 2636, 2641, 2639}"
1194,1,251,0,0.5967416,"\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","Int[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]","\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}","\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,"(4*a^2*(7*A + 8*B + 9*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(50*A + 55*B + 66*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^2*(50*A + 55*B + 66*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^2*(7*A + 8*B + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a^2*(89*A + 121*B + 99*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + (2*(4*A + 11*B)*Cos[c + d*x]^(5/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(99*d)","A",10,9,43,0.2093,1,"{4112, 3045, 2976, 2968, 3023, 2748, 2635, 2641, 2639}"
1195,1,215,0,0.5539753,"\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","Int[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{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}","\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,"(4*a^2*(8*A + 9*B + 12*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(5*A + 6*B + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^2*(5*A + 6*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(19*A + 27*B + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + (2*(4*A + 9*B)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d)","A",9,9,43,0.2093,1,"{4112, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1196,1,179,0,0.5461426,"\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","Int[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]","\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}","\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,"(4*a^2*(3*A + 4*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(6*A + 7*B + 14*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(33*A + 49*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(4*A + 7*B)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",8,8,43,0.1860,1,"{4112, 3045, 2976, 2968, 3023, 2748, 2641, 2639}"
1197,1,170,0,0.5293526,"\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","Int[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{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)}}","\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,"(4*a^2*(4*A + 5*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(A + 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(7*A + 5*B - 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 5*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(5*d)","A",8,8,43,0.1860,1,"{4112, 3043, 2976, 2968, 3023, 2748, 2641, 2639}"
1198,1,170,0,0.5375063,"\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","Int[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{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)}","\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,"(4*a^2*(A - C)*EllipticE[(c + d*x)/2, 2])/d + (4*a^2*(2*A + 3*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(A - 3*B - 5*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(3*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",8,8,43,0.1860,1,"{4112, 3043, 2975, 2968, 3023, 2748, 2641, 2639}"
1199,1,174,0,0.541798,"\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","Int[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{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)}","\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,"(-4*a^2*(5*B + 4*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(3*A + 2*B + C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a^2*(15*A + 25*B + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",8,8,43,0.1860,1,"{4112, 3043, 2975, 2968, 3021, 2748, 2641, 2639}"
1200,1,215,0,0.5734181,"\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","Int[((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]","\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)}","\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,"(-4*a^2*(5*A + 4*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^2*(14*A + 7*B + 6*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(35*A + 49*B + 33*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^2*(5*A + 4*B + 3*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(7*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2))","A",9,9,43,0.2093,1,"{4112, 3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1201,1,251,0,0.5990835,"\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","Int[((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{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)}","\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,"(-4*a^2*(12*A + 9*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^2*(7*A + 6*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a^2*(21*A + 27*B + 19*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(5/2)) + (4*a^2*(7*A + 6*B + 5*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (4*a^2*(12*A + 9*B + 8*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(9*B + 4*C)*(a^2 + a^2*Cos[c + d*x])*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2))","A",10,9,43,0.2093,1,"{4112, 3043, 2975, 2968, 3021, 2748, 2636, 2641, 2639}"
1202,1,267,0,0.7319354,"\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","Int[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]","\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}","\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,"(4*a^3*(15*A + 17*B + 21*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(105*A + 121*B + 143*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^3*(105*A + 121*B + 143*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^3*(210*A + 253*B + 264*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + (2*(6*A + 11*B)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*a*d) + (2*(105*A + 143*B + 99*C)*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(693*d)","A",10,9,43,0.2093,1,"{4112, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1203,1,231,0,0.7165298,"\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","Int[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{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}","\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,"(4*a^3*(17*A + 21*B + 27*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(11*A + 13*B + 21*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(32*A + 41*B + 42*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*(2*A + 3*B)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d) + (2*(73*A + 99*B + 63*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d)","A",9,8,43,0.1860,1,"{4112, 3045, 2976, 2968, 3023, 2748, 2641, 2639}"
1204,1,227,0,0.7045556,"\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","Int[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{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)}}","\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,"(4*a^3*(7*A + 9*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(13*A + 21*B + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(41*A + 42*B - 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(A - 7*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*a*d) + (2*(11*A + 7*B - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(35*d)","A",9,8,43,0.1860,1,"{4112, 3043, 2976, 2968, 3023, 2748, 2641, 2639}"
1205,1,226,0,0.7337706,"\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","Int[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{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)}","\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,"(4*a^3*(9*A + 5*B - 5*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(3*A + 5*(B + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^3*(6*A - 5*B - 20*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(B + 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) + (2*(3*A - 15*B - 35*C)*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d)","A",9,9,43,0.2093,1,"{4112, 3043, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
1206,1,231,0,0.7093554,"\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","Int[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{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)}","\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,"(4*a^3*(5*A - 5*B - 9*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(5*A + 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*d) - (4*a^3*(5*A + 20*B + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(15*a*d*Cos[c + d*x]^(3/2)) + (2*(15*A + 35*B + 33*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]])","A",9,8,43,0.1860,1,"{4112, 3043, 2975, 2968, 3023, 2748, 2641, 2639}"
1207,1,231,0,0.7270957,"\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","Int[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{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)}","\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,"(-4*a^3*(5*A + 9*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (4*a^3*(35*A + 21*B + 13*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(140*A + 147*B + 106*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(7*B + 6*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(35*a*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 9*B + 7*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",9,8,43,0.1860,1,"{4112, 3043, 2975, 2968, 3021, 2748, 2641, 2639}"
1208,1,267,0,0.7506423,"\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","Int[((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{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)}","\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,"(-4*a^3*(27*A + 21*B + 17*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (4*a^3*(21*A + 13*B + 11*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^3*(42*A + 41*B + 32*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*a^3*(27*A + 21*B + 17*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(3*B + 2*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*a*d*Cos[c + d*x]^(7/2)) + (2*(63*A + 99*B + 73*C)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2))","A",10,9,43,0.2093,1,"{4112, 3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1209,1,310,0,0.9091668,"\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","Int[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]","\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{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{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 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}","\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{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{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 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,"(8*a^4*(185*A + 208*B + 247*C)*EllipticE[(c + d*x)/2, 2])/(195*d) + (8*a^4*(100*A + 113*B + 132*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (8*a^4*(100*A + 113*B + 132*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (4*a^4*(5255*A + 6019*B + 6721*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d) + (2*a*(8*A + 13*B)*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(143*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(13*d) + (2*(13*A + 17*B + 11*C)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (4*(1355*A + 1612*B + 1573*C)*Cos[c + d*x]^(3/2)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(9009*d)","A",11,9,43,0.2093,1,"{4112, 3045, 2976, 2968, 3023, 2748, 2639, 2635, 2641}"
1210,1,274,0,0.8829757,"\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","Int[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{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{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{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 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}","\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{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{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 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,"(8*a^4*(16*A + 19*B + 24*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(113*A + 132*B + 187*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^4*(667*A + 803*B + 913*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(1155*d) + (2*a*(8*A + 11*B)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d) + (2*(43*A + 55*B + 33*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (4*(769*A + 946*B + 891*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(3465*d)","A",10,8,43,0.1860,1,"{4112, 3045, 2976, 2968, 3023, 2748, 2641, 2639}"
1211,1,270,0,0.8877952,"\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","Int[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{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{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{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 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)}}","\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{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{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 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,"(8*a^4*(19*A + 24*B + 21*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(12*A + 17*B + 28*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^4*(73*A + 83*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(A - 9*C)*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*(5*A + 3*B - 21*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (4*(86*A + 81*B - 126*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(315*d)","A",10,8,43,0.1860,1,"{4112, 3043, 2976, 2968, 3023, 2748, 2641, 2639}"
1212,1,269,0,0.8791202,"\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","Int[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{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{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{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 (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)}","\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{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{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 (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,"(8*a^4*(8*A + 7*B)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(17*A + 28*B + 35*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^4*(83*A + 7*B - 175*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(3*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(A - 7*B - 21*C)*Sqrt[Cos[c + d*x]]*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (4*(27*A - 42*B - 175*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(105*d)","A",10,9,43,0.2093,1,"{4112, 3043, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
1213,1,267,0,0.8745183,"\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","Int[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{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{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{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 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)}","\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{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{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 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,"(56*a^4*(A - C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(4*A + 5*B + 4*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (4*a^4*(A - 25*B - 41*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*a*(5*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(5*A + 15*B + 19*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) - (4*(6*A + 25*B + 34*C)*Sqrt[Cos[c + d*x]]*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(15*d)","A",10,9,43,0.2093,1,"{4112, 3043, 2975, 2976, 2968, 3023, 2748, 2641, 2639}"
1214,1,271,0,0.8705235,"\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","Int[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{8 a^4 (35 A+28 B+17 C) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 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{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 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)}","\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{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{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 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,"(-8*a^4*(7*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (8*a^4*(35*A + 28*B + 17*C)*EllipticF[(c + d*x)/2, 2])/(21*d) - (4*a^4*(175*A + 287*B + 253*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(7*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(35*A + 77*B + 73*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (4*(175*A + 238*B + 197*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]])","A",10,8,43,0.1860,1,"{4112, 3043, 2975, 2968, 3023, 2748, 2641, 2639}"
1215,1,274,0,0.8943102,"\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","Int[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{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{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{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 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)}","\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{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{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 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,"(-8*a^4*(21*A + 24*B + 19*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(28*A + 17*B + 12*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (4*a^4*(287*A + 253*B + 193*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) + (2*a*(9*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2)) + (2*(63*A + 117*B + 97*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (4*(21*A + 24*B + 19*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(45*d*Cos[c + d*x]^(3/2))","A",10,8,43,0.1860,1,"{4112, 3043, 2975, 2968, 3021, 2748, 2641, 2639}"
1216,1,310,0,0.9246625,"\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","Int[((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{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{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{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 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)}","\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{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{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 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,"(-8*a^4*(24*A + 19*B + 16*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (8*a^4*(187*A + 132*B + 113*C)*EllipticF[(c + d*x)/2, 2])/(231*d) + (4*a^4*(913*A + 803*B + 667*C)*Sin[c + d*x])/(1155*d*Cos[c + d*x]^(3/2)) + (8*a^4*(24*A + 19*B + 16*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(11*B + 8*C)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(99*d*Cos[c + d*x]^(9/2)) + (2*C*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d*Cos[c + d*x]^(11/2)) + (2*(33*A + 55*B + 43*C)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/(231*d*Cos[c + d*x]^(7/2)) + (4*(891*A + 946*B + 769*C)*(a^4 + a^4*Cos[c + d*x])*Sin[c + d*x])/(3465*d*Cos[c + d*x]^(5/2))","A",11,9,43,0.2093,1,"{4112, 3043, 2975, 2968, 3021, 2748, 2636, 2639, 2641}"
1217,1,210,0,0.3299613,"\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","Int[(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]","\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}","\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,"(-3*(7*A - 7*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) + (5*(9*A - 7*B + 7*C)*EllipticF[(c + d*x)/2, 2])/(21*a*d) + (5*(9*A - 7*B + 7*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a*d) - ((7*A - 7*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) + ((9*A - 7*B + 7*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",8,6,43,0.1395,1,"{4112, 3041, 2748, 2635, 2639, 2641}"
