1,1,99,171,0.4780315,"\int (b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[(b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{(b \sec (c+d x))^{5/2} \left(10 A \sin (2 (c+d x))+20 A \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 B \sin (c+d x)+9 B \sin (3 (c+d x))-36 B \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 A b^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 d}+\frac{2 A b \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}-\frac{6 b^3 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{6 b^2 B \sin (c+d x) \sqrt{b \sec (c+d x)}}{5 d}+\frac{2 B \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 d}",1,"((b*Sec[c + d*x])^(5/2)*(-36*B*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*A*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 21*B*Sin[c + d*x] + 10*A*Sin[2*(c + d*x)] + 9*B*Sin[3*(c + d*x)]))/(30*d)","A",1
2,1,87,136,0.2727154,"\int (b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[(b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{(b \sec (c+d x))^{3/2} \left(2 \sin (c+d x) (3 A \cos (c+d x)+B)-6 A \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 B \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","-\frac{2 A b^2 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A b \sin (c+d x) \sqrt{b \sec (c+d x)}}{d}+\frac{2 B \sin (c+d x) (b \sec (c+d x))^{3/2}}{3 d}+\frac{2 b B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 d}",1,"((b*Sec[c + d*x])^(3/2)*(-6*A*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*B*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(B + 3*A*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
3,1,73,104,0.121667,"\int \sqrt{b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{b \sec (c+d x)} \left(A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \sin (c+d x)-B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{d}+\frac{2 B \sin (c+d x) \sqrt{b \sec (c+d x)}}{d}-\frac{2 b B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}",1,"(2*Sqrt[b*Sec[c + d*x]]*(-(B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2]) + A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + B*Sin[c + d*x]))/d","A",1
4,1,54,82,0.0900083,"\int \frac{A+B \sec (c+d x)}{\sqrt{b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/Sqrt[b*Sec[c + d*x]],x]","\frac{2 \left(A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}","\frac{2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{b d}",1,"(2*(A*EllipticE[(c + d*x)/2, 2] + B*EllipticF[(c + d*x)/2, 2]))/(d*Sqrt[Cos[c + d*x]]*Sqrt[b*Sec[c + d*x]])","A",1
5,1,86,116,0.1846816,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) \left(A \left(\sin (2 (c+d x))+2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)+6 B \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d (b \sec (c+d x))^{3/2}}","\frac{2 A \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 b^2 d}+\frac{2 A \sin (c+d x)}{3 b d \sqrt{b \sec (c+d x)}}+\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}",1,"(Sec[c + d*x]^2*(6*B*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + A*(2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + Sin[2*(c + d*x)])))/(3*d*(b*Sec[c + d*x])^(3/2))","A",1
6,1,88,147,0.5107551,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(5/2),x]","\frac{2 \left(\sin (c+d x) \sqrt{\cos (c+d x)} (3 A \cos (c+d x)+5 B)+9 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+5 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d \cos ^{\frac{5}{2}}(c+d x) (b \sec (c+d x))^{5/2}}","\frac{6 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^2 d \sqrt{\cos (c+d x)} \sqrt{b \sec (c+d x)}}+\frac{2 A \sin (c+d x)}{5 b d (b \sec (c+d x))^{3/2}}+\frac{2 B \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sqrt{b \sec (c+d x)}}{3 b^3 d}+\frac{2 B \sin (c+d x)}{3 b^2 d \sqrt{b \sec (c+d x)}}",1,"(2*(9*A*EllipticE[(c + d*x)/2, 2] + 5*B*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*B + 3*A*Cos[c + d*x])*Sin[c + d*x]))/(15*d*Cos[c + d*x]^(5/2)*(b*Sec[c + d*x])^(5/2))","A",1
7,1,90,119,0.2620581,"\int \sec ^2(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","-\frac{3 \left(-\tan ^2(c+d x)\right)^{3/2} \csc ^3(c+d x) (b \sec (c+d x))^{2/3} \left(11 A \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\sec ^2(c+d x)\right)+8 B \, _2F_1\left(\frac{1}{2},\frac{11}{6};\frac{17}{6};\sec ^2(c+d x)\right)\right)}{88 d}","\frac{3 A \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 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)}}",1,"(-3*Csc[c + d*x]^3*(11*A*Cos[c + d*x]*Hypergeometric2F1[1/2, 4/3, 7/3, Sec[c + d*x]^2] + 8*B*Hypergeometric2F1[1/2, 11/6, 17/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(2/3)*(-Tan[c + d*x]^2)^(3/2))/(88*d)","A",1
8,1,91,116,0.1366901,"\int \sec (c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{5/3} \left(8 A \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sec ^2(c+d x)\right)+5 B \, _2F_1\left(\frac{1}{2},\frac{4}{3};\frac{7}{3};\sec ^2(c+d x)\right)\right)}{40 b d}","\frac{3 A \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 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)}}",1,"(3*Csc[c + d*x]*(8*A*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/6, 11/6, Sec[c + d*x]^2] + 5*B*Hypergeometric2F1[1/2, 4/3, 7/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(5/3)*Sqrt[-Tan[c + d*x]^2])/(40*b*d)","A",1
9,1,88,112,0.0881478,"\int (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{2/3} \left(5 A \cos (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)+2 B \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sec ^2(c+d x)\right)\right)}{10 d}","\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 A 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)}}",1,"(3*Csc[c + d*x]*(5*A*Cos[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2] + 2*B*Hypergeometric2F1[1/2, 5/6, 11/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(2/3)*Sqrt[-Tan[c + d*x]^2])/(10*d)","A",1
10,1,88,115,0.0960053,"\int \cos (c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{2/3} \left(2 A \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)-B \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)\right)}{2 d}","-\frac{3 A b^2 \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 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)}}",1,"(-3*Cot[c + d*x]*(2*A*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2] - B*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(2/3)*Sqrt[-Tan[c + d*x]^2])/(2*d)","A",1
11,1,88,119,0.1408862,"\int \cos ^2(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","-\frac{3 b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left(A \cos (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\sec ^2(c+d x)\right)+4 B \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)\right)}{4 d \sqrt[3]{b \sec (c+d x)}}","-\frac{3 A b^3 \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 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*Cot[c + d*x]*(A*Cos[c + d*x]*Hypergeometric2F1[-2/3, 1/2, 1/3, Sec[c + d*x]^2] + 4*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(4*d*(b*Sec[c + d*x])^(1/3))","A",1
12,1,90,119,0.3040216,"\int \sec ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","-\frac{3 \left(-\tan ^2(c+d x)\right)^{3/2} \csc ^3(c+d x) (b \sec (c+d x))^{4/3} \left(13 A \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\sec ^2(c+d x)\right)+10 B \, _2F_1\left(\frac{1}{2},\frac{13}{6};\frac{19}{6};\sec ^2(c+d x)\right)\right)}{130 d}","\frac{3 A \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 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)}}",1,"(-3*Csc[c + d*x]^3*(13*A*Cos[c + d*x]*Hypergeometric2F1[1/2, 5/3, 8/3, Sec[c + d*x]^2] + 10*B*Hypergeometric2F1[1/2, 13/6, 19/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(4/3)*(-Tan[c + d*x]^2)^(3/2))/(130*d)","A",1
13,1,91,116,0.1507883,"\int \sec (c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{7/3} \left(10 A \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sec ^2(c+d x)\right)+7 B \, _2F_1\left(\frac{1}{2},\frac{5}{3};\frac{8}{3};\sec ^2(c+d x)\right)\right)}{70 b d}","\frac{3 A \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 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)}}",1,"(3*Csc[c + d*x]*(10*A*Cos[c + d*x]*Hypergeometric2F1[1/2, 7/6, 13/6, Sec[c + d*x]^2] + 7*B*Hypergeometric2F1[1/2, 5/3, 8/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(7/3)*Sqrt[-Tan[c + d*x]^2])/(70*b*d)","A",1
14,1,88,112,0.1124301,"\int (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Integrate[(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{4/3} \left(7 A \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)+4 B \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sec ^2(c+d x)\right)\right)}{28 d}","\frac{3 A 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 \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)}}",1,"(3*Csc[c + d*x]*(7*A*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2] + 4*B*Hypergeometric2F1[1/2, 7/6, 13/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(4/3)*Sqrt[-Tan[c + d*x]^2])/(28*d)","A",1
15,1,87,115,0.1156938,"\int \cos (c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{4/3} \left(4 A \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)+B \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)\right)}{4 d}","\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 A b^2 \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}}",1,"(3*Cot[c + d*x]*(4*A*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2] + B*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(4/3)*Sqrt[-Tan[c + d*x]^2])/(4*d)","A",1