1218,1,174,0,0.3094126,"\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","Int[(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]","-\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}","-\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,"(3*(7*A - 5*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((5*A - 5*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) - ((5*A - 5*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) + ((7*A - 5*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{4112, 3041, 2748, 2635, 2641, 2639}"
1219,1,134,0,0.2891297,"\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","Int[(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]","\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}","\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*A - 3*B + C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((5*A - 3*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",6,6,43,0.1395,1,"{4112, 3041, 2748, 2639, 2635, 2641}"
1220,1,93,0,0.2659246,"\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","Int[(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{(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)}","-\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*A - B + C)*EllipticE[(c + d*x)/2, 2])/(a*d) - ((A - B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(d*(a + a*Cos[c + d*x]))","A",5,5,43,0.1163,1,"{4112, 3041, 2748, 2641, 2639}"
1221,1,122,0,0.2849461,"\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","Int[(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2)/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])),x]","\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)}","\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,"-(((A - B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d)) + ((A + B - C)*EllipticF[(c + d*x)/2, 2])/(a*d) + ((A - B + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x]))","A",6,6,43,0.1395,1,"{4112, 3041, 2748, 2636, 2639, 2641}"
1222,1,165,0,0.3058229,"\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","Int[(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]","\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)}}","\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,"((A - 3*B + 3*C)*EllipticE[(c + d*x)/2, 2])/(a*d) + ((3*A - 3*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((3*A - 3*B + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) - ((A - 3*B + 3*C)*Sin[c + d*x])/(a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{4112, 3041, 2748, 2636, 2641, 2639}"
1223,1,210,0,0.3226261,"\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","Int[(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]","-\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)}}","-\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*(5*A - 5*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(5*a*d) - ((3*A - 5*B + 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a*d) + ((5*A - 5*B + 7*C)*Sin[c + d*x])/(5*a*d*Cos[c + d*x]^(5/2)) - ((3*A - 5*B + 5*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)) + (3*(5*A - 5*B + 7*C)*Sin[c + d*x])/(5*a*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x]))","A",8,6,43,0.1395,1,"{4112, 3041, 2748, 2636, 2639, 2641}"
1224,1,258,0,0.4992477,"\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","Int[(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]","\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}","\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,"(-7*(11*A - 8*B + 5*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) + (5*(30*A - 21*B + 14*C)*EllipticF[(c + d*x)/2, 2])/(21*a^2*d) + (5*(30*A - 21*B + 14*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*a^2*d) - (7*(11*A - 8*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) + ((30*A - 21*B + 14*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*a^2*d) - ((11*A - 8*B + 5*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",9,7,43,0.1628,1,"{4112, 3041, 2977, 2748, 2635, 2639, 2641}"
1225,1,214,0,0.4700867,"\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","Int[(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]","-\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}","-\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*A - 35*B + 20*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(3*A - 2*B + C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - (5*(3*A - 2*B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + ((56*A - 35*B + 20*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a^2*d) - ((3*A - 2*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",8,7,43,0.1628,1,"{4112, 3041, 2977, 2748, 2635, 2641, 2639}"
1226,1,180,0,0.4505328,"\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","Int[(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]","\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}","\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*A - 4*B + C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((10*A - 5*B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((10*A - 5*B + 2*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) - ((7*A - 4*B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",7,7,43,0.1628,1,"{4112, 3041, 2977, 2748, 2639, 2635, 2641}"
1227,1,144,0,0.40982,"\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","Int[(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{(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}","-\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*A - B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) - ((5*A - 2*B - C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((5*A - 2*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,43,0.1395,1,"{4112, 3041, 2977, 2748, 2641, 2639}"
1228,1,133,0,0.4138587,"\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","Int[(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{(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}","\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,"-(((A - C)*EllipticE[(c + d*x)/2, 2])/(a^2*d)) + ((2*A + B + 2*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a^2*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x])^2)","A",6,6,43,0.1395,1,"{4112, 3041, 2978, 2748, 2641, 2639}"
1229,1,167,0,0.4394943,"\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","Int[(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{(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}","\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,"((B - 4*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((A + 2*B - 5*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) - ((B - 4*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) + ((A + 2*B - 5*C)*Sin[c + d*x])/(3*a^2*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2)","A",7,7,43,0.1628,1,"{4112, 3041, 2978, 2748, 2636, 2639, 2641}"
1230,1,211,0,0.4681533,"\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","Int[(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]","\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}","\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,"((A - 4*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + ((2*A - 5*B + 10*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((2*A - 5*B + 10*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) - ((A - 4*B + 7*C)*Sin[c + d*x])/(a^2*d*Sqrt[Cos[c + d*x]]) - ((A - 4*B + 7*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2)","A",8,7,43,0.1628,1,"{4112, 3041, 2978, 2748, 2636, 2641, 2639}"
1231,1,250,0,0.4897803,"\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","Int[(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]","-\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}","-\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,"-((20*A - 35*B + 56*C)*EllipticE[(c + d*x)/2, 2])/(5*a^2*d) - (5*(A - 2*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(3*a^2*d) + ((20*A - 35*B + 56*C)*Sin[c + d*x])/(15*a^2*d*Cos[c + d*x]^(5/2)) - (5*(A - 2*B + 3*C)*Sin[c + d*x])/(3*a^2*d*Cos[c + d*x]^(3/2)) + ((20*A - 35*B + 56*C)*Sin[c + d*x])/(5*a^2*d*Sqrt[Cos[c + d*x]]) - ((A - 2*B + 3*C)*Sin[c + d*x])/(a^2*d*Cos[c + d*x]^(5/2)*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2)","A",9,7,43,0.1628,1,"{4112, 3041, 2978, 2748, 2636, 2639, 2641}"
1232,1,273,0,0.6599556,"\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","Int[(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]","-\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}","-\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,"(7*(33*A - 17*B + 7*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((63*A - 33*B + 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((63*A - 33*B + 13*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) + (7*(33*A - 17*B + 7*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^3*d) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((12*A - 7*B + 2*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((63*A - 33*B + 13*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{4112, 3041, 2977, 2748, 2635, 2641, 2639}"
1233,1,234,0,0.631445,"\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","Int[(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]","\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}","\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*A - 49*B + 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((33*A - 13*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((33*A - 13*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((2*A - B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(3*a*d*(a + a*Cos[c + d*x])^2) - ((119*A - 49*B + 9*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*d*(a^3 + a^3*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{4112, 3041, 2977, 2748, 2639, 2635, 2641}"
1234,1,201,0,0.61,"\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","Int[(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]","-\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}","-\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*A - 9*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) - ((13*A - 3*B - C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((8*A - 3*B - 2*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) - ((13*A - 3*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{4112, 3041, 2977, 2748, 2641, 2639}"
1235,1,193,0,0.593453,"\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","Int[(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]","\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}","\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*A + B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A + B + C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) - ((6*A - B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((9*A + B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",7,7,43,0.1628,1,"{4112, 3041, 2977, 2978, 2748, 2641, 2639}"
1236,1,191,0,0.5909697,"\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","Int[(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]","\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}","\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,"-((A - B - 9*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + B + 3*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ((4*A + B - 6*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*a*d*(a + a*Cos[c + d*x])^2) + ((A - B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*d*(a^3 + a^3*Cos[c + d*x]))","A",7,6,43,0.1395,1,"{4112, 3041, 2978, 2748, 2641, 2639}"
1237,1,229,0,0.6331212,"\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","Int[(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]","\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}","\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,"((A + 9*B - 49*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((A + 3*B - 13*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) - ((A + 9*B - 49*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3) + ((2*A + 3*B - 8*C)*Sin[c + d*x])/(15*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^2) + ((A + 3*B - 13*C)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]*(a^3 + a^3*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{4112, 3041, 2978, 2748, 2636, 2639, 2641}"
1238,1,268,0,0.6646757,"\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","Int[(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]","\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}","\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*A - 49*B + 119*C)*EllipticE[(c + d*x)/2, 2])/(10*a^3*d) + ((3*A - 13*B + 33*C)*EllipticF[(c + d*x)/2, 2])/(6*a^3*d) + ((3*A - 13*B + 33*C)*Sin[c + d*x])/(6*a^3*d*Cos[c + d*x]^(3/2)) - ((9*A - 49*B + 119*C)*Sin[c + d*x])/(10*a^3*d*Sqrt[Cos[c + d*x]]) - ((A - B + C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3) + ((B - 2*C)*Sin[c + d*x])/(3*a*d*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^2) - ((9*A - 49*B + 119*C)*Sin[c + d*x])/(30*d*Cos[c + d*x]^(3/2)*(a^3 + a^3*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{4112, 3041, 2978, 2748, 2636, 2641, 2639}"
1239,1,278,0,0.8326646,"\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","Int[(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]","\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}","\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,"-((176*A - 57*B + 8*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((339*A - 108*B + 17*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) + ((339*A - 108*B + 17*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(42*a^4*d) - ((43*A - 15*B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(42*a^4*d*(1 + Cos[c + d*x])^2) - ((176*A - 57*B + 8*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(30*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((13*A - 6*B - C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",9,7,43,0.1628,1,"{4112, 3041, 2977, 2748, 2639, 2635, 2641}"
1240,1,244,0,0.7901961,"\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","Int[(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]","-\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}","-\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,"((57*A - 8*B - C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) - ((108*A - 17*B - 4*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) - ((141*A - 29*B - 13*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) - ((108*A - 17*B - 4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(42*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((11*A - 4*B - 3*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,6,43,0.1395,1,"{4112, 3041, 2977, 2748, 2641, 2639}"
1241,1,232,0,0.7741192,"\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","Int[(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{(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}","\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,"-((8*A + B)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((17*A + 4*B + 3*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) - ((83*A + B - 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) + ((8*A + B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((9*A - 2*B - 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,7,43,0.1628,1,"{4112, 3041, 2977, 2978, 2748, 2641, 2639}"
1242,1,229,0,0.7652885,"\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","Int[(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{(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}","\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,"-((A - C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((4*A + 3*B + 4*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) + ((41*A + 15*B - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) + ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) - ((A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(5*a*d*(a + a*Cos[c + d*x])^3)","A",8,7,43,0.1628,1,"{4112, 3041, 2977, 2978, 2748, 2641, 2639}"
1243,1,234,0,0.7631878,"\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","Int[(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{(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}","\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,"((B + 8*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((3*A + 4*B + 17*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) + ((15*A - B - 83*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(210*a^4*d*(1 + Cos[c + d*x])^2) - ((B + 8*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(10*a^4*d*(1 + Cos[c + d*x])) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(7*d*(a + a*Cos[c + d*x])^4) + ((5*A + 2*B - 9*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(35*a*d*(a + a*Cos[c + d*x])^3)","A",8,6,43,0.1395,1,"{4112, 3041, 2978, 2748, 2641, 2639}"
1244,1,276,0,0.8263892,"\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","Int[(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]","\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}","\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,"((A + 8*B - 57*C)*EllipticE[(c + d*x)/2, 2])/(10*a^4*d) + ((4*A + 17*B - 108*C)*EllipticF[(c + d*x)/2, 2])/(42*a^4*d) - ((A + 8*B - 57*C)*Sin[c + d*x])/(10*a^4*d*Sqrt[Cos[c + d*x]]) + ((13*A + 29*B - 141*C)*Sin[c + d*x])/(210*a^4*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^2) + ((4*A + 17*B - 108*C)*Sin[c + d*x])/(42*a^4*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])) - ((A - B + C)*Sin[c + d*x])/(7*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^4) + ((3*A + 4*B - 11*C)*Sin[c + d*x])/(35*a*d*Sqrt[Cos[c + d*x]]*(a + a*Cos[c + d*x])^3)","A",9,7,43,0.1628,1,"{4112, 3041, 2978, 2748, 2636, 2639, 2641}"
1245,1,226,0,0.628749,"\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","Int[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{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}","\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,"(16*a*(16*A + 18*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a*(16*A + 18*B + 21*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(16*A + 18*B + 21*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 9*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(9*d)","A",6,5,45,0.1111,1,"{4265, 4086, 4015, 3805, 3804}"
1246,1,178,0,0.5505026,"\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","Int[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{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}","\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,"(4*a*(24*A + 28*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(24*A + 28*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",5,5,45,0.1111,1,"{4265, 4086, 4015, 3805, 3804}"
1247,1,129,0,0.4684946,"\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","Int[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 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}","\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*a*(7*A + 5*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",4,4,45,0.08889,1,"{4265, 4086, 4013, 3804}"
1248,1,140,0,0.4468476,"\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","Int[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{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}","\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,"(2*Sqrt[a]*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a*(A + 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",5,5,45,0.1111,1,"{4265, 4086, 4015, 3801, 215}"
1249,1,139,0,0.4504001,"\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","Int[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{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)}}","\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[a]*(2*B + C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a*(2*A - C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",5,5,45,0.1111,1,"{4265, 4088, 4015, 3801, 215}"
1250,1,151,0,0.4559263,"\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","Int[(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{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)}","\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[a]*(8*A + 4*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a*(4*B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2))","A",5,5,45,0.1111,1,"{4265, 4088, 4016, 3801, 215}"
1251,1,199,0,0.5479866,"\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","Int[(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{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)}","\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,"(Sqrt[a]*(8*A + 6*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a*(6*B + C)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*A + 6*B + 5*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2))","A",6,6,45,0.1333,1,"{4265, 4088, 4016, 3803, 3801, 215}"
1252,1,247,0,0.6327173,"\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","Int[(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{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)}","\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,"(Sqrt[a]*(48*A + 40*B + 35*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a*(8*B + C)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(48*A + 40*B + 35*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (C*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2))","A",7,6,45,0.1333,1,"{4265, 4088, 4016, 3803, 3801, 215}"
1253,1,284,0,0.8854881,"\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","Int[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{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}","\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,"(16*a^2*(336*A + 374*B + 429*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^2*(336*A + 374*B + 429*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(336*A + 374*B + 429*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(1155*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(84*A + 110*B + 99*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 11*B)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(11*d)","A",7,6,45,0.1333,1,"{4265, 4086, 4017, 4015, 3805, 3804}"
1254,1,232,0,0.787625,"\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","Int[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{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}","\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,"(4*a^2*(136*A + 156*B + 189*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 156*B + 189*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(52*A + 72*B + 63*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(A + 3*B)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)","A",6,6,45,0.1333,1,"{4265, 4086, 4017, 4015, 3805, 3804}"
1255,1,181,0,0.6007658,"\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","Int[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{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}","\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,"(8*a^2*(19*A + 21*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(19*A + 21*B + 35*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(3*A + 7*B)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)","A",5,5,45,0.1111,1,"{4265, 4086, 4013, 3809, 3804}"
1256,1,192,0,0.6388647,"\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","Int[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{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^{3/2} 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 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}","\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^{3/2} 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 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,"(2*a^(3/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^2*(12*A + 20*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,45,0.1333,1,"{4265, 4086, 4017, 4015, 3801, 215}"
1257,1,197,0,0.6487143,"\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","Int[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^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^{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 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}","\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^{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 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^(3/2)*(2*B + 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^2*(8*A + 6*B - 3*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a*(2*A - 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",6,6,45,0.1333,1,"{4265, 4086, 4018, 4015, 3801, 215}"
1258,1,203,0,0.661237,"\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","Int[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^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^{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 (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)}}","\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^{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 (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^(3/2)*(8*A + 12*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^2*(8*A - 4*B - 5*C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a*(4*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])","A",6,6,45,0.1333,1,"{4265, 4088, 4018, 4015, 3801, 215}"
1259,1,201,0,0.6779797,"\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","Int[((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^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^{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 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)}","\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^{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 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^(3/2)*(24*A + 14*B + 11*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^2*(24*A + 30*B + 19*C)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(2*B + C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",6,6,45,0.1333,1,"{4265, 4088, 4018, 4016, 3801, 215}"
1260,1,253,0,0.790027,"\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","Int[((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^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^{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 (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)}","\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^{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 (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^(3/2)*(112*A + 88*B + 75*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^2*(48*A + 56*B + 39*C)*Sin[c + d*x])/(96*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(112*A + 88*B + 75*C)*Sin[c + d*x])/(64*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(8*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))","A",7,7,45,0.1556,1,"{4265, 4088, 4018, 4016, 3803, 3801, 215}"
1261,1,303,0,0.8685423,"\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","Int[((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^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^{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 (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)}","\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^{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 (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^(3/2)*(176*A + 150*B + 133*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^2*(80*A + 90*B + 67*C)*Sin[c + d*x])/(240*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(176*A + 150*B + 133*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a*(10*B + 3*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(40*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(7/2))","A",8,7,45,0.1556,1,"{4265, 4088, 4018, 4016, 3803, 3801, 215}"
1262,1,334,0,1.0975989,"\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","Int[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{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^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 (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}","\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^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 (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,"(16*a^3*(8368*A + 9230*B + 10439*C)*Sin[c + d*x])/(45045*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (8*a^3*(8368*A + 9230*B + 10439*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(45045*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(8368*A + 9230*B + 10439*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15015*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2224*A + 2522*B + 2717*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(9009*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(136*A + 182*B + 143*C)*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(1287*d) + (2*a*(5*A + 13*B)*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(143*d) + (2*A*Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(13*d)","A",8,6,45,0.1333,1,"{4265, 4086, 4017, 4015, 3805, 3804}"
1263,1,284,0,1.0140604,"\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","Int[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{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^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 (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}","\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^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 (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,"(4*a^3*(2840*A + 3212*B + 3795*C)*Sin[c + d*x])/(3465*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(2840*A + 3212*B + 3795*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^3*(1160*A + 1364*B + 1485*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(32*A + 44*B + 33*C)*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(231*d) + (2*a*(5*A + 11*B)*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d)","A",7,6,45,0.1333,1,"{4265, 4086, 4017, 4015, 3805, 3804}"
1264,1,231,0,0.7061256,"\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","Int[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{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{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{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}","\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{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{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,"(64*a^3*(13*A + 15*B + 21*C)*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (16*a^2*(13*A + 15*B + 21*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*a*(13*A + 15*B + 21*C)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(105*d) + (2*(5*A + 9*B)*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d)","A",6,5,45,0.1111,1,"{4265, 4086, 4013, 3809, 3804}"
1265,1,242,0,0.8363494,"\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","Int[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{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^{5/2} 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 (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}","\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^{5/2} 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 (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,"(2*a^(5/2)*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (2*a^3*(160*A + 224*B + 245*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*a^2*(40*A + 56*B + 35*C)*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*a*(5*A + 7*B)*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",7,6,45,0.1333,1,"{4265, 4086, 4017, 4015, 3801, 215}"
1266,1,243,0,0.8752878,"\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","Int[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^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{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{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}","\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{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{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^(5/2)*(2*B + 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/d + (a^3*(64*A + 70*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(16*A + 10*B - 15*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a*(A + B)*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",7,7,45,0.1556,1,"{4265, 4086, 4017, 4018, 4015, 3801, 215}"
1267,1,253,0,0.8805849,"\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","Int[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^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^{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 (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}","\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^{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 (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^(5/2)*(8*A + 20*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*d) + (a^3*(56*A + 12*B - 27*C)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (a^2*(8*A - 12*B - 21*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (a*(4*A - 3*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",7,6,45,0.1333,1,"{4265, 4086, 4018, 4015, 3801, 215}"
1268,1,253,0,0.8733646,"\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","Int[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^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^{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 (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)}}","\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^{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 (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^(5/2)*(40*A + 38*B + 25*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*d) + (a^3*(24*A - 54*B - 49*C)*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(24*A + 42*B + 31*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + (a*(6*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",7,6,45,0.1333,1,"{4265, 4088, 4018, 4015, 3801, 215}"
1269,1,253,0,0.8910924,"\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","Int[((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^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^{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 (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)}","\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^{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 (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^(5/2)*(304*A + 200*B + 163*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(64*d) + (a^3*(432*A + 392*B + 299*C)*Sin[c + d*x])/(192*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(16*A + 24*B + 17*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(32*d*Cos[c + d*x]^(3/2)) + (a*(8*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2))","A",7,6,45,0.1333,1,"{4265, 4088, 4018, 4016, 3801, 215}"
1270,1,301,0,1.0060392,"\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","Int[((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^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^{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 (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)}","\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^{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 (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^(5/2)*(400*A + 326*B + 283*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(128*d) + (a^3*(1040*A + 950*B + 787*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(400*A + 326*B + 283*C)*Sin[c + d*x])/(128*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(80*A + 110*B + 79*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + (a*(2*B + C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",8,7,45,0.1556,1,"{4265, 4088, 4018, 4016, 3803, 3801, 215}"
1271,1,353,0,1.1037454,"\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","Int[((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{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^{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 (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)}","\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^{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 (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,"(a^(5/2)*(1304*A + 1132*B + 1015*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(512*d) + (a^3*(680*A + 628*B + 545*C)*Sin[c + d*x])/(960*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(768*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^3*(1304*A + 1132*B + 1015*C)*Sin[c + d*x])/(512*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]) + (a^2*(120*A + 156*B + 115*C)*Sqrt[a + a*Sec[c + d*x]]*Sin[c + d*x])/(480*d*Cos[c + d*x]^(7/2)) + (a*(12*B + 5*C)*(a + a*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(60*d*Cos[c + d*x]^(7/2)) + (C*(a + a*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(6*d*Cos[c + d*x]^(7/2))","A",9,7,45,0.1556,1,"{4265, 4088, 4018, 4016, 3803, 3801, 215}"
1272,1,257,0,0.8768496,"\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","Int[(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{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}}","\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,"(Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(43*A - 91*B + 35*C)*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*(31*A - 7*B + 35*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 7*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,45,0.1333,1,"{4265, 4086, 4022, 4013, 3808, 206}"
1273,1,211,0,0.6673698,"\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","Int[(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{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}}","\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,"-((Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d)) + (2*(13*A - 5*B + 15*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - (2*(A - 5*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4265, 4086, 4022, 4013, 3808, 206}"
1274,1,163,0,0.4864418,"\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","Int[(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{\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}}","\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,"(Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (2*(A - 3*B)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[a + a*Sec[c + d*x]])","A",5,5,45,0.1111,1,"{4265, 4086, 4013, 3808, 206}"
1275,1,178,0,0.5196863,"\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","Int[(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{\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}","-\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*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4265, 4086, 4023, 3808, 206, 3801, 215}"
1276,1,181,0,0.5257217,"\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","Int[(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{\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}}","\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,"((2*B - C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4265, 4088, 4023, 3808, 206, 3801, 215}"
1277,1,235,0,0.7351909,"\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","Int[(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{\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}}","-\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,"((8*A - 4*B + 7*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*Sqrt[a]*d) - (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((4*B - C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4265, 4088, 4021, 4023, 3808, 206, 3801, 215}"