16,1,88,119,0.0959707,"\int \cos ^2(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","-\frac{3 b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \sqrt[3]{b \sec (c+d x)} \left(A \cos (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sec ^2(c+d x)\right)-2 B \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)\right)}{2 d}","-\frac{3 A b^3 \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}}",1,"(-3*b*Cot[c + d*x]*(A*Cos[c + d*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[c + d*x]^2] - 2*B*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2])/(2*d)","A",1
17,1,90,117,0.255187,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","-\frac{3 \left(-\tan ^2(c+d x)\right)^{3/2} \csc ^3(c+d x) \left(7 A \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)+4 B \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sec ^2(c+d x)\right)\right)}{28 d (b \sec (c+d x))^{2/3}}","\frac{3 A \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 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)}}",1,"(-3*Csc[c + d*x]^3*(7*A*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2] + 4*B*Hypergeometric2F1[1/2, 7/6, 13/6, Sec[c + d*x]^2])*(-Tan[c + d*x]^2)^(3/2))/(28*d*(b*Sec[c + d*x])^(2/3))","A",1
18,1,90,114,0.1034056,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sqrt[3]{b \sec (c+d x)} \left(4 A \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)+B \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)\right)}{4 b d}","\frac{3 B \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 A \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}}",1,"(3*Csc[c + d*x]*(4*A*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2] + B*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2])/(4*b*d)","A",1
19,1,87,114,0.09043,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(2/3),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \left(A \cos (c+d x) \, _2F_1\left(-\frac{1}{3},\frac{1}{2};\frac{2}{3};\sec ^2(c+d x)\right)-2 B \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)\right)}{2 d (b \sec (c+d x))^{2/3}}","-\frac{3 A 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 \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}}",1,"(-3*Csc[c + d*x]*(A*Cos[c + d*x]*Hypergeometric2F1[-1/3, 1/2, 2/3, Sec[c + d*x]^2] - 2*B*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(2*d*(b*Sec[c + d*x])^(2/3))","A",1
20,1,90,114,0.0247372,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sqrt[3]{b \sec (c+d x)} \left(4 A \cos (c+d x) \, _2F_1\left(\frac{1}{6},\frac{1}{2};\frac{7}{6};\sec ^2(c+d x)\right)+B \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)\right)}{4 b d}","\frac{3 B \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 A \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}}",1,"(3*Csc[c + d*x]*(4*A*Cos[c + d*x]*Hypergeometric2F1[1/6, 1/2, 7/6, Sec[c + d*x]^2] + B*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2])/(4*b*d)","A",1
21,1,90,117,0.0236394,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","-\frac{3 \left(-\tan ^2(c+d x)\right)^{3/2} \csc ^3(c+d x) \left(7 A \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{2}{3};\frac{5}{3};\sec ^2(c+d x)\right)+4 B \, _2F_1\left(\frac{1}{2},\frac{7}{6};\frac{13}{6};\sec ^2(c+d x)\right)\right)}{28 d (b \sec (c+d x))^{2/3}}","\frac{3 A \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 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)}}",1,"(-3*Csc[c + d*x]^3*(7*A*Cos[c + d*x]*Hypergeometric2F1[1/2, 2/3, 5/3, Sec[c + d*x]^2] + 4*B*Hypergeometric2F1[1/2, 7/6, 13/6, Sec[c + d*x]^2])*(-Tan[c + d*x]^2)^(3/2))/(28*d*(b*Sec[c + d*x])^(2/3))","A",1
22,1,91,117,0.2324979,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{2/3} \left(5 A \cos (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)+2 B \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sec ^2(c+d x)\right)\right)}{10 b^2 d}","\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 A \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)}}",1,"(3*Csc[c + d*x]*(5*A*Cos[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2] + 2*B*Hypergeometric2F1[1/2, 5/6, 11/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(2/3)*Sqrt[-Tan[c + d*x]^2])/(10*b^2*d)","A",1
23,1,91,114,0.1012853,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \left(2 A \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)-B \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)\right)}{2 b d \sqrt[3]{b \sec (c+d x)}}","-\frac{3 A \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 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)}}",1,"(-3*Csc[c + d*x]*(2*A*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2] - B*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(2*b*d*(b*Sec[c + d*x])^(1/3))","A",1
24,1,87,114,0.1425174,"\int \frac{A+B \sec (c+d x)}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(A + B*Sec[c + d*x])/(b*Sec[c + d*x])^(4/3),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \left(A \cos (c+d x) \, _2F_1\left(-\frac{2}{3},\frac{1}{2};\frac{1}{3};\sec ^2(c+d x)\right)+4 B \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)\right)}{4 d (b \sec (c+d x))^{4/3}}","-\frac{3 A 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 \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*Csc[c + d*x]*(A*Cos[c + d*x]*Hypergeometric2F1[-2/3, 1/2, 1/3, Sec[c + d*x]^2] + 4*B*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(4*d*(b*Sec[c + d*x])^(4/3))","A",1
25,1,91,114,0.0854056,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","-\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \left(2 A \cos (c+d x) \, _2F_1\left(-\frac{1}{6},\frac{1}{2};\frac{5}{6};\sec ^2(c+d x)\right)-B \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)\right)}{2 b d \sqrt[3]{b \sec (c+d x)}}","-\frac{3 A \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 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)}}",1,"(-3*Csc[c + d*x]*(2*A*Cos[c + d*x]*Hypergeometric2F1[-1/6, 1/2, 5/6, Sec[c + d*x]^2] - B*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2])*Sqrt[-Tan[c + d*x]^2])/(2*b*d*(b*Sec[c + d*x])^(1/3))","A",1
26,1,91,117,0.1878381,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{2/3} \left(5 A \cos (c+d x) \, _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\sec ^2(c+d x)\right)+2 B \, _2F_1\left(\frac{1}{2},\frac{5}{6};\frac{11}{6};\sec ^2(c+d x)\right)\right)}{10 b^2 d}","\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 A \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)}}",1,"(3*Csc[c + d*x]*(5*A*Cos[c + d*x]*Hypergeometric2F1[1/3, 1/2, 4/3, Sec[c + d*x]^2] + 2*B*Hypergeometric2F1[1/2, 5/6, 11/6, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(2/3)*Sqrt[-Tan[c + d*x]^2])/(10*b^2*d)","A",1
27,1,140,167,0.3854371,"\int \sec ^m(c+d x) (b \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{4/3} \sec ^m(c+d x) \left(A (3 m+7) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\sec ^2(c+d x)\right)+B (3 m+4) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+7);\frac{1}{6} (3 m+13);\sec ^2(c+d x)\right)\right)}{d (3 m+4) (3 m+7)}","\frac{3 A 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 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)}}",1,"(3*Csc[c + d*x]*(A*(7 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Sec[c + d*x]^2] + B*(4 + 3*m)*Hypergeometric2F1[1/2, (7 + 3*m)/6, (13 + 3*m)/6, Sec[c + d*x]^2])*Sec[c + d*x]^m*(b*Sec[c + d*x])^(4/3)*Sqrt[-Tan[c + d*x]^2])/(d*(4 + 3*m)*(7 + 3*m))","A",1
28,1,140,165,0.2221758,"\int \sec ^m(c+d x) (b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^{2/3} \sec ^m(c+d x) \left(A (3 m+5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\sec ^2(c+d x)\right)+B (3 m+2) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+5);\frac{1}{6} (3 m+11);\sec ^2(c+d x)\right)\right)}{d (3 m+2) (3 m+5)}","\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 A \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) \sqrt{\sin ^2(c+d x)}}",1,"(3*Csc[c + d*x]*(A*(5 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Sec[c + d*x]^2] + B*(2 + 3*m)*Hypergeometric2F1[1/2, (5 + 3*m)/6, (11 + 3*m)/6, Sec[c + d*x]^2])*Sec[c + d*x]^m*(b*Sec[c + d*x])^(2/3)*Sqrt[-Tan[c + d*x]^2])/(d*(2 + 3*m)*(5 + 3*m))","A",1
29,1,140,165,0.2752803,"\int \sec ^m(c+d x) \sqrt[3]{b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x]),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sqrt[3]{b \sec (c+d x)} \sec ^m(c+d x) \left(A (3 m+4) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\sec ^2(c+d x)\right)+B (3 m+1) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+4);\frac{m}{2}+\frac{5}{3};\sec ^2(c+d x)\right)\right)}{d (3 m+1) (3 m+4)}","\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 A \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) \sqrt{\sin ^2(c+d x)}}",1,"(3*Csc[c + d*x]*(A*(4 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Sec[c + d*x]^2] + B*(1 + 3*m)*Hypergeometric2F1[1/2, (4 + 3*m)/6, 5/3 + m/2, Sec[c + d*x]^2])*Sec[c + d*x]^m*(b*Sec[c + d*x])^(1/3)*Sqrt[-Tan[c + d*x]^2])/(d*(1 + 3*m)*(4 + 3*m))","A",1
30,1,140,165,0.2498328,"\int \frac{\sec ^m(c+d x) (A+B \sec (c+d x))}{\sqrt[3]{b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(1/3),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^m(c+d x) \left(A (3 m+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\sec ^2(c+d x)\right)+B (3 m-1) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+2);\frac{1}{6} (3 m+8);\sec ^2(c+d x)\right)\right)}{d (3 m-1) (3 m+2) \sqrt[3]{b \sec (c+d x)}}","-\frac{3 A \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) \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)}}",1,"(3*Csc[c + d*x]*(A*(2 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Sec[c + d*x]^2] + B*(-1 + 3*m)*Hypergeometric2F1[1/2, (2 + 3*m)/6, (8 + 3*m)/6, Sec[c + d*x]^2])*Sec[c + d*x]^m*Sqrt[-Tan[c + d*x]^2])/(d*(-1 + 3*m)*(2 + 3*m)*(b*Sec[c + d*x])^(1/3))","A",1