1278,1,281,0,0.9267591,"\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","Int[(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{(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}}","\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,"-((8*A - 14*B + 9*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(8*Sqrt[a]*d) + (Sqrt[2]*(A - B + C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]) + ((6*B - C)*Sin[c + d*x])/(12*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) + ((8*A - 2*B + 7*C)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,8,45,0.1778,1,"{4265, 4088, 4021, 4023, 3808, 206, 3801, 215}"
1279,1,184,0,0.6022957,"\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","Int[(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{\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}","-\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*b*B*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) - (Sqrt[2]*(a - b)*(A - B)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(Sqrt[a]*d) + (2*a*A*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",7,7,54,0.1296,1,"{4265, 4086, 4023, 3808, 206, 3801, 215}"
1280,1,283,0,0.9175052,"\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","Int[(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{(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}}","-\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*A - 11*B + 7*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) + ((147*A - 95*B + 75*C)*Sin[c + d*x])/(30*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((39*A - 35*B + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(30*a*d*Sqrt[a + a*Sec[c + d*x]]) + ((9*A - 5*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(10*a*d*Sqrt[a + a*Sec[c + d*x]])","A",7,6,45,0.1333,1,"{4265, 4084, 4022, 4013, 3808, 206}"
1281,1,233,0,0.7248636,"\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","Int[(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{(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}}","\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,"((11*A - 7*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*d*(a + a*Sec[c + d*x])^(3/2)) - ((19*A - 15*B + 3*C)*Sin[c + d*x])/(6*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((7*A - 3*B + 3*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(6*a*d*Sqrt[a + a*Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4265, 4084, 4022, 4013, 3808, 206}"
1282,1,181,0,0.5247163,"\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","Int[(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{(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}}","-\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[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((5*A - B + C)*Sin[c + d*x])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",5,5,45,0.1111,1,"{4265, 4084, 4013, 3808, 206}"
1283,1,189,0,0.5572543,"\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","Int[(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{(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}}","\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,"(2*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((3*A + B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",7,7,45,0.1556,1,"{4265, 4084, 4023, 3808, 206, 3801, 215}"
1284,1,242,0,0.7537437,"\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","Int[(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{(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}}","\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,"((2*B - 3*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(3/2)*d) + ((A - 5*B + 9*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 3*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",8,8,45,0.1778,1,"{4265, 4084, 4021, 4023, 3808, 206, 3801, 215}"
1285,1,300,0,0.9854119,"\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","Int[(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{(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}}","-\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,"((8*A - 12*B + 19*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(4*a^(3/2)*d) - ((5*A - 9*B + 13*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(2*Sqrt[2]*a^(3/2)*d) - ((A - B + C)*Sin[c + d*x])/(2*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)) + ((A - B + 2*C)*Sin[c + d*x])/(2*a*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]) - ((2*A - 6*B + 7*C)*Sin[c + d*x])/(4*a*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,8,45,0.1778,1,"{4265, 4084, 4021, 4023, 3808, 206, 3801, 215}"
1286,1,333,0,1.1534711,"\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","Int[(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{(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{(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{(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}}","\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{(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{(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,"-((283*A - 163*B + 75*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((21*A - 13*B + 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) + ((2671*A - 1495*B + 735*C)*Sin[c + d*x])/(240*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) - ((787*A - 475*B + 195*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(240*a^2*d*Sqrt[a + a*Sec[c + d*x]]) + ((157*A - 85*B + 45*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(80*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",8,7,45,0.1556,1,"{4265, 4084, 4020, 4022, 4013, 3808, 206}"
1287,1,281,0,0.9559201,"\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","Int[(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{(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{(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{(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}}","\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{(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{(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,"((163*A - 75*B + 19*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*d*(a + a*Sec[c + d*x])^(5/2)) - ((17*A - 9*B + C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(16*a*d*(a + a*Sec[c + d*x])^(3/2)) - ((299*A - 147*B + 27*C)*Sin[c + d*x])/(48*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]) + ((95*A - 39*B + 15*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(48*a^2*d*Sqrt[a + a*Sec[c + d*x]])","A",7,7,45,0.1556,1,"{4265, 4084, 4020, 4022, 4013, 3808, 206}"
1288,1,231,0,0.7392616,"\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","Int[(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{(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{(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{(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}}","\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{(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{(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,"-((75*A - 19*B - 5*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)) - ((13*A - 5*B - 3*C)*Sin[c + d*x])/(16*a*d*Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)) + ((49*A - 9*B + C)*Sin[c + d*x])/(16*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]])","A",6,6,45,0.1333,1,"{4265, 4084, 4020, 4013, 3808, 206}"
1289,1,183,0,0.559329,"\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","Int[(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{(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}}","\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,"((19*A + 5*B + 3*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)) - ((9*A - B - 7*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",5,5,45,0.1111,1,"{4265, 4084, 4012, 3808, 206}"
1290,1,241,0,0.7490329,"\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","Int[(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{(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}}","\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*C*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((5*A + 3*B - 43*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)) + ((5*A + 3*B - 11*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2))","A",8,8,45,0.1778,1,"{4265, 4084, 4019, 4023, 3808, 206, 3801, 215}"
1291,1,294,0,0.9974167,"\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","Int[(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{(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{(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{(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}}","\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{(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{(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,"((2*B - 5*C)*ArcSinh[(Sqrt[a]*Tan[c + d*x])/Sqrt[a + a*Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(a^(5/2)*d) + ((3*A - 43*B + 115*C)*ArcTanh[(Sqrt[a]*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(Sqrt[2]*Sqrt[a + a*Sec[c + d*x]])]*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(16*Sqrt[2]*a^(5/2)*d) - ((A - B + C)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)) + ((A + 7*B - 15*C)*Sin[c + d*x])/(16*a*d*Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)) + ((3*A - 11*B + 35*C)*Sin[c + d*x])/(16*a^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]])","A",9,9,45,0.2000,1,"{4265, 4084, 4019, 4021, 4023, 3808, 206, 3801, 215}"
1292,1,190,0,0.2970418,"\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","Int[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{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}","\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,"(2*(7*a*A + 9*b*B + 9*a*C)*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*A*b + 5*a*B + 7*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*A*b + 5*a*B + 7*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(7*a*A + 9*b*B + 9*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*(A*b + a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*a*A*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d)","A",8,7,41,0.1707,1,"{4112, 3033, 3023, 2748, 2635, 2641, 2639}"
1293,1,154,0,0.2691834,"\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","Int[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{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}","\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,"(2*(3*A*b + 3*a*B + 5*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(5*a*A + 7*b*B + 7*a*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(5*a*A + 7*b*B + 7*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(A*b + a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*a*A*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d)","A",7,7,41,0.1707,1,"{4112, 3033, 3023, 2748, 2639, 2635, 2641}"
1294,1,116,0,0.2518216,"\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","Int[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{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}","\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,"(2*(3*a*A + 5*b*B + 5*a*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(A*b + a*B + 3*b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)","A",6,6,41,0.1463,1,"{4112, 3033, 3023, 2748, 2641, 2639}"
1295,1,106,0,0.2606551,"\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","Int[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{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)}}","\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,"(2*(A*b + a*B - b*C)*EllipticE[(c + d*x)/2, 2])/d + (2*(3*b*B + a*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*C*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)","A",6,6,41,0.1463,1,"{4112, 3031, 3023, 2748, 2641, 2639}"
1296,1,112,0,0.2780498,"\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","Int[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{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)}","\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,"(-2*(b*B - a*(A - C))*EllipticE[(c + d*x)/2, 2])/d + (2*(3*A*b + 3*a*B + b*C)*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*C*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",6,6,41,0.1463,1,"{4112, 3031, 3021, 2748, 2641, 2639}"
1297,1,152,0,0.3046024,"\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","Int[((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{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)}","\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,"(-2*(5*A*b + 5*a*B + 3*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(b*B + a*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*C*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b + 5*a*B + 3*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",7,7,41,0.1707,1,"{4112, 3031, 3021, 2748, 2636, 2639, 2641}"
1298,1,190,0,0.3190866,"\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","Int[((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{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)}","\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,"(-2*(5*a*A + 3*b*B + 3*a*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(7*A*b + 7*a*B + 5*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*C*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2)) + (2*(b*B + a*C)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b + 7*a*B + 5*b*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(5*a*A + 3*b*B + 3*a*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]])","A",8,7,41,0.1707,1,"{4112, 3031, 3021, 2748, 2636, 2641, 2639}"
1299,1,250,0,0.6032227,"\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","Int[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{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}","\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,"(2*(18*a*b*B + 3*b^2*(3*A + 5*C) + a^2*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(10*a*A*b + 5*a^2*B + 7*b^2*B + 14*a*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*(4*A*b^2 + 18*a*b*B + a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(45*d) + (2*a*(4*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(9*d)","A",8,8,43,0.1860,1,"{4112, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1300,1,202,0,0.5697672,"\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","Int[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]","\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}","\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,"(2*(6*a*A*b + 3*a^2*B + 5*b^2*B + 10*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(14*a*b*B + 7*b^2*(A + 3*C) + a^2*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(4*A*b^2 + 14*a*b*B + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a*(4*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d)","A",7,7,43,0.1628,1,"{4112, 3049, 3033, 3023, 2748, 2641, 2639}"
1301,1,186,0,0.5517694,"\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","Int[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]","\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)}}","\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,"(2*(10*a*b*B + 5*b^2*(A - C) + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^2*B + 3*b^2*B + 2*a*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(2*A*b + a*B - 6*b*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(A - 5*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",7,7,43,0.1628,1,"{4112, 3047, 3033, 3023, 2748, 2641, 2639}"
1302,1,180,0,0.5454937,"\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","Int[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]","\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)}","\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*(a^2*B - b^2*B + 2*a*b*(A - C))*EllipticE[(c + d*x)/2, 2])/d + (2*(6*a*b*B + b^2*(3*A + C) + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*(3*b*B + 4*a*C)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(A - C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",7,7,43,0.1628,1,"{4112, 3047, 3031, 3023, 2748, 2641, 2639}"
1303,1,201,0,0.5710354,"\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","Int[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]","\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)}","\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,"(-2*(10*a*b*B - 5*a^2*(A - C) + b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(3*a^2*B + b^2*B + 2*a*b*(3*A + C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*(5*b*B + 4*a*C)*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 + 10*a*b*B + 4*a^2*C + 3*b^2*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",7,7,43,0.1628,1,"{4112, 3047, 3031, 3021, 2748, 2641, 2639}"
1304,1,249,0,0.6052096,"\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","Int[((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 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)}","\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*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(14*a*b*B + 7*a^2*(3*A + C) + b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(7*b*B + 4*a*C)*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*(7*A*b^2 + 14*a*b*B + 4*a^2*C + 5*b^2*C)*Sin[c + d*x])/(21*d*Cos[c + d*x]^(3/2)) + (2*(10*a*A*b + 5*a^2*B + 3*b^2*B + 6*a*b*C)*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,8,43,0.1860,1,"{4112, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1305,1,361,0,0.9550485,"\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","Int[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{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(3 a^2 b (7 A+9 C)+7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)\right)}{15 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 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(33 a^2 b (7 A+9 C)+77 a^3 B+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}","\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(3 a^2 b (7 A+9 C)+7 a^3 B+27 a b^2 B+3 b^3 (3 A+5 C)\right)}{15 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 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(33 a^2 b (7 A+9 C)+77 a^3 B+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,"(2*(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])/(15*d) + (2*(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])/(231*d) + (2*(165*a^2*b*B + 77*b^3*B + 33*a*b^2*(5*A + 7*C) + 5*a^3*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(24*A*b^3 + 77*a^3*B + 242*a*b^2*B + 33*a^2*b*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(495*d) + (2*a*(24*A*b^2 + 143*a*b*B + 9*a^2*(9*A + 11*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(693*d) + (2*(6*A*b + 11*a*B)*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d)","A",9,8,43,0.1860,1,"{4112, 3049, 3033, 3023, 2748, 2639, 2635, 2641}"
1306,1,296,0,0.9103542,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (5 A+7 C)+5 a^3 B+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 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 \sin (c+d x) \sqrt{\cos (c+d x)} \left(9 a^2 b (5 A+7 C)+15 a^3 B+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}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (5 A+7 C)+5 a^3 B+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 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 \sin (c+d x) \sqrt{\cos (c+d x)} \left(9 a^2 b (5 A+7 C)+15 a^3 B+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,"(2*(27*a^2*b*B + 15*b^3*B + 9*a*b^2*(3*A + 5*C) + a^3*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a^3*B + 21*a*b^2*B + 7*b^3*(A + 3*C) + 3*a^2*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*(8*A*b^3 + 15*a^3*B + 54*a*b^2*B + 9*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a*(24*A*b^2 + 99*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*(2*A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d)","A",8,7,43,0.1628,1,"{4112, 3049, 3033, 3023, 2748, 2641, 2639}"
1307,1,277,0,0.9056238,"\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","Int[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]","\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^2 b (3 A+5 C)+3 a^3 B+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\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 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)}}","\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^2 b (3 A+5 C)+3 a^3 B+15 a b^2 B+5 b^3 (A-C)\right)}{5 d}+\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 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,"(2*(3*a^3*B + 15*a*b^2*B + 5*b^3*(A - C) + 3*a^2*b*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^2*b*B + 21*b^3*B + 21*a*b^2*(A + 3*C) + a^3*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(21*a*b*B + 6*b^2*(3*A - 7*C) + a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(11*A*b + 7*a*B - 35*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(35*d) + (2*a*(A - 7*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",8,8,43,0.1860,1,"{4112, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1308,1,267,0,0.8788596,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (A+3 C)+a^3 B+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 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 (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)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b (A+3 C)+a^3 B+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 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 (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,"(2*(15*a^2*b*B - 5*b^3*B + 15*a*b^2*(A - C) + a^3*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^3*B + 9*a*b^2*B + b^3*(3*A + C) + 3*a^2*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(a^2*B - 6*b^2*B + 3*a*b*(A - 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d) + (2*a^2*(3*a*A - 15*b*B - 35*a*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(b*B + 2*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",8,7,43,0.1628,1,"{4112, 3047, 3033, 3023, 2748, 2641, 2639}"
1309,1,274,0,0.8711396,"\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","Int[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]","\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(15 a^2 b (A-C)+5 a^3 B-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}+\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 (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)}","\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(15 a^2 b (A-C)+5 a^3 B-15 a b^2 B-b^3 (5 A+3 C)\right)}{5 d}+\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 (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,"(2*(5*a^3*B - 15*a*b^2*B + 15*a^2*b*(A - C) - b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(9*a^2*b*B + b^3*B + 3*a*b^2*(3*A + C) + a^3*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*b*(15*A*b^2 + 35*a*b*B + 24*a^2*C + 9*b^2*C)*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*a^2*(5*a*A - 5*b*B - 9*a*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) + (2*(5*b*B + 6*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",8,7,43,0.1628,1,"{4112, 3047, 3031, 3023, 2748, 2641, 2639}"
1310,1,294,0,0.8959405,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^2 b (3 A+C)+21 a^3 B+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 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 \sin (c+d x) \left(98 a^2 b B+24 a^3 C+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)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(21 a^2 b (3 A+C)+21 a^3 B+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 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 \sin (c+d x) \left(98 a^2 b B+24 a^3 C+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,"(-2*(15*a^2*b*B + 3*b^3*B - 5*a^3*(A - C) + 3*a*b^2*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(21*a^3*B + 21*a*b^2*B + 21*a^2*b*(3*A + C) + b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(35*A*b^2 + 63*a*b*B + 24*a^2*C + 25*b^2*C)*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(98*a^2*b*B + 21*b^3*B + 24*a^3*C + 21*a*b^2*(5*A + 3*C))*Sin[c + d*x])/(35*d*Sqrt[Cos[c + d*x]]) + (2*(7*b*B + 6*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",8,7,43,0.1628,1,"{4112, 3047, 3031, 3021, 2748, 2641, 2639}"
1311,1,357,0,0.9740864,"\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","Int[((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]","\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(9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\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 \sin (c+d x) \left(9 a^2 b (5 A+3 C)+15 a^3 B+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)}","\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(9 a^2 b (5 A+3 C)+15 a^3 B+27 a b^2 B+b^3 (9 A+7 C)\right)}{15 d}+\frac{2 \sin (c+d x) \left(54 a^2 b B+8 a^3 C+9 a b^2 (7 A+5 C)+15 b^3 B\right)}{63 d \cos ^{\frac{3}{2}}(c+d x)}+\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 \sin (c+d x) \left(9 a^2 b (5 A+3 C)+15 a^3 B+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,"(-2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^2*b*B + 5*b^3*B + 7*a^3*(3*A + C) + 3*a*b^2*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(63*A*b^2 + 99*a*b*B + 24*a^2*C + 49*b^2*C)*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(54*a^2*b*B + 15*b^3*B + 8*a^3*C + 9*a*b^2*(7*A + 5*C))*Sin[c + d*x])/(63*d*Cos[c + d*x]^(3/2)) + (2*(15*a^3*B + 27*a*b^2*B + 9*a^2*b*(5*A + 3*C) + b^3*(9*A + 7*C))*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(3*b*B + 2*a*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d*Cos[c + d*x]^(7/2)) + (2*C*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",9,8,43,0.1860,1,"{4112, 3047, 3031, 3021, 2748, 2636, 2639, 2641}"
1312,1,404,0,1.3203393,"\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","Int[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{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+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(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right)}{3465 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 \sin (c+d x) \sqrt{\cos (c+d x)} \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+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}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+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(4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right)}{15 d}+\frac{2 a \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right)}{3465 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 \sin (c+d x) \sqrt{\cos (c+d x)} \left(9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+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,"(2*(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])/(15*d) + (2*(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])/(231*d) + (2*(64*A*b^4 + 660*a^3*b*B + 682*a*b^3*B + 15*a^4*(9*A + 11*C) + 9*a^2*b^2*(101*A + 143*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(693*d) + (2*a*(192*A*b^3 + 539*a^3*B + 1353*a*b^2*B + 2*a^2*b*(673*A + 891*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3465*d) + (2*(16*A*b^2 + 55*a*b*B + 3*a^2*(9*A + 11*C))*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(231*d) + (2*(8*A*b + 11*a*B)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(99*d) + (2*A*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(11*d)","A",9,7,43,0.1628,1,"{4112, 3049, 3033, 3023, 2748, 2641, 2639}"
1313,1,377,0,1.3051899,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 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(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\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(12 a^2 b (5 A+7 C)+15 a^3 B+117 a b^2 B+2 b^3 (31 A-63 C)\right)}{63 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)}}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (5 A+7 C)+42 a^2 b^2 B+5 a^4 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(18 a^2 b^2 (3 A+5 C)+a^4 (7 A+9 C)+36 a^3 b B+60 a b^3 B+15 b^4 (A-C)\right)}{15 d}+\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(12 a^2 b (5 A+7 C)+15 a^3 B+117 a b^2 B+2 b^3 (31 A-63 C)\right)}{63 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,"(2*(36*a^3*b*B + 60*a*b^3*B + 15*b^4*(A - C) + 18*a^2*b^2*(3*A + 5*C) + a^4*(7*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(5*a^4*B + 42*a^2*b^2*B + 21*b^4*B + 28*a*b^3*(A + 3*C) + 4*a^3*b*(5*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(15*a^3*B + 117*a*b^2*B + 2*b^3*(31*A - 63*C) + 12*a^2*b*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(63*d) + (2*a^2*(162*a*b*B + 3*b^2*(41*A - 105*C) + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(315*d) + (2*a*(5*A*b + 3*a*B - 21*b*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(21*d) + (2*a*(A - 9*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",9,8,43,0.1860,1,"{4112, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1314,1,371,0,1.2951593,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+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(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\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 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)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+28 a^3 b B+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(4 a^3 b (3 A+5 C)+30 a^2 b^2 B+3 a^4 B+20 a b^3 (A-C)-5 b^4 B\right)}{5 d}+\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 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,"(2*(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))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(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))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*a*(28*a^2*b*B - 42*b^3*B + 3*a*b^2*(13*A - 49*C) + a^3*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(21*d) + (2*a^2*(54*a*A*b + 21*a^2*B - 105*b^2*B - 350*a*b*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(105*d) + (2*a*(a*A - 7*b*B - 21*a*C)*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(7*d) + (2*(3*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",9,8,43,0.1860,1,"{4112, 3047, 3049, 3033, 3023, 2748, 2641, 2639}"
1315,1,388,0,1.3029335,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 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(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}-\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 (-(3 A-59 C))+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(4 a^2 b (5 A-33 C)+5 a^3 B-105 a b^2 B-6 b^3 (5 A+3 C)\right)}{15 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)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(4 a^3 b (A+3 C)+18 a^2 b^2 B+a^4 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(30 a^2 b^2 (A-C)+a^4 (3 A+5 C)+20 a^3 b B-20 a b^3 B-b^4 (5 A+3 C)\right)}{5 d}-\frac{2 a^2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 (-(3 A-59 C))+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(4 a^2 b (5 A-33 C)+5 a^3 B-105 a b^2 B-6 b^3 (5 A+3 C)\right)}{15 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,"(2*(20*a^3*b*B - 20*a*b^3*B + 30*a^2*b^2*(A - C) - b^4*(5*A + 3*C) + a^4*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(a^4*B + 18*a^2*b^2*B + b^4*B + 4*a*b^3*(3*A + C) + 4*a^3*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*a*(5*a^3*B - 105*a*b^2*B + 4*a^2*b*(5*A - 33*C) - 6*b^3*(5*A + 3*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(15*d) - (2*a^2*(50*a*b*B - a^2*(3*A - 59*C) + 3*b^2*(5*A + 3*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(15*d) + (2*(5*A*b^2 + 15*a*b*B + 16*a^2*C + 3*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(5*d*Sqrt[Cos[c + d*x]]) + (2*(5*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(15*d*Cos[c + d*x]^(3/2)) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",9,7,43,0.1628,1,"{4112, 3047, 3033, 3023, 2748, 2641, 2639}"
1316,1,384,0,1.3164825,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+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(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}+\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(a^2 (-(35 A-87 C))+98 a b B+5 b^2 (7 A+5 C)\right)}{105 d}+\frac{2 b \sin (c+d x) \left(413 a^2 b B+192 a^3 C+2 a b^2 (175 A+101 C)+63 b^3 B\right)}{105 d \sqrt{\cos (c+d x)}}+\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)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(42 a^2 b^2 (3 A+C)+7 a^4 (A+3 C)+84 a^3 b B+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(20 a^3 b (A-C)-30 a^2 b^2 B+5 a^4 B-4 a b^3 (5 A+3 C)-3 b^4 B\right)}{5 d}+\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(a^2 (-(35 A-87 C))+98 a b B+5 b^2 (7 A+5 C)\right)}{105 d}+\frac{2 b \sin (c+d x) \left(413 a^2 b B+192 a^3 C+2 a b^2 (175 A+101 C)+63 b^3 B\right)}{105 d \sqrt{\cos (c+d x)}}+\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,"(2*(5*a^4*B - 30*a^2*b^2*B - 3*b^4*B + 20*a^3*b*(A - C) - 4*a*b^3*(5*A + 3*C))*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*(84*a^3*b*B + 28*a*b^3*B + 42*a^2*b^2*(3*A + C) + 7*a^4*(A + 3*C) + b^4*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(413*a^2*b*B + 63*b^3*B + 192*a^3*C + 2*a*b^2*(175*A + 101*C))*Sin[c + d*x])/(105*d*Sqrt[Cos[c + d*x]]) - (2*a^2*(98*a*b*B - a^2*(35*A - 87*C) + 5*b^2*(7*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(35*A*b^2 + 77*a*b*B + 48*a^2*C + 25*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(105*d*Cos[c + d*x]^(3/2)) + (2*(7*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(35*d*Cos[c + d*x]^(5/2)) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(7*d*Cos[c + d*x]^(7/2))","A",9,7,43,0.1628,1,"{4112, 3047, 3031, 3023, 2748, 2641, 2639}"
1317,1,401,0,1.3306271,"\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","Int[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]","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 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(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}+\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(261 a^2 b B+64 a^3 C+2 a b^2 (147 A+101 C)+75 b^3 B\right)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 b^2 (261 A+155 C)+1098 a^3 b B+192 a^4 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)}","\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(28 a^3 b (3 A+C)+42 a^2 b^2 B+21 a^4 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(18 a^2 b^2 (5 A+3 C)-15 a^4 (A-C)+60 a^3 b B+36 a b^3 B+b^4 (9 A+7 C)\right)}{15 d}+\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(261 a^2 b B+64 a^3 C+2 a b^2 (147 A+101 C)+75 b^3 B\right)}{315 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \sin (c+d x) \left(7 a^2 b^2 (261 A+155 C)+1098 a^3 b B+192 a^4 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,"(-2*(60*a^3*b*B + 36*a*b^3*B - 15*a^4*(A - C) + 18*a^2*b^2*(5*A + 3*C) + b^4*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/(15*d) + (2*(21*a^4*B + 42*a^2*b^2*B + 5*b^4*B + 28*a^3*b*(3*A + C) + 4*a*b^3*(7*A + 5*C))*EllipticF[(c + d*x)/2, 2])/(21*d) + (2*b*(261*a^2*b*B + 75*b^3*B + 64*a^3*C + 2*a*b^2*(147*A + 101*C))*Sin[c + d*x])/(315*d*Cos[c + d*x]^(3/2)) + (2*(1098*a^3*b*B + 756*a*b^3*B + 192*a^4*C + 21*b^4*(9*A + 7*C) + 7*a^2*b^2*(261*A + 155*C))*Sin[c + d*x])/(315*d*Sqrt[Cos[c + d*x]]) + (2*(63*A*b^2 + 117*a*b*B + 48*a^2*C + 49*b^2*C)*(b + a*Cos[c + d*x])^2*Sin[c + d*x])/(315*d*Cos[c + d*x]^(5/2)) + (2*(9*b*B + 8*a*C)*(b + a*Cos[c + d*x])^3*Sin[c + d*x])/(63*d*Cos[c + d*x]^(7/2)) + (2*C*(b + a*Cos[c + d*x])^4*Sin[c + d*x])/(9*d*Cos[c + d*x]^(9/2))","A",9,7,43,0.1628,1,"{4112, 3047, 3031, 3021, 2748, 2641, 2639}"
1318,1,209,0,0.9653891,"\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","Int[(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{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b (A+3 C)+a^3 (-B)-3 a b^2 B+3 A b^3\right)}{3 a^4 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 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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}","-\frac{2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b (A+3 C)+a^3 (-B)-3 a b^2 B+3 A b^3\right)}{3 a^4 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 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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}",1,"(2*(5*A*b^2 - 5*a*b*B + a^2*(3*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(5*a^3*d) - (2*(3*A*b^3 - a^3*B - 3*a*b^2*B + a^2*b*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*a^4*d) + (2*b^2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^4*(a + b)*d) - (2*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*a*d)","A",8,7,43,0.1628,1,"{4112, 3049, 3059, 2639, 3002, 2641, 2805}"
1319,1,147,0,0.6683075,"\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","Int[(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{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 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 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a 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 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 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(-2*(A*b - a*B)*EllipticE[(c + d*x)/2, 2])/(a^2*d) + (2*(3*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*a^3*d) - (2*b*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a + b)*d) + (2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",7,7,43,0.1628,1,"{4112, 3049, 3059, 2639, 3002, 2641, 2805}"
1320,1,97,0,0.391676,"\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","Int[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2))/(a + b*Sec[c + d*x]),x]","\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}","\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,"(2*A*EllipticE[(c + d*x)/2, 2])/(a*d) - (2*(A*b - a*B)*EllipticF[(c + d*x)/2, 2])/(a^2*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a + b)*d)","A",6,6,43,0.1395,1,"{4112, 3059, 2639, 3002, 2641, 2805}"