31,1,140,165,0.2358415,"\int \frac{\sec ^m(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{2/3}} \, dx","Integrate[(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(2/3),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^m(c+d x) \left(A (3 m+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-2);\frac{1}{6} (3 m+4);\sec ^2(c+d x)\right)+B (3 m-2) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m+1);\frac{1}{6} (3 m+7);\sec ^2(c+d x)\right)\right)}{d (3 m-2) (3 m+1) (b \sec (c+d x))^{2/3}}","-\frac{3 A \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) \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}}",1,"(3*Csc[c + d*x]*(A*(1 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-2 + 3*m)/6, (4 + 3*m)/6, Sec[c + d*x]^2] + B*(-2 + 3*m)*Hypergeometric2F1[1/2, (1 + 3*m)/6, (7 + 3*m)/6, Sec[c + d*x]^2])*Sec[c + d*x]^m*Sqrt[-Tan[c + d*x]^2])/(d*(-2 + 3*m)*(1 + 3*m)*(b*Sec[c + d*x])^(2/3))","A",1
32,1,140,173,0.3402807,"\int \frac{\sec ^m(c+d x) (A+B \sec (c+d x))}{(b \sec (c+d x))^{4/3}} \, dx","Integrate[(Sec[c + d*x]^m*(A + B*Sec[c + d*x]))/(b*Sec[c + d*x])^(4/3),x]","\frac{3 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^m(c+d x) \left(A (3 m-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-4);\frac{1}{6} (3 m+2);\sec ^2(c+d x)\right)+B (3 m-4) \, _2F_1\left(\frac{1}{2},\frac{1}{6} (3 m-1);\frac{1}{6} (3 m+5);\sec ^2(c+d x)\right)\right)}{d (3 m-4) (3 m-1) (b \sec (c+d x))^{4/3}}","-\frac{3 A \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 (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)}}",1,"(3*Csc[c + d*x]*(A*(-1 + 3*m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-4 + 3*m)/6, (2 + 3*m)/6, Sec[c + d*x]^2] + B*(-4 + 3*m)*Hypergeometric2F1[1/2, (-1 + 3*m)/6, (5 + 3*m)/6, Sec[c + d*x]^2])*Sec[c + d*x]^m*Sqrt[-Tan[c + d*x]^2])/(d*(-4 + 3*m)*(-1 + 3*m)*(b*Sec[c + d*x])^(4/3))","A",1
33,1,126,172,0.2225894,"\int \sec ^m(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^m(c+d x) (b \sec (c+d x))^n \left(A (m+n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+n}{2};\frac{1}{2} (m+n+2);\sec ^2(c+d x)\right)+B (m+n) \, _2F_1\left(\frac{1}{2},\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);\sec ^2(c+d x)\right)\right)}{d (m+n) (m+n+1)}","\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{A \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) \sqrt{\sin ^2(c+d x)}}",1,"(Csc[c + d*x]*(A*(1 + m + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (m + n)/2, (2 + m + n)/2, Sec[c + d*x]^2] + B*(m + n)*Hypergeometric2F1[1/2, (1 + m + n)/2, (3 + m + n)/2, Sec[c + d*x]^2])*Sec[c + d*x]^m*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(m + n)*(1 + m + n))","A",1
34,1,119,143,0.2276118,"\int \sec ^2(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec (c+d x) (b \sec (c+d x))^n \left(A (n+3) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sec ^2(c+d x)\right)+B (n+2) \sec (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+3}{2};\frac{n+5}{2};\sec ^2(c+d x)\right)\right)}{d (n+2) (n+3)}","\frac{A \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{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)}}",1,"(Csc[c + d*x]*Sec[c + d*x]*(b*Sec[c + d*x])^n*(A*(3 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sec[c + d*x]^2] + B*(2 + n)*Hypergeometric2F1[1/2, (3 + n)/2, (5 + n)/2, Sec[c + d*x]^2]*Sec[c + d*x])*Sqrt[-Tan[c + d*x]^2])/(d*(2 + n)*(3 + n))","A",1
35,1,119,136,0.257045,"\int \sec (c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec (c+d x) (b \sec (c+d x))^n \left(A (n+2) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(c+d x)\right)+B (n+1) \, _2F_1\left(\frac{1}{2},\frac{n+2}{2};\frac{n+4}{2};\sec ^2(c+d x)\right)\right)}{d (n+1) (n+2)}","\frac{A \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{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)}}",1,"(Csc[c + d*x]*(A*(2 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[c + d*x]^2] + B*(1 + n)*Hypergeometric2F1[1/2, (2 + n)/2, (4 + n)/2, Sec[c + d*x]^2])*Sec[c + d*x]*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(1 + n)*(2 + n))","A",1
36,1,107,137,0.1562668,"\int (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^n \left(A (n+1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(c+d x)\right)+B n \, _2F_1\left(\frac{1}{2},\frac{n+1}{2};\frac{n+3}{2};\sec ^2(c+d x)\right)\right)}{d n (n+1)}","\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{A 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)}}",1,"(Csc[c + d*x]*(A*(1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[c + d*x]^2] + B*n*Hypergeometric2F1[1/2, (1 + n)/2, (3 + n)/2, Sec[c + d*x]^2])*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*n*(1 + n))","A",1
37,1,107,151,0.1611402,"\int \cos (c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^n \left(A n \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\sec ^2(c+d x)\right)+B (n-1) \, _2F_1\left(\frac{1}{2},\frac{n}{2};\frac{n+2}{2};\sec ^2(c+d x)\right)\right)}{d (n-1) n}","-\frac{A b^2 \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 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)}}",1,"(Cot[c + d*x]*(A*n*Cos[c + d*x]*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Sec[c + d*x]^2] + B*(-1 + n)*Hypergeometric2F1[1/2, n/2, (2 + n)/2, Sec[c + d*x]^2])*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(-1 + n)*n)","A",1
38,1,114,153,0.3106921,"\int \cos ^2(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{b \sqrt{-\tan ^2(c+d x)} \cot (c+d x) (b \sec (c+d x))^{n-1} \left(A (n-1) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{n-2}{2};\frac{n}{2};\sec ^2(c+d x)\right)+B (n-2) \, _2F_1\left(\frac{1}{2},\frac{n-1}{2};\frac{n+1}{2};\sec ^2(c+d x)\right)\right)}{d (n-2) (n-1)}","-\frac{A b^3 \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^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)}}",1,"(b*Cot[c + d*x]*(A*(-1 + n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (-2 + n)/2, n/2, Sec[c + d*x]^2] + B*(-2 + n)*Hypergeometric2F1[1/2, (-1 + n)/2, (1 + n)/2, Sec[c + d*x]^2])*(b*Sec[c + d*x])^(-1 + n)*Sqrt[-Tan[c + d*x]^2])/(d*(-2 + n)*(-1 + n))","A",1
39,1,140,163,0.3293495,"\int \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{\frac{3}{2}}(c+d x) (b \sec (c+d x))^n \left(A (2 n+5) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\sec ^2(c+d x)\right)+B (2 n+3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+5);\frac{1}{4} (2 n+9);\sec ^2(c+d x)\right)\right)}{d (2 n+3) (2 n+5)}","\frac{2 A \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 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)}}",1,"(2*Csc[c + d*x]*(A*(5 + 2*n)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Sec[c + d*x]^2] + B*(3 + 2*n)*Hypergeometric2F1[1/2, (5 + 2*n)/4, (9 + 2*n)/4, Sec[c + d*x]^2])*Sec[c + d*x]^(3/2)*(b*Sec[c + d*x])^n*Sqrt[-Tan[c + d*x]^2])/(d*(3 + 2*n)*(5 + 2*n))","A",1
40,1,140,163,0.244042,"\int \sqrt{\sec (c+d x)} (b \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^n \left(A (2 n+3) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\sec ^2(c+d x)\right)+B (2 n+1) \sec (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+3);\frac{1}{4} (2 n+7);\sec ^2(c+d x)\right)\right)}{d (2 n+1) (2 n+3) \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 A \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)}}",1,"(2*Csc[c + d*x]*(b*Sec[c + d*x])^n*(A*(3 + 2*n)*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Sec[c + d*x]^2] + B*(1 + 2*n)*Hypergeometric2F1[1/2, (3 + 2*n)/4, (7 + 2*n)/4, Sec[c + d*x]^2]*Sec[c + d*x])*Sqrt[-Tan[c + d*x]^2])/(d*(1 + 2*n)*(3 + 2*n)*Sqrt[Sec[c + d*x]])","A",1
41,1,135,163,0.364279,"\int \frac{(b \sec (c+d x))^n (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^n \left(A (2 n+1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\sec ^2(c+d x)\right)+B (2 n-1) \sec (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n+1);\frac{1}{4} (2 n+5);\sec ^2(c+d x)\right)\right)}{d \left(4 n^2-1\right) \sec ^{\frac{3}{2}}(c+d x)}","-\frac{2 A \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 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)}}",1,"(2*Csc[c + d*x]*(b*Sec[c + d*x])^n*(A*(1 + 2*n)*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Sec[c + d*x]^2] + B*(-1 + 2*n)*Hypergeometric2F1[1/2, (1 + 2*n)/4, (5 + 2*n)/4, Sec[c + d*x]^2]*Sec[c + d*x])*Sqrt[-Tan[c + d*x]^2])/(d*(-1 + 4*n^2)*Sec[c + d*x]^(3/2))","A",1
42,1,140,163,0.3271837,"\int \frac{(b \sec (c+d x))^n (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((b*Sec[c + d*x])^n*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \csc (c+d x) (b \sec (c+d x))^n \left(A (2 n-1) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-3);\frac{1}{4} (2 n+1);\sec ^2(c+d x)\right)+B (2 n-3) \sec (c+d x) \, _2F_1\left(\frac{1}{2},\frac{1}{4} (2 n-1);\frac{1}{4} (2 n+3);\sec ^2(c+d x)\right)\right)}{d (2 n-3) (2 n-1) \sec ^{\frac{5}{2}}(c+d x)}","-\frac{2 A \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 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)}",1,"(2*Csc[c + d*x]*(b*Sec[c + d*x])^n*(A*(-1 + 2*n)*Hypergeometric2F1[1/2, (-3 + 2*n)/4, (1 + 2*n)/4, Sec[c + d*x]^2] + B*(-3 + 2*n)*Hypergeometric2F1[1/2, (-1 + 2*n)/4, (3 + 2*n)/4, Sec[c + d*x]^2]*Sec[c + d*x])*Sqrt[-Tan[c + d*x]^2])/(d*(-3 + 2*n)*(-1 + 2*n)*Sec[c + d*x]^(5/2))","A",1