1321,1,118,0,0.5999092,"\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","Int[(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{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)}}","-\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*C*EllipticE[(c + d*x)/2, 2])/(b*d) + (2*A*EllipticF[(c + d*x)/2, 2])/(a*d) - (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*b*(a + b)*d) + (2*C*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",7,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1322,1,158,0,0.9346318,"\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","Int[(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{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)}","\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*B - a*C)*EllipticE[(c + d*x)/2, 2])/(b^2*d) + (2*C*EllipticF[(c + d*x)/2, 2])/(3*b*d) + (2*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b^2*(a + b)*d) + (2*C*Sin[c + d*x])/(3*b*d*Cos[c + d*x]^(3/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(b^2*d*Sqrt[Cos[c + d*x]])","A",8,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1323,1,236,0,1.2781984,"\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","Int[(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{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)}","-\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,"(-2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*EllipticE[(c + d*x)/2, 2])/(5*b^3*d) + (2*(b*B - a*C)*EllipticF[(c + d*x)/2, 2])/(3*b^2*d) - (2*a*(A*b^2 - a*(b*B - a*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(b^3*(a + b)*d) + (2*C*Sin[c + d*x])/(5*b*d*Cos[c + d*x]^(5/2)) + (2*(b*B - a*C)*Sin[c + d*x])/(3*b^2*d*Cos[c + d*x]^(3/2)) + (2*(5*A*b^2 - 5*a*b*B + 5*a^2*C + 3*b^2*C)*Sin[c + d*x])/(5*b^3*d*Sqrt[Cos[c + d*x]])","A",9,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1324,1,346,0,1.2145302,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (16 A-3 C)-2 a^4 (A+3 C)+12 a^3 b B-9 a b^3 B+15 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 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}+\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(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}","-\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (16 A-3 C)-2 a^4 (A+3 C)+12 a^3 b B-9 a b^3 B+15 A b^4\right)}{3 a^4 d \left(a^2-b^2\right)}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A-C)+2 a^3 B-3 a b^2 B+5 A b^3\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(-a^2 b^2 (7 A-C)+5 a^3 b B-3 a^4 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}+\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(a^2 (-(2 A-3 C))-3 a b B+5 A b^2\right)}{3 a^2 d \left(a^2-b^2\right)}",1,"((5*A*b^3 + 2*a^3*B - 3*a*b^2*B - a^2*b*(4*A - C))*EllipticE[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) - ((15*A*b^4 + 12*a^3*b*B - 9*a*b^3*B - a^2*b^2*(16*A - 3*C) - 2*a^4*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(3*a^4*(a^2 - b^2)*d) + (b*(5*A*b^4 + 5*a^3*b*B - 3*a*b^3*B - a^2*b^2*(7*A - C) - 3*a^4*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^4*(a - b)*(a + b)^2*d) - ((5*A*b^2 - 3*a*b*B - a^2*(2*A - 3*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",8,8,43,0.1860,1,"{4112, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1325,1,257,0,0.800583,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b (4 A+C)+2 a^3 B-a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-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}+\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(-a^2 b (4 A+C)+2 a^3 B-a b^2 B+3 A b^3\right)}{a^3 d \left(a^2-b^2\right)}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A-C))-a b B+3 A b^2\right)}{a^2 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (5 A+C)+3 a^3 b B+a^4 (-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}+\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)}",1,"-(((3*A*b^2 - a*b*B - a^2*(2*A - C))*EllipticE[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d)) + ((3*A*b^3 + 2*a^3*B - a*b^2*B - a^2*b*(4*A + C))*EllipticF[(c + d*x)/2, 2])/(a^3*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - a*b^3*B - a^4*C - a^2*b^2*(5*A + C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^3*(a - b)*(a + b)^2*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",7,7,43,0.1628,1,"{4112, 3047, 3059, 2639, 3002, 2641, 2805}"
1326,1,239,0,0.7879574,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A+C))+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{\left(-3 a^2 b^2 (A+C)+a^3 b B+a^4 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}-\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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 (-(2 A+C))+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{\left(-3 a^2 b^2 (A+C)+a^3 b B+a^4 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}-\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)}",1,"((A*b^2 - a*(b*B - a*C))*EllipticE[(c + d*x)/2, 2])/(a*b*(a^2 - b^2)*d) - ((A*b^2 + a*b*B - a^2*(2*A + C))*EllipticF[(c + d*x)/2, 2])/(a^2*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B + a*b^3*B + a^4*C - 3*a^2*b^2*(A + C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a^2*(a - b)*b*(a + b)^2*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(b*(a^2 - b^2)*d*(b + a*Cos[c + d*x]))","A",7,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1327,1,307,0,1.1022043,"\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","Int[(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{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{\left(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 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)}+\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{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{\left(a^2 b^2 (A+5 C)+a^3 b B-3 a^4 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)}+\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)}",1,"-(((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*EllipticE[(c + d*x)/2, 2])/(b^2*(a^2 - b^2)*d)) - ((A*b^2 - a*(b*B - a*C))*EllipticF[(c + d*x)/2, 2])/(a*b*(a^2 - b^2)*d) + ((A*b^4 + a^3*b*B - 3*a*b^3*B - 3*a^4*C + a^2*b^2*(A + 5*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a*b^2*(a + b)*(a^2 - b^2)*d) + ((A*b^2 - a*b*B + 3*a^2*C - 2*b^2*C)*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1328,1,387,0,1.4961022,"\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","Int[(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{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{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-5 a^3 C-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 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}-\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{\sin (c+d x) \left(3 a^2 b B-5 a^3 C-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{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{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(3 a^2 b B-5 a^3 C-a b^2 (A-4 C)-2 b^3 B\right)}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-a^2 b^2 (A-7 C)+3 a^3 b B-5 a^4 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}-\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{\sin (c+d x) \left(3 a^2 b B-5 a^3 C-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)}}",1,"-(((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*EllipticE[(c + d*x)/2, 2])/(b^3*(a^2 - b^2)*d)) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*EllipticF[(c + d*x)/2, 2])/(3*b^2*(a^2 - b^2)*d) - ((3*A*b^4 + 3*a^3*b*B - 5*a*b^3*B - a^2*b^2*(A - 7*C) - 5*a^4*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/((a - b)*b^3*(a + b)^2*d) + ((3*A*b^2 - 3*a*b*B + 5*a^2*C - 2*b^2*C)*Sin[c + d*x])/(3*b^2*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)) + ((3*a^2*b*B - 2*b^3*B - a*b^2*(A - 4*C) - 5*a^3*C)*Sin[c + d*x])/(b^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1329,1,538,0,2.043284,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)+99 a^3 b^3 B-72 a^5 b B-45 a b^5 B+105 A b^6\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (65 A-3 C)+a^4 (24 A b-9 b C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b \left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 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}+\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(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 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^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}","\frac{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^4 b^2 (128 A-15 C)-a^2 b^4 (223 A-9 C)+8 a^6 (A+3 C)+99 a^3 b^3 B-72 a^5 b B-45 a b^5 B+105 A b^6\right)}{12 a^5 d \left(a^2-b^2\right)^2}-\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (65 A-3 C)+a^4 (24 A b-9 b C)+29 a^3 b^2 B-8 a^5 B-15 a b^4 B+35 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b \left(3 a^4 b^2 (21 A-2 C)-a^2 b^4 (86 A-3 C)+38 a^3 b^3 B-35 a^5 b B+15 a^6 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}+\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(-a^2 b^2 (13 A+C)+9 a^3 b B-5 a^4 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^2 b^2 (61 A-3 C)+a^4 (8 A-21 C)+33 a^3 b B-15 a b^3 B+35 A b^4\right)}{12 a^3 d \left(a^2-b^2\right)^2}",1,"-((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))*EllipticE[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((105*A*b^6 - 72*a^5*b*B + 99*a^3*b^3*B - 45*a*b^5*B + a^4*b^2*(128*A - 15*C) - a^2*b^4*(223*A - 9*C) + 8*a^6*(A + 3*C))*EllipticF[(c + d*x)/2, 2])/(12*a^5*(a^2 - b^2)^2*d) - (b*(35*A*b^6 - 35*a^5*b*B + 38*a^3*b^3*B - 15*a*b^5*B - a^2*b^4*(86*A - 3*C) + 3*a^4*b^2*(21*A - 2*C) + 15*a^6*C)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^5*(a - b)^2*(a + b)^3*d) + ((35*A*b^4 + 33*a^3*b*B - 15*a*b^3*B + a^4*(8*A - 21*C) - a^2*b^2*(61*A - 3*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(12*a^3*(a^2 - b^2)^2*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - ((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]^(3/2)*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",9,8,43,0.1860,1,"{4112, 3047, 3049, 3059, 2639, 3002, 2641, 2805}"
1330,1,426,0,1.4376771,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^3 (33 A+C)+a^4 b (24 A+7 C)+5 a^3 b^2 B-8 a^5 B-3 a b^4 B+15 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 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}+\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{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 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(-a^2 b^3 (33 A+C)+a^4 b (24 A+7 C)+5 a^3 b^2 B-8 a^5 B-3 a b^4 B+15 A b^5\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{E\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (29 A+C)+a^4 (8 A-5 C)+9 a^3 b B-3 a b^3 B+15 A b^4\right)}{4 a^3 d \left(a^2-b^2\right)^2}+\frac{\left(5 a^4 b^2 (7 A+2 C)-a^2 b^4 (38 A+C)+6 a^3 b^3 B-15 a^5 b B+3 a^6 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}+\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{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (11 A+3 C)+7 a^3 b B-3 a^4 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)}",1,"((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))*EllipticE[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) - ((15*A*b^5 - 8*a^5*B + 5*a^3*b^2*B - 3*a*b^4*B - a^2*b^3*(33*A + C) + a^4*b*(24*A + 7*C))*EllipticF[(c + d*x)/2, 2])/(4*a^4*(a^2 - b^2)^2*d) + ((15*A*b^6 - 15*a^5*b*B + 6*a^3*b^3*B - 3*a*b^5*B + 3*a^6*C - a^2*b^4*(38*A + C) + 5*a^4*b^2*(7*A + 2*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^4*(a - b)^2*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) - ((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))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{4112, 3047, 3059, 2639, 3002, 2641, 2805}"
1331,1,423,0,1.35867,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-a^2 b^2 (5 A-3 C)+a^4 (8 A+3 C)-7 a^3 b B+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^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-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}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-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{\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^2 b^2 (5 A-3 C)+a^4 (8 A+3 C)-7 a^3 b B+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^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-C)+a b^3 B+3 A b^4\right)}{4 a^2 b d \left(a^2-b^2\right)^2}-\frac{\left(5 a^4 b^2 (3 A+2 C)-3 a^2 b^4 (2 A-C)-10 a^3 b^3 B-3 a^5 b B+a^6 (-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}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (9 A+5 C)+5 a^3 b B+a^4 (-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{\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}",1,"-((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*EllipticE[(c + d*x)/2, 2])/(4*a^2*b*(a^2 - b^2)^2*d) + ((3*A*b^4 - 7*a^3*b*B + a*b^3*B - a^2*b^2*(5*A - 3*C) + a^4*(8*A + 3*C))*EllipticF[(c + d*x)/2, 2])/(4*a^3*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B - 10*a^3*b^3*B + a*b^5*B - 3*a^2*b^4*(2*A - C) - a^6*C + 5*a^4*b^2*(3*A + 2*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^3*(a - b)^2*b*(a + b)^3*d) + ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*a*(a^2 - b^2)*d*(b + a*Cos[c + d*x])^2) + ((3*A*b^4 + 5*a^3*b*B + a*b^3*B - a^4*C - a^2*b^2*(9*A + 5*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*a*b*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",8,8,43,0.1860,1,"{4112, 3047, 3055, 3059, 2639, 3002, 2641, 2805}"
1332,1,409,0,1.3539553,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 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(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 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}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 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{\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(-7 a^2 b^2 (A+C)+3 a^3 b B+a^4 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(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 C-5 a b^3 B+A b^4\right)}{4 a b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^4 b^2 (A-2 C)-5 a^2 b^4 (2 A+3 C)+10 a^3 b^3 B-a^5 b B-3 a^6 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}+\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 b^2 (5 A+9 C)-a^3 b B-3 a^4 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{\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}",1,"-((A*b^4 - a^3*b*B - 5*a*b^3*B - 3*a^4*C + a^2*b^2*(5*A + 9*C))*EllipticE[(c + d*x)/2, 2])/(4*a*b^2*(a^2 - b^2)^2*d) + ((A*b^4 + 3*a^3*b*B + 3*a*b^3*B + a^4*C - 7*a^2*b^2*(A + C))*EllipticF[(c + d*x)/2, 2])/(4*a^2*b*(a^2 - b^2)^2*d) - ((A*b^6 - a^5*b*B + 10*a^3*b^3*B + 3*a*b^5*B - 3*a^4*b^2*(A - 2*C) - 3*a^6*C - 5*a^2*b^4*(2*A + 3*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a^2*(a - b)^2*b^2*(a + b)^3*d) - ((A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*(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))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*(b + a*Cos[c + d*x]))","A",8,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1333,1,496,0,1.8811798,"\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","Int[(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{F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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{\left(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 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}-\frac{\sin (c+d x) \left(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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{\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(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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{\left(-a^4 b^2 (A+38 C)+5 a^2 b^4 (2 A+7 C)+6 a^3 b^3 B-3 a^5 b B+15 a^6 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}-\frac{\sin (c+d x) \left(a^2 b^2 (A+29 C)+3 a^3 b B-15 a^4 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(a^2 b^2 (3 A+11 C)+a^3 b B-5 a^4 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{\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}",1,"((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*EllipticE[(c + d*x)/2, 2])/(4*b^3*(a^2 - b^2)^2*d) + ((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))*EllipticF[(c + d*x)/2, 2])/(4*a*b^2*(a^2 - b^2)^2*d) - ((3*A*b^6 - 3*a^5*b*B + 6*a^3*b^3*B - 15*a*b^5*B + 15*a^6*C + 5*a^2*b^4*(2*A + 7*C) - a^4*b^2*(A + 38*C))*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(4*a*(a - b)^2*b^3*(a + b)^3*d) - ((3*a^3*b*B - 9*a*b^3*B + b^4*(5*A - 8*C) - 15*a^4*C + a^2*b^2*(A + 29*C))*Sin[c + d*x])/(4*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]) - ((A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(2*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x])^2) + ((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))*Sin[c + d*x])/(4*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*(b + a*Cos[c + d*x]))","A",9,7,43,0.1628,1,"{4112, 3055, 3059, 2639, 3002, 2641, 2805}"
1334,1,457,0,1.7983867,"\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","Int[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]","-\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(a^2 b (13 A+21 C)+75 a^3 B-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(6 a^2 b (6 A+7 C)-75 a^3 B-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(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-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}","-\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(a^2 b (13 A+21 C)+75 a^3 B-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(6 a^2 b (6 A+7 C)-75 a^3 B-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(6 a^2 b^2 (4 A+7 C)-21 a^4 (7 A+9 C)-57 a^3 b B-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,"(-2*(a^2 - b^2)*(16*A*b^3 - 75*a^3*B - 24*a*b^2*B + 6*a^2*b*(6*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(8*A*b^3 + 75*a^3*B - 12*a*b^2*B + a^2*b*(13*A + 21*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^3*d) - (2*(6*A*b^2 - 9*a*b*B - 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d) + (2*(A*b + 9*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(63*a*d) + (2*A*Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(9*d)","A",12,10,45,0.2222,1,"{4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1335,1,360,0,1.2994309,"\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","Int[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]","-\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(a^2 b (19 A+35 C)+63 a^3 B-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}","-\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(a^2 b (19 A+35 C)+63 a^3 B-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,"(2*(a^2 - b^2)*(25*a^2*A + 8*A*b^2 - 14*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 63*a^3*B - 14*a*b^2*B + a^2*b*(19*A + 35*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^2 - 7*a*b*B - 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^2*d) + (2*(A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*d)","A",11,10,45,0.2222,1,"{4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1336,1,273,0,0.9381425,"\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","Int[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{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}","-\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,"(-2*(a^2 - b^2)*(2*A*b - 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^2 - 5*a*b*B - 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*d)","A",10,10,45,0.2222,1,"{4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1337,1,277,0,1.028838,"\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","Int[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]","-\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)}}","-\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,"(-2*(A*b^2 - a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b + 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d)","A",13,13,45,0.2889,1,"{4265, 4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1338,1,258,0,0.9644,"\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","Int[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]","\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)}}","\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,"((2*a*B + b*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B + a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*A - C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(d*Sqrt[Cos[c + d*x]])","A",13,13,45,0.2889,1,"{4265, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1339,1,346,0,1.327046,"\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","Int[(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]","\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)}","\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,"((8*a*A + 4*b*B + 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 4*a*b*B - a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B + a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*d*Cos[c + d*x]^(3/2)) + ((4*b*B + a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b*d*Sqrt[Cos[c + d*x]])","A",14,14,45,0.3111,1,"{4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1340,1,447,0,1.7715581,"\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","Int[(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]","\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(2 a^2 b B+a^3 (-C)-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)}","\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(2 a^2 b B+a^3 (-C)-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,"((24*A*b^2 + 18*a*b*B - a^2*C + 16*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((2*a^2*b*B - 8*b^3*B - a^3*C - 