43,1,87,134,0.7767586,"\int \sec ^4(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \left(45 (A+B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 (A+2 B) \tan ^2(c+d x)+15 (A+B)+3 B \tan ^4(c+d x)\right)+30 (A+B) \sec ^3(c+d x)+45 (A+B) \sec (c+d x)\right)\right)}{120 d}","\frac{a (5 A+4 B) \tan ^3(c+d x)}{15 d}+\frac{a (5 A+4 B) \tan (c+d x)}{5 d}+\frac{3 a (A+B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (A+B) \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 a (A+B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^4(c+d x)}{5 d}",1,"(a*(45*(A + B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(45*(A + B)*Sec[c + d*x] + 30*(A + B)*Sec[c + d*x]^3 + 8*(15*(A + B) + 5*(A + 2*B)*Tan[c + d*x]^2 + 3*B*Tan[c + d*x]^4))))/(120*d)","A",1
44,1,77,106,0.4064809,"\int \sec ^3(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \left(3 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(8 (A+B) (\cos (2 (c+d x))+2) \sec (c+d x)+12 A+6 B \sec ^2(c+d x)+9 B\right)\right)}{24 d}","\frac{a (A+B) \tan ^3(c+d x)}{3 d}+\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a (4 A+3 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(a*(3*(4*A + 3*B)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(12*A + 9*B + 8*(A + B)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*B*Sec[c + d*x]^2)*Tan[c + d*x]))/(24*d)","A",1
45,1,56,86,0.3446132,"\int \sec ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \left(3 (A+B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (A+B) \sec (c+d x)+6 (A+B)+2 B \tan ^2(c+d x)\right)\right)}{6 d}","\frac{a (3 A+2 B) \tan (c+d x)}{3 d}+\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a (A+B) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{a B \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(a*(3*(A + B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*(A + B) + 3*(A + B)*Sec[c + d*x] + 2*B*Tan[c + d*x]^2)))/(6*d)","A",1
46,1,75,56,0.026927,"\int \sec (c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a A \tan (c+d x)}{d}+\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}","\frac{a (A+B) \tan (c+d x)}{d}+\frac{a (2 A+B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a B \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/(2*d) + (a*A*Tan[c + d*x])/d + (a*B*Tan[c + d*x])/d + (a*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
47,1,43,32,0.0163572,"\int (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{a B \tan (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}","\frac{a (A+B) \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{a B \tan (c+d x)}{d}",1,"a*A*x + (a*A*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*B*Tan[c + d*x])/d","A",1
48,1,46,32,0.0268533,"\int \cos (c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+a B x","a x (A+B)+\frac{a A \sin (c+d x)}{d}+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}",1,"a*A*x + a*B*x + (a*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Cos[d*x]*Sin[c])/d + (a*A*Cos[c]*Sin[d*x])/d","A",1
49,1,44,47,0.0999086,"\int \cos ^2(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a (4 (A+B) \sin (c+d x)+A \sin (2 (c+d x))+2 A c+2 A d x+4 B d x)}{4 d}","\frac{a (A+B) \sin (c+d x)}{d}+\frac{1}{2} a x (A+2 B)+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}",1,"(a*(2*A*c + 2*A*d*x + 4*B*d*x + 4*(A + B)*Sin[c + d*x] + A*Sin[2*(c + d*x)]))/(4*d)","A",1
50,1,65,77,0.1812635,"\int \cos ^3(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a (3 (3 A+4 B) \sin (c+d x)+3 (A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+6 A c+6 A d x+6 B c+6 B d x)}{12 d}","\frac{a (2 A+3 B) \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)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(a*(6*A*c + 6*B*c + 6*A*d*x + 6*B*d*x + 3*(3*A + 4*B)*Sin[c + d*x] + 3*(A + B)*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)","A",1
51,1,75,97,0.2422672,"\int \cos ^4(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \left(-32 (A+B) \sin ^3(c+d x)+96 (A+B) \sin (c+d x)+24 (A+B) \sin (2 (c+d x))+3 A \sin (4 (c+d x))+36 A c+36 A d x+48 B c+48 B d x\right)}{96 d}","-\frac{a (A+B) \sin ^3(c+d x)}{3 d}+\frac{a (A+B) \sin (c+d x)}{d}+\frac{a (3 A+4 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a x (3 A+4 B)+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(a*(36*A*c + 48*B*c + 36*A*d*x + 48*B*d*x + 96*(A + B)*Sin[c + d*x] - 32*(A + B)*Sin[c + d*x]^3 + 24*(A + B)*Sin[2*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
52,1,77,125,0.2480064,"\int \cos ^5(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \left(-160 (2 A+B) \sin ^3(c+d x)+480 (A+B) \sin (c+d x)+15 (A+B) (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))+96 A \sin ^5(c+d x)\right)}{480 d}","-\frac{a (4 A+5 B) \sin ^3(c+d x)}{15 d}+\frac{a (4 A+5 B) \sin (c+d x)}{5 d}+\frac{a (A+B) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 a (A+B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3}{8} a x (A+B)+\frac{a A \sin (c+d x) \cos ^4(c+d x)}{5 d}",1,"(a*(480*(A + B)*Sin[c + d*x] - 160*(2*A + B)*Sin[c + d*x]^3 + 96*A*Sin[c + d*x]^5 + 15*(A + B)*(12*(c + d*x) + 8*Sin[2*(c + d*x)] + Sin[4*(c + d*x)])))/(480*d)","A",1
53,1,280,169,1.3985339,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(240 (7 A+6 B) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (2 A+B) \sin (2 c+d x)+80 (14 A+15 B) \sin (d x)+330 A \sin (c+2 d x)+330 A \sin (3 c+2 d x)+800 A \sin (2 c+3 d x)+105 A \sin (3 c+4 d x)+105 A \sin (5 c+4 d x)+160 A \sin (4 c+5 d x)+420 B \sin (c+2 d x)+420 B \sin (3 c+2 d x)+720 B \sin (2 c+3 d x)+90 B \sin (3 c+4 d x)+90 B \sin (5 c+4 d x)+144 B \sin (4 c+5 d x))\right)}{7680 d}","\frac{a^2 (10 A+9 B) \tan ^3(c+d x)}{15 d}+\frac{a^2 (10 A+9 B) \tan (c+d x)}{5 d}+\frac{a^2 (7 A+6 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (5 A+6 B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{a^2 (7 A+6 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) \sec ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d}",1,"-1/7680*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^5*(240*(7*A + 6*B)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(14*A + 15*B)*Sin[d*x] - 240*(2*A + B)*Sin[2*c + d*x] + 330*A*Sin[c + 2*d*x] + 420*B*Sin[c + 2*d*x] + 330*A*Sin[3*c + 2*d*x] + 420*B*Sin[3*c + 2*d*x] + 800*A*Sin[2*c + 3*d*x] + 720*B*Sin[2*c + 3*d*x] + 105*A*Sin[3*c + 4*d*x] + 90*B*Sin[3*c + 4*d*x] + 105*A*Sin[5*c + 4*d*x] + 90*B*Sin[5*c + 4*d*x] + 160*A*Sin[4*c + 5*d*x] + 144*B*Sin[4*c + 5*d*x])))/d","A",1
54,1,262,138,1.327741,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","-\frac{a^2 (\cos (c+d x)+1)^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(24 (8 A+7 B) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-24 (5 A+4 B) \sin (c)+3 (8 A+15 B) \sin (d x)+24 A \sin (2 c+d x)+136 A \sin (c+2 d x)-24 A \sin (3 c+2 d x)+24 A \sin (2 c+3 d x)+24 A \sin (4 c+3 d x)+40 A \sin (3 c+4 d x)+45 B \sin (2 c+d x)+128 B \sin (c+2 d x)+21 B \sin (2 c+3 d x)+21 B \sin (4 c+3 d x)+32 B \sin (3 c+4 d x))\right)}{768 d}","\frac{a^2 (8 A+7 B) \tan (c+d x)}{6 d}+\frac{a^2 (8 A+7 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^2 (8 A+7 B) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{(4 A-B) \tan (c+d x) (a \sec (c+d x)+a)^2}{12 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 a d}",1,"-1/768*(a^2*(1 + Cos[c + d*x])^2*Sec[(c + d*x)/2]^4*Sec[c + d*x]^4*(24*(8*A + 7*B)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-24*(5*A + 4*B)*Sin[c] + 3*(8*A + 15*B)*Sin[d*x] + 24*A*Sin[2*c + d*x] + 45*B*Sin[2*c + d*x] + 136*A*Sin[c + 2*d*x] + 128*B*Sin[c + 2*d*x] - 24*A*Sin[3*c + 2*d*x] + 24*A*Sin[2*c + 3*d*x] + 21*B*Sin[2*c + 3*d*x] + 24*A*Sin[4*c + 3*d*x] + 21*B*Sin[4*c + 3*d*x] + 40*A*Sin[3*c + 4*d*x] + 32*B*Sin[3*c + 4*d*x])))/d","A",1
55,1,481,103,6.1925645,"\int \sec (c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 \cos ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left(\frac{4 (6 A+5 B) \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (6 A+5 B) \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{(3 A+7 B) \cos \left(\frac{c}{2}\right)-(3 A+5 B) \sin \left(\frac{c}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{(3 A+5 B) \sin \left(\frac{c}{2}\right)+(3 A+7 B) \cos \left(\frac{c}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}-6 (3 A+2 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 (3 A+2 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{2 B \sin \left(\frac{d x}{2}\right)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 B \sin \left(\frac{d x}{2}\right)}{\left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}\right)}{48 d (A \cos (c+d x)+B)}","\frac{2 a^2 (3 A+2 B) \tan (c+d x)}{3 d}+\frac{a^2 (3 A+2 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (3 A+2 B) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^2*Cos[c + d*x]^3*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(-6*(3*A + 2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*(3*A + 2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*B*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((3*A + 7*B)*Cos[c/2] - (3*A + 5*B)*Sin[c/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(6*A + 5*B)*Sin[(d*x)/2])/((Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*B*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - ((3*A + 7*B)*Cos[c/2] + (3*A + 5*B)*Sin[c/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(6*A + 5*B)*Sin[(d*x)/2])/((Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(48*d*(B + A*Cos[c + d*x]))","B",1
56,1,307,82,1.3227077,"\int (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 \cos ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left(\frac{4 (A+2 B) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (A+2 B) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 (4 A+3 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (4 A+3 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+4 A x+\frac{B}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{16 (A \cos (c+d x)+B)}","\frac{a^2 (2 A+3 B) \tan (c+d x)}{2 d}+\frac{a^2 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac{B \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}",1,"(a^2*Cos[c + d*x]^3*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(4*A*x - (2*(4*A + 3*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(4*A + 3*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + B/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(A + 2*B)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - B/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(A + 2*B)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(16*(B + A*Cos[c + d*x]))","B",1
57,1,258,73,1.6616684,"\int \cos (c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 \cos ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left(-\frac{(A+2 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{(A+2 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+x (2 A+B)+\frac{A \sin (c) \cos (d x)}{d}+\frac{A \cos (c) \sin (d x)}{d}+\frac{B \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{B \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{4 (A \cos (c+d x)+B)}","\frac{a^2 (A-B) \sin (c+d x)}{d}+\frac{a^2 (A+2 B) \tanh ^{-1}(\sin (c+d x))}{d}+a^2 x (2 A+B)+\frac{B \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{d}",1,"(a^2*Cos[c + d*x]^3*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((2*A + B)*x - ((A + 2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + ((A + 2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (A*Cos[d*x]*Sin[c])/d + (A*Cos[c]*Sin[d*x])/d + (B*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (B*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(4*(B + A*Cos[c + d*x]))","B",1
58,1,96,88,0.1575254,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 \left(4 (2 A+B) \sin (c+d x)+A \sin (2 (c+d x))+6 A d x-4 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+8 B d x\right)}{4 d}","\frac{a^2 (3 A+2 B) \sin (c+d x)}{2 d}+\frac{1}{2} a^2 x (3 A+4 B)+\frac{A \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{2 d}+\frac{a^2 B \tanh ^{-1}(\sin (c+d x))}{d}",1,"(a^2*(6*A*d*x + 8*B*d*x - 4*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*(2*A + B)*Sin[c + d*x] + A*Sin[2*(c + d*x)]))/(4*d)","A",1
59,1,61,102,0.1740204,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 (3 (7 A+8 B) \sin (c+d x)+3 (2 A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+12 A d x+18 B d x)}{12 d}","\frac{2 a^2 (2 A+3 B) \sin (c+d x)}{3 d}+\frac{a^2 (2 A+3 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (2 A+3 B)+\frac{A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(a^2*(12*A*d*x + 18*B*d*x + 3*(7*A + 8*B)*Sin[c + d*x] + 3*(2*A + B)*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)","A",1
60,1,86,135,0.3540854,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 (24 (6 A+7 B) \sin (c+d x)+48 (A+B) \sin (2 (c+d x))+16 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+84 A c+84 A d x+8 B \sin (3 (c+d x))+96 B d x)}{96 d}","\frac{a^2 (4 A+5 B) \sin (c+d x)}{3 d}+\frac{a^2 (5 A+4 B) \sin (c+d x) \cos ^2(c+d x)}{12 d}+\frac{a^2 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (7 A+8 B)+\frac{A \sin (c+d x) \cos ^3(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{4 d}",1,"(a^2*(84*A*c + 84*A*d*x + 96*B*d*x + 24*(6*A + 7*B)*Sin[c + d*x] + 48*(A + B)*Sin[2*(c + d*x)] + 16*A*Sin[3*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
61,1,108,160,0.4464804,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 (60 (11 A+12 B) \sin (c+d x)+240 (A+B) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+30 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+360 A c+360 A d x+80 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+420 B d x)}{480 d}","-\frac{a^2 (9 A+10 B) \sin ^3(c+d x)}{15 d}+\frac{a^2 (9 A+10 B) \sin (c+d x)}{5 d}+\frac{a^2 (6 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{a^2 (6 A+7 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} a^2 x (6 A+7 B)+\frac{A \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d}",1,"(a^2*(360*A*c + 360*A*d*x + 420*B*d*x + 60*(11*A + 12*B)*Sin[c + d*x] + 240*(A + B)*Sin[2*(c + d*x)] + 90*A*Sin[3*(c + d*x)] + 80*B*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)","A",1
62,1,346,210,1.8853004,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(480 (26 A+23 B) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-320 (19 A+17 B) \sin (c)+750 (2 A+3 B) \sin (d x)+1500 A \sin (2 c+d x)+7680 A \sin (c+2 d x)-1440 A \sin (3 c+2 d x)+1890 A \sin (2 c+3 d x)+1890 A \sin (4 c+3 d x)+3648 A \sin (3 c+4 d x)+390 A \sin (4 c+5 d x)+390 A \sin (6 c+5 d x)+608 A \sin (5 c+6 d x)+2250 B \sin (2 c+d x)+7680 B \sin (c+2 d x)-480 B \sin (3 c+2 d x)+1955 B \sin (2 c+3 d x)+1955 B \sin (4 c+3 d x)+3264 B \sin (3 c+4 d x)+345 B \sin (4 c+5 d x)+345 B \sin (6 c+5 d x)+544 B \sin (5 c+6 d x))\right)}{61440 d}","\frac{a^3 (19 A+17 B) \tan ^3(c+d x)}{15 d}+\frac{a^3 (19 A+17 B) \tan (c+d x)}{5 d}+\frac{a^3 (26 A+23 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^3 (22 A+21 B) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{(3 A+4 B) \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) \tan (c+d x) \sec (c+d x)}{16 d}+\frac{a B \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^2}{6 d}",1,"-1/61440*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^6*(480*(26*A + 23*B)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-320*(19*A + 17*B)*Sin[c] + 750*(2*A + 3*B)*Sin[d*x] + 1500*A*Sin[2*c + d*x] + 2250*B*Sin[2*c + d*x] + 7680*A*Sin[c + 2*d*x] + 7680*B*Sin[c + 2*d*x] - 1440*A*Sin[3*c + 2*d*x] - 480*B*Sin[3*c + 2*d*x] + 1890*A*Sin[2*c + 3*d*x] + 1955*B*Sin[2*c + 3*d*x] + 1890*A*Sin[4*c + 3*d*x] + 1955*B*Sin[4*c + 3*d*x] + 3648*A*Sin[3*c + 4*d*x] + 3264*B*Sin[3*c + 4*d*x] + 390*A*Sin[4*c + 5*d*x] + 345*B*Sin[4*c + 5*d*x] + 390*A*Sin[6*c + 5*d*x] + 345*B*Sin[6*c + 5*d*x] + 608*A*Sin[5*c + 6*d*x] + 544*B*Sin[5*c + 6*d*x])))/d","A",1
63,1,294,163,1.4571566,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(240 (15 A+13 B) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-240 (5 A+3 B) \sin (2 c+d x)+80 (30 A+29 B) \sin (d x)+570 A \sin (c+2 d x)+570 A \sin (3 c+2 d x)+1680 A \sin (2 c+3 d x)-120 A \sin (4 c+3 d x)+225 A \sin (3 c+4 d x)+225 A \sin (5 c+4 d x)+360 A \sin (4 c+5 d x)+750 B \sin (c+2 d x)+750 B \sin (3 c+2 d x)+1520 B \sin (2 c+3 d x)+195 B \sin (3 c+4 d x)+195 B \sin (5 c+4 d x)+304 B \sin (4 c+5 d x))\right)}{15360 d}","\frac{a^3 (15 A+13 B) \tan ^3(c+d x)}{60 d}+\frac{a^3 (15 A+13 B) \tan (c+d x)}{5 d}+\frac{a^3 (15 A+13 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (15 A+13 B) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{(5 A-B) \tan (c+d x) (a \sec (c+d x)+a)^3}{20 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^4}{5 a d}",1,"-1/15360*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^5*(240*(15*A + 13*B)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(30*A + 29*B)*Sin[d*x] - 240*(5*A + 3*B)*Sin[2*c + d*x] + 570*A*Sin[c + 2*d*x] + 750*B*Sin[c + 2*d*x] + 570*A*Sin[3*c + 2*d*x] + 750*B*Sin[3*c + 2*d*x] + 1680*A*Sin[2*c + 3*d*x] + 1520*B*Sin[2*c + 3*d*x] - 120*A*Sin[4*c + 3*d*x] + 225*A*Sin[3*c + 4*d*x] + 195*B*Sin[3*c + 4*d*x] + 225*A*Sin[5*c + 4*d*x] + 195*B*Sin[5*c + 4*d*x] + 360*A*Sin[4*c + 5*d*x] + 304*B*Sin[4*c + 5*d*x])))/d","A",1
64,1,273,125,1.2955746,"\int \sec (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","-\frac{a^3 (\cos (c+d x)+1)^3 \sec ^6\left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) \left(120 (4 A+3 B) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-24 (11 A+9 B) \sin (c)+(36 A+69 B) \sin (d x)+36 A \sin (2 c+d x)+280 A \sin (c+2 d x)-72 A \sin (3 c+2 d x)+36 A \sin (2 c+3 d x)+36 A \sin (4 c+3 d x)+88 A \sin (3 c+4 d x)+69 B \sin (2 c+d x)+264 B \sin (c+2 d x)-24 B \sin (3 c+2 d x)+45 B \sin (2 c+3 d x)+45 B \sin (4 c+3 d x)+72 B \sin (3 c+4 d x))\right)}{1536 d}","\frac{a^3 (4 A+3 B) \tan ^3(c+d x)}{12 d}+\frac{a^3 (4 A+3 B) \tan (c+d x)}{d}+\frac{5 a^3 (4 A+3 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{3 a^3 (4 A+3 B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"-1/1536*(a^3*(1 + Cos[c + d*x])^3*Sec[(c + d*x)/2]^6*Sec[c + d*x]^4*(120*(4*A + 3*B)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-24*(11*A + 9*B)*Sin[c] + (36*A + 69*B)*Sin[d*x] + 36*A*Sin[2*c + d*x] + 69*B*Sin[2*c + d*x] + 280*A*Sin[c + 2*d*x] + 264*B*Sin[c + 2*d*x] - 72*A*Sin[3*c + 2*d*x] - 24*B*Sin[3*c + 2*d*x] + 36*A*Sin[2*c + 3*d*x] + 45*B*Sin[2*c + 3*d*x] + 36*A*Sin[4*c + 3*d*x] + 45*B*Sin[4*c + 3*d*x] + 88*A*Sin[3*c + 4*d*x] + 72*B*Sin[3*c + 4*d*x])))/d","B",1