4*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Cos[c + d*x]^(5/2)) + ((6*b*B + a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*b*d*Cos[c + d*x]^(3/2)) + ((24*A*b^2 + 6*a*b*B - 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b^2*d*Sqrt[Cos[c + d*x]])","A",15,14,45,0.3111,1,"{4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1341,1,455,0,1.8568161,"\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","Int[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]","\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(-2 a^2 b (44 A+63 C)-75 a^3 B-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(a^2 (39 A b+63 b C)+75 a^3 B-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(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-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}","\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(-2 a^2 b (44 A+63 C)-75 a^3 B-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(a^2 (39 A b+63 b C)+75 a^3 B-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(3 a^2 b^2 (11 A+21 C)+21 a^4 (7 A+9 C)+246 a^3 b B-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,"(2*(a^2 - b^2)*(8*A*b^3 + 75*a^3*B - 18*a*b^2*B + a^2*(39*A*b + 63*b*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b^3 - 75*a^3*B - 9*a*b^2*B - 2*a^2*b*(44*A + 63*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a^2*d) + (2*(3*A*b^2 + 72*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(A*b + 3*a*B)*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(21*d) + (2*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(9*d)","A",12,10,45,0.2222,1,"{4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1342,1,359,0,1.3829517,"\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","Int[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]","\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(-2 a^2 b (41 A+70 C)-63 a^3 B-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}","\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(-2 a^2 b (41 A+70 C)-63 a^3 B-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,"(2*(a^2 - b^2)*(25*a^2*A - 6*A*b^2 + 21*a*b*B + 35*a^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(6*A*b^3 - 63*a^3*B - 21*a*b^2*B - 2*a^2*b*(41*A + 70*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(3*A*b^2 + 42*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a*d) + (2*(3*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(7*d)","A",11,10,45,0.2222,1,"{4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1343,1,356,0,1.3990655,"\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","Int[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]","-\frac{2 \left(-3 a^2 b (A+5 C)-5 a^3 B+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 \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 (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)}}","-\frac{2 \left(-3 a^2 b (A+5 C)-5 a^3 B+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 \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 (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,"(-2*(3*A*b^3 - 5*a^3*B + 5*a*b^2*B - 3*a^2*b*(A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(3*A*b^2 + 20*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(3*A*b + 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d) + (2*A*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(5*d)","A",14,13,45,0.2889,1,"{4265, 4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1344,1,340,0,1.3848057,"\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","Int[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]","\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)}}","\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,"((6*a*b*B - b^2*(2*A - 3*C) + 2*a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(2*b*B + 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b + 6*a*B - 3*b*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(2*A - 3*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d)","A",14,14,45,0.3111,1,"{4265, 4094, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1345,1,353,0,1.3407467,"\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","Int[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]","\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)}}","\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,"((8*a^2*B + 4*b^2*B + a*b*(8*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 + 12*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*a*A - 4*b*B - 5*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((4*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(2*d*Sqrt[Cos[c + d*x]])","A",14,13,45,0.2889,1,"{4265, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1346,1,446,0,1.8107894,"\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","Int[((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]","\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(6 a^2 b B+a^3 (-C)+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)}","\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(6 a^2 b B+a^3 (-C)+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,"((42*a*b*B + 8*b^2*(3*A + 2*C) + a^2*(48*A + 17*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((6*a^2*b*B + 8*b^3*B - a^3*C + 12*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((2*b*B + a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2)) + ((24*A*b^2 + 30*a*b*B + 3*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*b*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d*Cos[c + d*x]^(3/2))","A",15,14,45,0.3111,1,"{4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1347,1,551,0,2.3483169,"\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","Int[((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]","\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(24 a^2 b B-9 a^3 C+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(136 a^2 b B-3 a^3 C+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(24 a^2 b B-9 a^3 C+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(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 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)}","\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(24 a^2 b B-9 a^3 C+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(136 a^2 b B-3 a^3 C+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(24 a^2 b B-9 a^3 C+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(-24 a^2 b^2 (2 A+C)+8 a^3 b B-3 a^4 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,"((136*a^2*b*B + 128*b^3*B - 3*a^3*C + 12*a*b^2*(28*A + 19*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(192*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((8*a^3*b*B - 96*a*b^3*B - 3*a^4*C - 24*a^2*b^2*(2*A + C) - 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(64*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((8*b*B + 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Cos[c + d*x]^(5/2)) + ((48*A*b^2 + 56*a*b*B + 3*a^2*C + 36*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(96*b*d*Cos[c + d*x]^(3/2)) + ((24*a^2*b*B + 128*b^3*B - 9*a^3*C + 12*a*b^2*(20*A + 13*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b^2*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(5/2))","A",16,14,45,0.3111,1,"{4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1348,1,565,0,2.4846332,"\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","Int[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]","\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(5 a^2 b (229 A+297 C)+539 a^3 B+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(-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-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(15 a^2 b^2 (19 A+33 C)+75 a^4 (9 A+11 C)+1254 a^3 b B-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(15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-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}","\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(5 a^2 b (229 A+297 C)+539 a^3 B+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(-5 a^2 b^2 (205 A+297 C)-75 a^4 (9 A+11 C)-1793 a^3 b B-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(15 a^2 b^2 (19 A+33 C)+75 a^4 (9 A+11 C)+1254 a^3 b B-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(15 a^2 b^3 (17 A+33 C)+15 a^4 b (247 A+319 C)+3069 a^3 b^2 B+1617 a^5 B-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,"(2*(a^2 - b^2)*(40*A*b^4 + 1254*a^3*b*B - 110*a*b^3*B + 75*a^4*(9*A + 11*C) + 15*a^2*b^2*(19*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3465*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3465*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(20*A*b^4 - 1793*a^3*b*B - 55*a*b^3*B - 75*a^4*(9*A + 11*C) - 5*a^2*b^2*(205*A + 297*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a^2*d) + (2*(15*A*b^3 + 539*a^3*B + 825*a*b^2*B + 5*a^2*b*(229*A + 297*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3465*a*d) + (2*(5*A*b^2 + 44*a*b*B + 3*a^2*(9*A + 11*C))*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(231*d) + (2*(5*A*b + 11*a*B)*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(99*d) + (2*A*Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(11*d)","A",13,10,45,0.2222,1,"{4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1349,1,452,0,1.910442,"\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","Int[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]","\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(a^2 b (163 A+231 C)+75 a^3 B+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(-6 a^2 b (19 A+28 C)-75 a^3 B-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(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-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}","\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(a^2 b (163 A+231 C)+75 a^3 B+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(-6 a^2 b (19 A+28 C)-75 a^3 B-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(-3 a^2 b^2 (93 A+161 C)-21 a^4 (7 A+9 C)-435 a^3 b B-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,"(-2*(a^2 - b^2)*(10*A*b^3 - 75*a^3*B - 45*a*b^2*B - 6*a^2*b*(19*A + 28*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(315*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(315*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(5*A*b^3 + 75*a^3*B + 135*a*b^2*B + a^2*b*(163*A + 231*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*a*d) + (2*(15*A*b^2 + 90*a*b*B + 7*a^2*(7*A + 9*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(315*d) + (2*(5*A*b + 9*a*B)*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(63*d) + (2*A*Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(9*d)","A",12,10,45,0.2222,1,"{4265, 4094, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1350,1,441,0,1.8234946,"\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","Int[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]","\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 \left(10 a^2 b^2 (A-7 C)-5 a^4 (5 A+7 C)-56 a^3 b B+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 \sqrt{\cos (c+d x)} \left(5 a^2 b (29 A+49 C)+63 a^3 B+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 (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)}}","\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 \left(10 a^2 b^2 (A-7 C)-5 a^4 (5 A+7 C)-56 a^3 b B+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 \sqrt{\cos (c+d x)} \left(5 a^2 b (29 A+49 C)+63 a^3 B+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 (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,"(-2*(15*A*b^4 - 56*a^3*b*B + 56*a*b^3*B + 10*a^2*b^2*(A - 7*C) - 5*a^4*(5*A + 7*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b^3*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(15*A*b^3 + 63*a^3*B + 161*a*b^2*B + 5*a^2*b*(29*A + 49*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(15*A*b^2 + 56*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*d) + (2*(5*A*b + 7*a*B)*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(35*d) + (2*A*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(7*d)","A",15,13,45,0.2889,1,"{4265, 4094, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1351,1,419,0,1.828003,"\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","Int[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]","\frac{\left(4 a^2 b (4 A+15 C)+10 a^3 B+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{\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{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)}}","\frac{\left(4 a^2 b (4 A+15 C)+10 a^3 B+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{\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{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,"((10*a^3*B + 20*a*b^2*B - b^3*(16*A - 15*C) + 4*a^2*b*(4*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b^2*(2*b*B + 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((70*a*b*B + b^2*(46*A - 15*C) + 6*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(16*A*b + 10*a*B - 15*b*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[Cos[c + d*x]]) + (2*(A*b + a*B)*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(3*d) + (2*A*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d)","A",15,14,45,0.3111,1,"{4265, 4094, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1352,1,427,0,1.8281729,"\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","Int[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]","\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{\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{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}","\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{\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{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,"((48*a^2*b*B + 12*b^3*B + 8*a^3*(A + 3*C) + a*b^2*(16*A + 33*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(12*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (b*(8*A*b^2 + 20*a*b*B + 15*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((24*a^2*B - 12*b^2*B + a*b*(56*A - 27*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(12*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (b*(8*a*A - 12*b*B - 21*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) - (b*(4*A - 3*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(6*d*Sqrt[Cos[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d)","A",15,14,45,0.3111,1,"{4265, 4094, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1353,1,453,0,1.8757034,"\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","Int[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]","\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{\left(a^2 b (96 A+59 C)+48 a^3 B+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{\sqrt{\cos (c+d x)} \left(a^2 (-(48 A-33 C))+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(30 a^2 b B+5 a^3 C+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)}}","\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{\left(a^2 b (96 A+59 C)+48 a^3 B+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{\sqrt{\cos (c+d x)} \left(a^2 (-(48 A-33 C))+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(30 a^2 b B+5 a^3 C+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,"((48*a^3*B + 66*a*b^2*B + 8*b^3*(3*A + 2*C) + a^2*b*(96*A + 59*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(24*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((30*a^2*b*B + 8*b^3*B + 5*a^3*C + 20*a*b^2*(2*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(8*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((54*a*b*B - a^2*(48*A - 33*C) + 8*b^2*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(24*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((24*A*b^2 + 42*a*b*B + 15*a^2*C + 16*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(24*d*Sqrt[Cos[c + d*x]]) + ((6*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(12*d*Sqrt[Cos[c + d*x]]) + (C*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(3*d*Sqrt[Cos[c + d*x]])","A",15,13,45,0.2889,1,"{4265, 4096, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1354,1,550,0,2.3861498,"\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","Int[((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]","\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(264 a^2 b B+15 a^3 C+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(264 a^2 b B+15 a^3 C+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(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 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)}","\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(264 a^2 b B+15 a^3 C+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(264 a^2 b B+15 a^3 C+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(120 a^2 b^2 (2 A+C)+40 a^3 b B-5 a^4 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,"((472*a^2*b*B + 128*b^3*B + 4*a*b^2*(132*A + 89*C) + a^3*(384*A + 133*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(192*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((40*a^3*b*B + 160*a*b^3*B - 5*a^4*C + 120*a^2*b^2*(2*A + C) + 16*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(64*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(192*b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((16*A*b^2 + 24*a*b*B + 5*a^2*C + 12*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(32*d*Cos[c + d*x]^(3/2)) + ((264*a^2*b*B + 128*b^3*B + 15*a^3*C + 4*a*b^2*(108*A + 71*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(192*b*d*Sqrt[Cos[c + d*x]]) + ((8*b*B + 5*a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(24*d*Cos[c + d*x]^(3/2)) + (C*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(4*d*Cos[c + d*x]^(3/2))","A",16,14,45,0.3111,1,"{4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1355,1,674,0,2.9736851,"\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","Int[((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]","\frac{\sin (c+d x) \left(590 a^2 b B+15 a^3 C+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(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(12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 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(4 a^2 b^2 (1180 A+809 C)+1330 a^3 b B-15 a^4 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(12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 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(-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B+10 a^4 b B-3 a^5 C-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)}","\frac{\sin (c+d x) \left(590 a^2 b B+15 a^3 C+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(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(12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 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(4 a^2 b^2 (1180 A+809 C)+1330 a^3 b B-15 a^4 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(12 a^2 b^2 (220 A+141 C)+150 a^3 b B-45 a^4 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(-40 a^3 b^2 (2 A+C)-240 a^2 b^3 B+10 a^4 b B-3 a^5 C-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,"((1330*a^3*b*B + 3560*a*b^3*B - 15*a^4*C + 256*b^4*(5*A + 4*C) + 4*a^2*b^2*(1180*A + 809*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(1920*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((10*a^4*b*B - 240*a^2*b^3*B - 96*b^5*B - 3*a^5*C - 40*a^3*b^2*(2*A + C) - 80*a*b^4*(4*A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(128*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(1920*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + ((80*A*b^2 + 110*a*b*B + 15*a^2*C + 64*b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(240*d*Cos[c + d*x]^(5/2)) + ((590*a^2*b*B + 360*b^3*B + 15*a^3*C + 4*a*b^2*(260*A + 193*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(960*b*d*Cos[c + d*x]^(3/2)) + ((150*a^3*b*B + 2840*a*b^3*B - 45*a^4*C + 256*b^4*(5*A + 4*C) + 12*a^2*b^2*(220*A + 141*C))*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(1920*b^2*d*Sqrt[Cos[c + d*x]]) + ((2*b*B + a*C)*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)) + (C*(a + b*Sec[c + d*x])^(5/2)*Sin[c + d*x])/(5*d*Cos[c + d*x]^(5/2))","A",17,14,45,0.3111,1,"{4265, 4096, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1356,1,380,0,1.3500837,"\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","Int[(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{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 \left(2 a^2 b^2 (16 A+35 C)+5 a^4 (5 A+7 C)-49 a^3 b B-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 \sqrt{\cos (c+d x)} \left(a^2 (44 A b+70 b C)-63 a^3 B-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 (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 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 a 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 \left(2 a^2 b^2 (16 A+35 C)+5 a^4 (5 A+7 C)-49 a^3 b B-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 \sqrt{\cos (c+d x)} \left(a^2 (44 A b+70 b C)-63 a^3 B-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 (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 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 a d}",1,"(2*(48*A*b^4 - 49*a^3*b*B - 56*a*b^3*B + 5*a^4*(5*A + 7*C) + 2*a^2*b^2*(16*A + 35*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(105*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(48*A*b^3 - 63*a^3*B - 56*a*b^2*B + a^2*(44*A*b + 70*b*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(105*a^4*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(24*A*b^2 - 28*a*b*B + 5*a^2*(5*A + 7*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(105*a^3*d) - (2*(6*A*b - 7*a*B)*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(35*a^2*d) + (2*A*Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(7*a*d)","A",11,9,45,0.2000,1,"{4265, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1357,1,291,0,0.9751894,"\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","Int[(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 \left(a^2 b (7 A+15 C)-5 a^3 B-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 \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 (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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}","-\frac{2 \left(a^2 b (7 A+15 C)-5 a^3 B-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 \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 (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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}",1,"(-2*(8*A*b^3 - 5*a^3*B - 10*a*b^2*B + a^2*b*(7*A + 15*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^2 - 10*a*b*B + 3*a^2*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^3*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(4*A*b - 5*a*B)*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^2*d) + (2*A*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a*d)","A",10,9,45,0.2000,1,"{4265, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1358,1,216,0,0.6652063,"\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","Int[(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{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}","\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,"(2*(2*A*b^2 - 3*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b - 3*a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*A*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a*d)","A",9,9,45,0.2000,1,"{4265, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1359,1,219,0,0.7560444,"\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","Int[(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]","-\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)}}","-\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,"(-2*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])","A",12,12,45,0.2667,1,"{4265, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1360,1,260,0,0.9798302,"\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","Int[(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]","\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}}}","\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,"((2*A + C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (C*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b*d*Sqrt[Cos[c + d*x]])","A",13,13,45,0.2889,1,"{4265, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1361,1,350,0,1.2848074,"\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","Int[(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]","\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)}","\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,"((4*b*B - a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(4*b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^2 - 4*a*b*B + 3*a^2*C + 4*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(4*b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((4*b*B - 3*a*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(4*b^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (C*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(2*b*d*Cos[c + d*x]^(3/2)) + ((4*b*B - 3*a*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(4*b^2*d*Sqrt[Cos[c + d*x]])","A",14,13,45,0.2889,1,"{4265, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1362,1,208,0,0.8976412,"\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","Int[(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]","\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)}}","\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,"(2*a*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*b*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*A*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(d*Sqrt[(b + a*Cos[c + d*x])/(a + b)])","A",13,13,54,0.2407,1,"{4265, 4072, 4037, 3854, 3858, 2663, 2661, 3859, 2807, 2805, 3856, 2655, 2653}"
1363,1,461,0,1.5672952,"\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","Int[(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]","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 (-(A-5 C))-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(-a^2 (9 A b-15 b C)+5 a^3 B-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(6 a^2 b (2 A+5 C)-5 a^3 B-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(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-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}}}","-\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 (-(A-5 C))-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(-a^2 (9 A b-15 b C)+5 a^3 B-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(6 a^2 b (2 A+5 C)-5 a^3 B-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(-6 a^2 b^2 (4 A-5 C)-3 a^4 (3 A+5 C)+25 a^3 b B-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,"(-2*(48*A*b^3 - 5*a^3*B - 40*a*b^2*B + 6*a^2*b*(2*A + 5*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^4*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^4*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(24*A*b^3 + 5*a^3*B - 20*a*b^2*B - a^2*(9*A*b - 15*b*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)*d) - (2*(6*A*b^2 - 5*a*b*B - a^2*(A - 5*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(5*a^2*(a^2 - b^2)*d)","A",11,10,45,0.2222,1,"{4265, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1364,1,350,0,1.1144964,"\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","Int[(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]","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 (-(A-3 C))-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(-a^2 (5 A b-3 b C)+3 a^3 B-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}}}","-\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 (-(A-3 C))-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(-a^2 (5 A b-3 b C)+3 a^3 B-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,"(2*(8*A*b^2 - 6*a*b*B + a^2*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^3 + 3*a^3*B - 6*a*b^2*B - a^2*(5*A*b - 3*b*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(4*A*b^2 - 3*a*b*B - a^2*(A - 3*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)*d)","A",10,10,45,0.2222,1,"{4265, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1365,1,249,0,0.7552814,"\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","Int[(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{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(a^2 (-(A-C))-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)}}","\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(a^2 (-(A-C))-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,"(-2*(2*A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^2 - a*b*B - a^2*(A - C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",9,9,45,0.2000,1,"{4265, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1366,1,311,0,1.1669845,"\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","Int[(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]","-\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)}}","-\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,"(2*A*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(a*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(a*b*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",13,13,45,0.2889,1,"{4265, 4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1367,1,393,0,1.4924798,"\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","Int[(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]","-\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)}}","-\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,"(C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - 3*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(b^2*(a^2 - b^2)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) + ((2*A*b^2 - 2*a*b*B + 3*a^2*C - b^2*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]])","A",14,14,45,0.3111,1,"{4265, 4098, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1368,1,663,0,2.4599711,"\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","Int[(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]","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (71 A-15 C)+a^4 (3 A-35 C)+50 a^3 b B-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(-2 a^2 b^2 (6 A-C)+9 a^3 b B-6 a^4 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) \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) \sqrt{\cos (c+d x)} \left(-2 a^2 b^3 (49 A-10 C)+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-5 a^5 B-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(-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)+80 a^3 b^2 B+5 a^5 B-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(5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)+140 a^3 b^3 B-40 a^5 b B-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}}}","\frac{2 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(-a^2 b^2 (71 A-15 C)+a^4 (3 A-35 C)+50 a^3 b B-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(-2 a^2 b^2 (6 A-C)+9 a^3 b B-6 a^4 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) \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) \sqrt{\cos (c+d x)} \left(-2 a^2 b^3 (49 A-10 C)+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-5 a^5 B-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(-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)+80 a^3 b^2 B+5 a^5 B-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(5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)+140 a^3 b^3 B-40 a^5 b B-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,"(2*(128*A*b^5 + 5*a^5*B + 80*a^3*b^2*B - 80*a*b^4*B - 4*a^2*b^3*(29*A - 10*C) - a^4*b*(17*A + 45*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(15*a^5*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (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) - 4*a^2*b^4*(53*A - 10*C) + 3*a^6*(3*A + 5*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(15*a^5*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) - (2*(8*A*b^4 + 9*a^3*b*B - 5*a*b^3*B - 2*a^2*b^2*(6*A - C) - 6*a^4*C)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) - (2*(64*A*b^5 - 5*a^5*B + 65*a^3*b^2*B - 40*a*b^4*B + 2*a^4*b*(7*A - 20*C) - 2*a^2*b^3*(49*A - 10*C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^4*(a^2 - b^2)^2*d) + (2*(48*A*b^4 + 50*a^3*b*B - 30*a*b^3*B + a^4*(3*A - 35*C) - a^2*b^2*(71*A - 15*C))*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(15*a^3*(a^2 - b^2)^2*d)","A",12,10,45,0.2222,1,"{4265, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1369,1,521,0,1.8131002,"\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","Int[(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]","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-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(10 a^2 A b^2-7 a^3 b B+4 a^4 C+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 \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 \left(-2 a^2 b^2 (8 A-C)+a^4 (-(A+3 C))+9 a^3 b B-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(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-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}}}","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(-a^2 b^2 (13 A-C)+a^4 (A-5 C)+8 a^3 b B-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(10 a^2 A b^2-7 a^3 b B+4 a^4 C+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 \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 \left(-2 a^2 b^2 (8 A-C)+a^4 (-(A+3 C))+9 a^3 b B-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(-2 a^2 b^3 (14 A-C)+a^4 (8 A b-6 b C)+15 a^3 b^2 B-3 a^5 B-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,"(-2*(16*A*b^4 + 9*a^3*b*B - 8*a*b^3*B - 2*a^2*b^2*(8*A - C) - a^4*(A + 3*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^4*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^4*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)) + (2*(10*a^2*A*b^2 - 6*A*b^4 - 7*a^3*b*B + 3*a*b^3*B + 4*a^4*C)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[a + b*Sec[c + d*x]]) + (2*(8*A*b^4 + 8*a^3*b*B - 4*a*b^3*B + a^4*(A - 5*C) - a^2*b^2*(13*A - C))*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*a^3*(a^2 - b^2)^2*d)","A",11,10,45,0.2222,1,"{4265, 4100, 4104, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1370,1,401,0,1.2289426,"\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","Int[(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]","-\frac{2 \sin (c+d x) \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 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 \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(-a^2 b (9 A+C)+3 a^3 B-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 \sqrt{\cos (c+d x)} \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-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}}}","-\frac{2 \sin (c+d x) \left(-2 a^2 b^2 (4 A+C)+5 a^3 b B-2 a^4 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 \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(-a^2 b (9 A+C)+3 a^3 B-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 \sqrt{\cos (c+d x)} \left(-a^2 b^2 (15 A+C)+3 a^4 (A-C)+6 a^3 b B-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,"(2*(8*A*b^3 + 3*a^3*B - 2*a*b^2*B - a^2*b*(9*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^3*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (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))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) - (2*(4*A*b^4 + 5*a^3*b*B - a*b^3*B - 2*a^4*C - 2*a^2*b^2*(4*A + C))*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,9,45,0.2000,1,"{4265, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1371,1,378,0,1.229336,"\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","Int[(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{2 \sin (c+d x) \left(-5 a^2 b^2 (A+C)+2 a^3 b B+a^4 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)}}-\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(a^2 (-(3 A+C))+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(-2 a^2 b (3 A+2 C)+3 a^3 B+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(-5 a^2 b^2 (A+C)+2 a^3 b B+a^4 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)}}-\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(a^2 (-(3 A+C))+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(-2 a^2 b (3 A+2 C)+3 a^3 B+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}}}",1,"(-2*(2*A*b^2 + a*b*B - a^2*(3*A + C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(2*A*b^3 + 3*a^3*B + a*b^2*B - 2*a^2*b*(3*A + 2*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 + 2*a^3*b*B + 2*a*b^3*B + a^4*C - 5*a^2*b^2*(A + C))*Sin[c + d*x])/(3*a*b*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",10,10,45,0.2222,1,"{4265, 4098, 4100, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1372,1,447,0,1.6481531,"\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","Int[(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]","-\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 \sin (c+d x) \left(a^2 b^2 (3 A+7 C)-3 a^4 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 \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 \sqrt{\cos (c+d x)} \left(a^2 b^2 (3 A+7 C)-3 a^4 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)}}","-\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 \sin (c+d x) \left(a^2 b^2 (3 A+7 C)-3 a^4 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 \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 \sqrt{\cos (c+d x)} \left(a^2 b^2 (3 A+7 C)-3 a^4 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,"(-2*(A*b^2 - a*(b*B - a*C))*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*a*b*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + (2*C*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) - (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*a*b^2*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*(A*b^4 - 4*a*b^3*B - 3*a^4*C + a^2*b^2*(3*A + 7*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",14,13,45,0.2889,1,"{4265, 4098, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"
1373,1,563,0,2.1634905,"\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","Int[(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]","\frac{2 \sin (c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 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{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{\sin (c+d x) \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 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{\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{\sqrt{\cos (c+d x)} \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 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)}}","\frac{2 \sin (c+d x) \left(a^2 b^2 (A+9 C)+2 a^3 b B-5 a^4 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{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{\sin (c+d x) \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 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{\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{\sqrt{\cos (c+d x)} \left(26 a^2 b^2 C+6 a^3 b B-15 a^4 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,"((2*A*b^2 - 2*a*b*B + 5*a^2*C - 3*b^2*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(3*b^2*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((2*b*B - 5*a*C)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]) + ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]*Sqrt[a + b*Sec[c + d*x]])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - (2*(A*b^2 - a*(b*B - a*C))*Sin[c + d*x])/(3*b*(a^2 - b^2)*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)) + (2*(3*A*b^4 + 2*a^3*b*B - 6*a*b^3*B - 5*a^4*C + a^2*b^2*(A + 9*C))*Sin[c + d*x])/(3*b^2*(a^2 - b^2)^2*d*Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]) - ((8*A*b^4 + 6*a^3*b*B - 14*a*b^3*B - 15*a^4*C + 26*a^2*b^2*C - 3*b^4*C)*Sqrt[a + b*Sec[c + d*x]]*Sin[c + d*x])/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Cos[c + d*x]])","A",15,14,45,0.3111,1,"{4265, 4098, 4102, 4108, 3859, 2807, 2805, 4035, 3856, 2655, 2653, 3858, 2663, 2661}"