65,1,1056,111,6.4064512,"\int (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{(-7 A-5 B) \cos ^4(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (B+A \cos (c+d x))}+\frac{(7 A+5 B) \cos ^4(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (B+A \cos (c+d x))}+\frac{A x \cos ^4(c+d x) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{8 (B+A \cos (c+d x))}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \left(9 A \sin \left(\frac{d x}{2}\right)+11 B \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \left(9 A \sin \left(\frac{d x}{2}\right)+11 B \sin \left(\frac{d x}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{24 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \left(3 A \cos \left(\frac{c}{2}\right)+10 B \cos \left(\frac{c}{2}\right)-3 A \sin \left(\frac{c}{2}\right)-8 B \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^4(c+d x) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \left(-3 A \cos \left(\frac{c}{2}\right)-10 B \cos \left(\frac{c}{2}\right)-3 A \sin \left(\frac{c}{2}\right)-8 B \sin \left(\frac{c}{2}\right)\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{B \cos ^4(c+d x) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{B \cos ^4(c+d x) (\sec (c+d x) a+a)^3 (A+B \sec (c+d x)) \sin \left(\frac{d x}{2}\right) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{48 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","\frac{5 a^3 (A+B) \tan (c+d x)}{2 d}+\frac{a^3 (7 A+5 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(3 A+5 B) \tan (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{6 d}+a^3 A x+\frac{a B \tan (c+d x) (a \sec (c+d x)+a)^2}{3 d}",1,"(A*x*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(8*(B + A*Cos[c + d*x])) + ((-7*A - 5*B)*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(16*d*(B + A*Cos[c + d*x])) + ((7*A + 5*B)*Cos[c + d*x]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(16*d*(B + A*Cos[c + d*x])) + (B*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*Sin[(d*x)/2])/(48*d*(B + A*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(3*A*Cos[c/2] + 10*B*Cos[c/2] - 3*A*Sin[c/2] - 8*B*Sin[c/2]))/(96*d*(B + A*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(9*A*Sin[(d*x)/2] + 11*B*Sin[(d*x)/2]))/(24*d*(B + A*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (B*Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*Sin[(d*x)/2])/(48*d*(B + A*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(-3*A*Cos[c/2] - 10*B*Cos[c/2] - 3*A*Sin[c/2] - 8*B*Sin[c/2]))/(96*d*(B + A*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^4*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(9*A*Sin[(d*x)/2] + 11*B*Sin[(d*x)/2]))/(24*d*(B + A*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
66,1,335,108,2.7086194,"\int \cos (c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 \cos ^4(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 (A+B \sec (c+d x)) \left(\frac{4 (A+3 B) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (A+3 B) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 (6 A+7 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (6 A+7 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+4 x (3 A+B)+\frac{4 A \sin (c) \cos (d x)}{d}+\frac{4 A \cos (c) \sin (d x)}{d}+\frac{B}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{32 (A \cos (c+d x)+B)}","\frac{a^3 (6 A+7 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(A+2 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{d}+a^3 x (3 A+B)-\frac{5 a^3 B \sin (c+d x)}{2 d}+\frac{a B \sin (c+d x) (a \sec (c+d x)+a)^2}{2 d}",1,"(a^3*Cos[c + d*x]^4*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(4*(3*A + B)*x - (2*(6*A + 7*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(6*A + 7*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*A*Cos[d*x]*Sin[c])/d + (4*A*Cos[c]*Sin[d*x])/d + B/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(A + 3*B)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - B/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(A + 3*B)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(32*(B + A*Cos[c + d*x]))","B",1
67,1,302,117,5.094592,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 \cos ^4(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 (A+B \sec (c+d x)) \left(\frac{4 (3 A+B) \sin (c) \cos (d x)}{d}+\frac{4 (3 A+B) \cos (c) \sin (d x)}{d}-\frac{4 (A+3 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{4 (A+3 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+2 x (7 A+6 B)+\frac{A \sin (2 c) \cos (2 d x)}{d}+\frac{A \cos (2 c) \sin (2 d x)}{d}+\frac{4 B \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 B \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}\right)}{32 (A \cos (c+d x)+B)}","\frac{a^3 (A+3 B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(A-2 B) \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)+\frac{5 a^3 A \sin (c+d x)}{2 d}+\frac{a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^2}{2 d}",1,"(a^3*Cos[c + d*x]^4*Sec[(c + d*x)/2]^6*(1 + Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(2*(7*A + 6*B)*x - (4*(A + 3*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (4*(A + 3*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(3*A + B)*Cos[d*x]*Sin[c])/d + (A*Cos[2*d*x]*Sin[2*c])/d + (4*(3*A + B)*Cos[c]*Sin[d*x])/d + (A*Cos[2*c]*Sin[2*d*x])/d + (4*B*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (4*B*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(32*(B + A*Cos[c + d*x]))","B",1
68,1,113,125,0.2407378,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 \left(9 (5 A+4 B) \sin (c+d x)+3 (3 A+B) \sin (2 (c+d x))+A \sin (3 (c+d x))+30 A d x-12 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+42 B d x\right)}{12 d}","\frac{5 a^3 (A+B) \sin (c+d x)}{2 d}+\frac{(5 A+3 B) \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 A+7 B)+\frac{a^3 B \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)^2}{3 d}",1,"(a^3*(30*A*d*x + 42*B*d*x - 12*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*(5*A + 4*B)*Sin[c + d*x] + 3*(3*A + B)*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(12*d)","A",1
69,1,86,124,0.2704016,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 (24 (13 A+15 B) \sin (c+d x)+24 (4 A+3 B) \sin (2 (c+d x))+24 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+180 A d x+8 B \sin (3 (c+d x))+240 B d x)}{96 d}","-\frac{a^3 (3 A+4 B) \sin ^3(c+d x)}{12 d}+\frac{a^3 (3 A+4 B) \sin (c+d x)}{d}+\frac{3 a^3 (3 A+4 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{5}{8} a^3 x (3 A+4 B)+\frac{A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^3*(180*A*d*x + 240*B*d*x + 24*(13*A + 15*B)*Sin[c + d*x] + 24*(4*A + 3*B)*Sin[2*(c + d*x)] + 24*A*Sin[3*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
70,1,108,176,0.4305419,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 (60 (23 A+26 B) \sin (c+d x)+480 (A+B) \sin (2 (c+d x))+170 A \sin (3 (c+d x))+45 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+780 A c+780 A d x+120 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+900 B d x)}{480 d}","\frac{a^3 (38 A+45 B) \sin (c+d x)}{15 d}+\frac{a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac{a^3 (13 A+15 B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{(7 A+5 B) \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 A+15 B)+\frac{a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}",1,"(a^3*(780*A*c + 780*A*d*x + 900*B*d*x + 60*(23*A + 26*B)*Sin[c + d*x] + 480*(A + B)*Sin[2*(c + d*x)] + 170*A*Sin[3*(c + d*x)] + 120*B*Sin[3*(c + d*x)] + 45*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)","A",1
71,1,134,201,0.556255,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 (120 (21 A+23 B) \sin (c+d x)+15 (63 A+64 B) \sin (2 (c+d x))+380 A \sin (3 (c+d x))+135 A \sin (4 (c+d x))+36 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+1380 A c+1380 A d x+340 B \sin (3 (c+d x))+90 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+1560 B d x)}{960 d}","-\frac{a^3 (17 A+19 B) \sin ^3(c+d x)}{15 d}+\frac{a^3 (17 A+19 B) \sin (c+d x)}{5 d}+\frac{a^3 (21 A+22 B) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac{a^3 (23 A+26 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(4 A+3 B) \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 A+26 B)+\frac{a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}",1,"(a^3*(1380*A*c + 1380*A*d*x + 1560*B*d*x + 120*(21*A + 23*B)*Sin[c + d*x] + 15*(63*A + 64*B)*Sin[2*(c + d*x)] + 380*A*Sin[3*(c + d*x)] + 340*B*Sin[3*(c + d*x)] + 135*A*Sin[4*(c + d*x)] + 90*B*Sin[4*(c + d*x)] + 36*A*Sin[5*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
72,1,358,194,2.3798217,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^6(c+d x) \left(3360 (8 A+7 B) \cos ^6(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-160 (83 A+72 B) \sin (c)+30 (88 A+125 B) \sin (d x)+2640 A \sin (2 c+d x)+15840 A \sin (c+2 d x)-4080 A \sin (3 c+2 d x)+3480 A \sin (2 c+3 d x)+3480 A \sin (4 c+3 d x)+7728 A \sin (3 c+4 d x)-240 A \sin (5 c+4 d x)+840 A \sin (4 c+5 d x)+840 A \sin (6 c+5 d x)+1328 A \sin (5 c+6 d x)+3750 B \sin (2 c+d x)+15360 B \sin (c+2 d x)-1920 B \sin (3 c+2 d x)+3845 B \sin (2 c+3 d x)+3845 B \sin (4 c+3 d x)+6912 B \sin (3 c+4 d x)+735 B \sin (4 c+5 d x)+735 B \sin (6 c+5 d x)+1152 B \sin (5 c+6 d x))\right)}{122880 d}","\frac{2 a^4 (8 A+7 B) \tan ^3(c+d x)}{15 d}+\frac{4 a^4 (8 A+7 B) \tan (c+d x)}{5 d}+\frac{7 a^4 (8 A+7 B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{a^4 (8 A+7 B) \tan (c+d x) \sec ^3(c+d x)}{40 d}+\frac{27 a^4 (8 A+7 B) \tan (c+d x) \sec (c+d x)}{80 d}+\frac{(6 A-B) \tan (c+d x) (a \sec (c+d x)+a)^4}{30 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^5}{6 a d}",1,"-1/122880*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^6*(3360*(8*A + 7*B)*Cos[c + d*x]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(-160*(83*A + 72*B)*Sin[c] + 30*(88*A + 125*B)*Sin[d*x] + 2640*A*Sin[2*c + d*x] + 3750*B*Sin[2*c + d*x] + 15840*A*Sin[c + 2*d*x] + 15360*B*Sin[c + 2*d*x] - 4080*A*Sin[3*c + 2*d*x] - 1920*B*Sin[3*c + 2*d*x] + 3480*A*Sin[2*c + 3*d*x] + 3845*B*Sin[2*c + 3*d*x] + 3480*A*Sin[4*c + 3*d*x] + 3845*B*Sin[4*c + 3*d*x] + 7728*A*Sin[3*c + 4*d*x] + 6912*B*Sin[3*c + 4*d*x] - 240*A*Sin[5*c + 4*d*x] + 840*A*Sin[4*c + 5*d*x] + 735*B*Sin[4*c + 5*d*x] + 840*A*Sin[6*c + 5*d*x] + 735*B*Sin[6*c + 5*d*x] + 1328*A*Sin[5*c + 6*d*x] + 1152*B*Sin[5*c + 6*d*x])))/d","A",1
73,1,306,159,1.7465928,"\int \sec (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","-\frac{a^4 (\cos (c+d x)+1)^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) \left(1680 (5 A+4 B) \cos ^5(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)-\sec (c) (-960 (3 A+2 B) \sin (2 c+d x)+80 (64 A+59 B) \sin (d x)+930 A \sin (c+2 d x)+930 A \sin (3 c+2 d x)+3520 A \sin (2 c+3 d x)-480 A \sin (4 c+3 d x)+405 A \sin (3 c+4 d x)+405 A \sin (5 c+4 d x)+800 A \sin (4 c+5 d x)+1320 B \sin (c+2 d x)+1320 B \sin (3 c+2 d x)+3200 B \sin (2 c+3 d x)-120 B \sin (4 c+3 d x)+420 B \sin (3 c+4 d x)+420 B \sin (5 c+4 d x)+664 B \sin (4 c+5 d x))\right)}{30720 d}","\frac{4 a^4 (5 A+4 B) \tan ^3(c+d x)}{15 d}+\frac{8 a^4 (5 A+4 B) \tan (c+d x)}{5 d}+\frac{7 a^4 (5 A+4 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{a^4 (5 A+4 B) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{27 a^4 (5 A+4 B) \tan (c+d x) \sec (c+d x)}{40 d}+\frac{B \tan (c+d x) (a \sec (c+d x)+a)^4}{5 d}",1,"-1/30720*(a^4*(1 + Cos[c + d*x])^4*Sec[(c + d*x)/2]^8*Sec[c + d*x]^5*(1680*(5*A + 4*B)*Cos[c + d*x]^5*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) - Sec[c]*(80*(64*A + 59*B)*Sin[d*x] - 960*(3*A + 2*B)*Sin[2*c + d*x] + 930*A*Sin[c + 2*d*x] + 1320*B*Sin[c + 2*d*x] + 930*A*Sin[3*c + 2*d*x] + 1320*B*Sin[3*c + 2*d*x] + 3520*A*Sin[2*c + 3*d*x] + 3200*B*Sin[2*c + 3*d*x] - 480*A*Sin[4*c + 3*d*x] - 120*B*Sin[4*c + 3*d*x] + 405*A*Sin[3*c + 4*d*x] + 420*B*Sin[3*c + 4*d*x] + 405*A*Sin[5*c + 4*d*x] + 420*B*Sin[5*c + 4*d*x] + 800*A*Sin[4*c + 5*d*x] + 664*B*Sin[4*c + 5*d*x])))/d","A",1
74,1,326,151,2.0814445,"\int (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 \left(\sec (c) (48 A \sin (2 c+d x)+496 A \sin (c+2 d x)-144 A \sin (3 c+2 d x)+48 A \sin (2 c+3 d x)+48 A \sin (4 c+3 d x)+160 A \sin (3 c+4 d x)+72 A d x \cos (c)+48 A d x \cos (c+2 d x)+48 A d x \cos (3 c+2 d x)+12 A d x \cos (3 c+4 d x)+12 A d x \cos (5 c+4 d x)-480 A \sin (c)+48 A \sin (d x)+105 B \sin (2 c+d x)+544 B \sin (c+2 d x)-96 B \sin (3 c+2 d x)+81 B \sin (2 c+3 d x)+81 B \sin (4 c+3 d x)+160 B \sin (3 c+4 d x)-480 B \sin (c)+105 B \sin (d x))-24 (48 A+35 B) \cos ^4(c+d x) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3072 d}","\frac{5 a^4 (8 A+7 B) \tan (c+d x)}{8 d}+\frac{a^4 (48 A+35 B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(32 A+35 B) \tan (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{24 d}+a^4 A x+\frac{(4 A+7 B) \tan (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 d}+\frac{a B \tan (c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^4*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(-24*(48*A + 35*B)*Cos[c + d*x]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Sec[c]*(72*A*d*x*Cos[c] + 48*A*d*x*Cos[c + 2*d*x] + 48*A*d*x*Cos[3*c + 2*d*x] + 12*A*d*x*Cos[3*c + 4*d*x] + 12*A*d*x*Cos[5*c + 4*d*x] - 480*A*Sin[c] - 480*B*Sin[c] + 48*A*Sin[d*x] + 105*B*Sin[d*x] + 48*A*Sin[2*c + d*x] + 105*B*Sin[2*c + d*x] + 496*A*Sin[c + 2*d*x] + 544*B*Sin[c + 2*d*x] - 144*A*Sin[3*c + 2*d*x] - 96*B*Sin[3*c + 2*d*x] + 48*A*Sin[2*c + 3*d*x] + 81*B*Sin[2*c + 3*d*x] + 48*A*Sin[4*c + 3*d*x] + 81*B*Sin[4*c + 3*d*x] + 160*A*Sin[3*c + 4*d*x] + 160*B*Sin[3*c + 4*d*x])))/(3072*d)","B",1
75,1,1202,151,6.4790389,"\int \cos (c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{(-13 A-12 B) \cos ^5(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (B+A \cos (c+d x))}+\frac{(13 A+12 B) \cos ^5(c+d x) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{32 d (B+A \cos (c+d x))}+\frac{(4 A+B) x \cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 (B+A \cos (c+d x))}+\frac{A \cos (d x) \cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \sin (c) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (B+A \cos (c+d x))}+\frac{A \cos (c) \cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \sin (d x) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{16 d (B+A \cos (c+d x))}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \left(3 A \sin \left(\frac{d x}{2}\right)+5 B \sin \left(\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \left(3 A \sin \left(\frac{d x}{2}\right)+5 B \sin \left(\frac{d x}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{12 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \left(3 A \cos \left(\frac{c}{2}\right)+13 B \cos \left(\frac{c}{2}\right)-3 A \sin \left(\frac{c}{2}\right)-11 B \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{192 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{\cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \left(-3 A \cos \left(\frac{c}{2}\right)-13 B \cos \left(\frac{c}{2}\right)-3 A \sin \left(\frac{c}{2}\right)-11 B \sin \left(\frac{c}{2}\right)\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{192 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^2}+\frac{B \cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}+\frac{B \cos ^5(c+d x) (\sec (c+d x) a+a)^4 (A+B \sec (c+d x)) \sin \left(\frac{d x}{2}\right) \sec ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{96 d (B+A \cos (c+d x)) \left(\cos \left(\frac{c}{2}\right)+\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)^3}","-\frac{5 a^4 (A+2 B) \sin (c+d x)}{2 d}+\frac{a^4 (13 A+12 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(9 A+11 B) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{3 d}+a^4 x (4 A+B)+\frac{(A+2 B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 d}+\frac{a B \sin (c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"((4*A + B)*x*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(16*(B + A*Cos[c + d*x])) + ((-13*A - 12*B)*Cos[c + d*x]^5*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(32*d*(B + A*Cos[c + d*x])) + ((13*A + 12*B)*Cos[c + d*x]^5*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(32*d*(B + A*Cos[c + d*x])) + (A*Cos[d*x]*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sin[c])/(16*d*(B + A*Cos[c + d*x])) + (A*Cos[c]*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sin[d*x])/(16*d*(B + A*Cos[c + d*x])) + (B*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sin[(d*x)/2])/(96*d*(B + A*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(3*A*Cos[c/2] + 13*B*Cos[c/2] - 3*A*Sin[c/2] - 11*B*Sin[c/2]))/(192*d*(B + A*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(3*A*Sin[(d*x)/2] + 5*B*Sin[(d*x)/2]))/(12*d*(B + A*Cos[c + d*x])*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])) + (B*Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sin[(d*x)/2])/(96*d*(B + A*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^3) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(-3*A*Cos[c/2] - 13*B*Cos[c/2] - 3*A*Sin[c/2] - 11*B*Sin[c/2]))/(192*d*(B + A*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2) + (Cos[c + d*x]^5*Sec[c/2 + (d*x)/2]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(3*A*Sin[(d*x)/2] + 5*B*Sin[(d*x)/2]))/(12*d*(B + A*Cos[c + d*x])*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]))","B",1
76,1,373,160,4.9692922,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 \cos ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 (A+B \sec (c+d x)) \left(\frac{4 (4 A+B) \sin (c) \cos (d x)}{d}+\frac{4 (4 A+B) \cos (c) \sin (d x)}{d}+\frac{4 (A+4 B) \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{4 (A+4 B) \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-\frac{2 (8 A+13 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{2 (8 A+13 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+2 x (13 A+8 B)+\frac{A \sin (2 c) \cos (2 d x)}{d}+\frac{A \cos (2 c) \sin (2 d x)}{d}+\frac{B}{d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}\right)}{64 (A \cos (c+d x)+B)}","\frac{5 a^4 (A-B) \sin (c+d x)}{2 d}+\frac{a^4 (8 A+13 B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(A+6 B) \sin (c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{2 d}+\frac{1}{2} a^4 x (13 A+8 B)-\frac{(A-B) \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 d}+\frac{a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^3}{2 d}",1,"(a^4*Cos[c + d*x]^5*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(2*(13*A + 8*B)*x - (2*(8*A + 13*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (2*(8*A + 13*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (4*(4*A + B)*Cos[d*x]*Sin[c])/d + (A*Cos[2*d*x]*Sin[2*c])/d + (4*(4*A + B)*Cos[c]*Sin[d*x])/d + (A*Cos[2*c]*Sin[2*d*x])/d + B/(d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (4*(A + 4*B)*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) - B/(d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (4*(A + 4*B)*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(64*(B + A*Cos[c + d*x]))","B",1
77,1,342,165,2.0122284,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 \cos ^5(c+d x) \sec ^8\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^4 (A+B \sec (c+d x)) \left(\frac{3 (27 A+16 B) \sin (c) \cos (d x)}{d}+\frac{3 (4 A+B) \sin (2 c) \cos (2 d x)}{d}+\frac{3 (27 A+16 B) \cos (c) \sin (d x)}{d}+\frac{3 (4 A+B) \cos (2 c) \sin (2 d x)}{d}-\frac{12 (A+4 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{12 (A+4 B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{d}+\frac{A \sin (3 c) \cos (3 d x)}{d}+\frac{A \cos (3 c) \sin (3 d x)}{d}+72 A x+\frac{12 B \sin \left(\frac{d x}{2}\right)}{d \left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{12 B \sin \left(\frac{d x}{2}\right)}{d \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+78 B x\right)}{192 (A \cos (c+d x)+B)}","\frac{5 a^4 (2 A+B) \sin (c+d x)}{2 d}+\frac{a^4 (A+4 B) \tanh ^{-1}(\sin (c+d x))}{d}-\frac{(8 A-3 B) \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)+\frac{(2 A+B) \sin (c+d x) \cos (c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{2 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^3}{3 d}",1,"(a^4*Cos[c + d*x]^5*Sec[(c + d*x)/2]^8*(1 + Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(72*A*x + 78*B*x - (12*(A + 4*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/d + (12*(A + 4*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/d + (3*(27*A + 16*B)*Cos[d*x]*Sin[c])/d + (3*(4*A + B)*Cos[2*d*x]*Sin[2*c])/d + (A*Cos[3*d*x]*Sin[3*c])/d + (3*(27*A + 16*B)*Cos[c]*Sin[d*x])/d + (3*(4*A + B)*Cos[2*c]*Sin[2*d*x])/d + (A*Cos[3*c]*Sin[3*d*x])/d + (12*B*Sin[(d*x)/2])/(d*(Cos[c/2] - Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (12*B*Sin[(d*x)/2])/(d*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))))/(192*(B + A*Cos[c + d*x]))","B",1
78,1,138,173,0.3872713,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 \left(24 (28 A+27 B) \sin (c+d x)+24 (7 A+4 B) \sin (2 (c+d x))+32 A \sin (3 (c+d x))+3 A \sin (4 (c+d x))+420 A d x+8 B \sin (3 (c+d x))-96 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+576 B d x\right)}{96 d}","\frac{5 a^4 (7 A+8 B) \sin (c+d x)}{8 d}+\frac{(35 A+32 B) \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 (35 A+48 B)+\frac{a^4 B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{(7 A+4 B) \sin (c+d x) \cos ^2(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{12 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^3}{4 d}",1,"(a^4*(420*A*d*x + 576*B*d*x - 96*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(28*A + 27*B)*Sin[c + d*x] + 24*(7*A + 4*B)*Sin[2*(c + d*x)] + 32*A*Sin[3*(c + d*x)] + 8*B*Sin[3*(c + d*x)] + 3*A*Sin[4*(c + d*x)]))/(96*d)","A",1
79,1,108,158,0.3593936,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 (420 (7 A+8 B) \sin (c+d x)+120 (8 A+7 B) \sin (2 (c+d x))+290 A \sin (3 (c+d x))+60 A \sin (4 (c+d x))+6 A \sin (5 (c+d x))+1680 A d x+160 B \sin (3 (c+d x))+15 B \sin (4 (c+d x))+2100 B d x)}{480 d}","-\frac{4 a^4 (4 A+5 B) \sin ^3(c+d x)}{15 d}+\frac{8 a^4 (4 A+5 B) \sin (c+d x)}{5 d}+\frac{a^4 (4 A+5 B) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{27 a^4 (4 A+5 B) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{7}{8} a^4 x (4 A+5 B)+\frac{A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{5 d}",1,"(a^4*(1680*A*d*x + 2100*B*d*x + 420*(7*A + 8*B)*Sin[c + d*x] + 120*(8*A + 7*B)*Sin[2*(c + d*x)] + 290*A*Sin[3*(c + d*x)] + 160*B*Sin[3*(c + d*x)] + 60*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)","A",1
80,1,134,220,0.6107321,"\int \cos ^6(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 (120 (44 A+49 B) \sin (c+d x)+15 (127 A+128 B) \sin (2 (c+d x))+720 A \sin (3 (c+d x))+225 A \sin (4 (c+d x))+48 A \sin (5 (c+d x))+5 A \sin (6 (c+d x))+2940 A c+2940 A d x+580 B \sin (3 (c+d x))+120 B \sin (4 (c+d x))+12 B \sin (5 (c+d x))+3360 B d x)}{960 d}","\frac{a^4 (72 A+83 B) \sin (c+d x)}{15 d}+\frac{a^4 (159 A+176 B) \sin (c+d x) \cos ^2(c+d x)}{120 d}+\frac{7 a^4 (7 A+8 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{(73 A+72 B) \sin (c+d x) \cos ^3(c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{120 d}+\frac{7}{16} a^4 x (7 A+8 B)+\frac{(3 A+2 B) \sin (c+d x) \cos ^4(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{10 d}+\frac{a A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^3}{6 d}",1,"(a^4*(2940*A*c + 2940*A*d*x + 3360*B*d*x + 120*(44*A + 49*B)*Sin[c + d*x] + 15*(127*A + 128*B)*Sin[2*(c + d*x)] + 720*A*Sin[3*(c + d*x)] + 580*B*Sin[3*(c + d*x)] + 225*A*Sin[4*(c + d*x)] + 120*B*Sin[4*(c + d*x)] + 48*A*Sin[5*(c + d*x)] + 12*B*Sin[5*(c + d*x)] + 5*A*Sin[6*(c + d*x)]))/(960*d)","A",1
81,1,156,241,0.7440573,"\int \cos ^7(c+d x) (a+a \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 (105 (323 A+352 B) \sin (c+d x)+105 (124 A+127 B) \sin (2 (c+d x))+5495 A \sin (3 (c+d x))+2100 A \sin (4 (c+d x))+651 A \sin (5 (c+d x))+140 A \sin (6 (c+d x))+15 A \sin (7 (c+d x))+18480 A c+18480 A d x+5040 B \sin (3 (c+d x))+1575 B \sin (4 (c+d x))+336 B \sin (5 (c+d x))+35 B \sin (6 (c+d x))+20580 B d x)}{6720 d}","-\frac{a^4 (227 A+252 B) \sin ^3(c+d x)}{105 d}+\frac{a^4 (227 A+252 B) \sin (c+d x)}{35 d}+\frac{a^4 (276 A+301 B) \sin (c+d x) \cos ^3(c+d x)}{280 d}+\frac{a^4 (44 A+49 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{7 (A+B) \sin (c+d x) \cos ^4(c+d x) \left(a^4 \sec (c+d x)+a^4\right)}{15 d}+\frac{1}{16} a^4 x (44 A+49 B)+\frac{(10 A+7 B) \sin (c+d x) \cos ^5(c+d x) \left(a^2 \sec (c+d x)+a^2\right)^2}{42 d}+\frac{a A \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}",1,"(a^4*(18480*A*c + 18480*A*d*x + 20580*B*d*x + 105*(323*A + 352*B)*Sin[c + d*x] + 105*(124*A + 127*B)*Sin[2*(c + d*x)] + 5495*A*Sin[3*(c + d*x)] + 5040*B*Sin[3*(c + d*x)] + 2100*A*Sin[4*(c + d*x)] + 1575*B*Sin[4*(c + d*x)] + 651*A*Sin[5*(c + d*x)] + 336*B*Sin[5*(c + d*x)] + 140*A*Sin[6*(c + d*x)] + 35*B*Sin[6*(c + d*x)] + 15*A*Sin[7*(c + d*x)]))/(6720*d)","A",1
82,1,635,131,6.3014979,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(12 A \sin \left(c-\frac{d x}{2}\right)+6 A \sin \left(c+\frac{d x}{2}\right)+24 A \sin \left(2 c+\frac{d x}{2}\right)-9 A \sin \left(c+\frac{3 d x}{2}\right)-9 A \sin \left(2 c+\frac{3 d x}{2}\right)+9 A \sin \left(3 c+\frac{3 d x}{2}\right)-3 A \sin \left(c+\frac{5 d x}{2}\right)+3 A \sin \left(2 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{5 d x}{2}\right)+9 A \sin \left(4 c+\frac{5 d x}{2}\right)-12 A \sin \left(2 c+\frac{7 d x}{2}\right)-6 A \sin \left(3 c+\frac{7 d x}{2}\right)-6 A \sin \left(4 c+\frac{7 d x}{2}\right)+6 A \sin \left(\frac{d x}{2}\right)-27 A \sin \left(\frac{3 d x}{2}\right)-24 B \sin \left(c-\frac{d x}{2}\right)-6 B \sin \left(c+\frac{d x}{2}\right)-24 B \sin \left(2 c+\frac{d x}{2}\right)+21 B \sin \left(c+\frac{3 d x}{2}\right)+9 B \sin \left(2 c+\frac{3 d x}{2}\right)-9 B \sin \left(3 c+\frac{3 d x}{2}\right)+7 B \sin \left(c+\frac{5 d x}{2}\right)+B \sin \left(2 c+\frac{5 d x}{2}\right)-3 B \sin \left(3 c+\frac{5 d x}{2}\right)-9 B \sin \left(4 c+\frac{5 d x}{2}\right)+16 B \sin \left(2 c+\frac{7 d x}{2}\right)+10 B \sin \left(3 c+\frac{7 d x}{2}\right)+6 B \sin \left(4 c+\frac{7 d x}{2}\right)+6 B \sin \left(\frac{d x}{2}\right)+39 B \sin \left(\frac{3 d x}{2}\right)\right) (A+B \sec (c+d x))}{48 d (a \sec (c+d x)+a) (A \cos (c+d x)+B)}+\frac{3 (B-A) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) (A+B \sec (c+d x)) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a) (A \cos (c+d x)+B)}-\frac{3 (B-A) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right) (A+B \sec (c+d x)) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a) (A \cos (c+d x)+B)}","-\frac{(3 A-4 B) \tan ^3(c+d x)}{3 a d}-\frac{(3 A-4 B) \tan (c+d x)}{a d}+\frac{3 (A-B) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 (A-B) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(3*(-A + B)*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - (3*(-A + B)*Cos[c/2 + (d*x)/2]^2*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*(6*A*Sin[(d*x)/2] + 6*B*Sin[(d*x)/2] - 27*A*Sin[(3*d*x)/2] + 39*B*Sin[(3*d*x)/2] + 12*A*Sin[c - (d*x)/2] - 24*B*Sin[c - (d*x)/2] + 6*A*Sin[c + (d*x)/2] - 6*B*Sin[c + (d*x)/2] + 24*A*Sin[2*c + (d*x)/2] - 24*B*Sin[2*c + (d*x)/2] - 9*A*Sin[c + (3*d*x)/2] + 21*B*Sin[c + (3*d*x)/2] - 9*A*Sin[2*c + (3*d*x)/2] + 9*B*Sin[2*c + (3*d*x)/2] + 9*A*Sin[3*c + (3*d*x)/2] - 9*B*Sin[3*c + (3*d*x)/2] - 3*A*Sin[c + (5*d*x)/2] + 7*B*Sin[c + (5*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] + B*Sin[2*c + (5*d*x)/2] + 3*A*Sin[3*c + (5*d*x)/2] - 3*B*Sin[3*c + (5*d*x)/2] + 9*A*Sin[4*c + (5*d*x)/2] - 9*B*Sin[4*c + (5*d*x)/2] - 12*A*Sin[2*c + (7*d*x)/2] + 16*B*Sin[2*c + (7*d*x)/2] - 6*A*Sin[3*c + (7*d*x)/2] + 10*B*Sin[3*c + (7*d*x)/2] - 6*A*Sin[4*c + (7*d*x)/2] + 6*B*Sin[4*c + (7*d*x)/2]))/(48*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x]))","B",1
83,1,311,108,3.8756654,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) (A+B \sec (c+d x)) \left(4 (A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{4 (A-B) \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+(4 A-6 B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-4 A \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+6 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{2 a d (\sec (c+d x)+1) (A \cos (c+d x)+B)}","\frac{2 (A-B) \tan (c+d x)}{a d}-\frac{(2 A-3 B) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{(2 A-3 B) \tan (c+d x) \sec (c+d x)}{2 a d}",1,"(Cos[(c + d*x)/2]*(A + B*Sec[c + d*x])*(4*(A - B)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((4*A - 6*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 4*A*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 6*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + B/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 - B/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*(A - B)*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(2*a*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))","B",1
84,1,224,62,1.4057138,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) (A+B \sec (c+d x)) \left((B-A) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos \left(\frac{1}{2} (c+d x)\right) \left(\frac{B \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-(A-B) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)\right)}{a d (\sec (c+d x)+1) (A \cos (c+d x)+B)}","\frac{(A-B) \tanh ^{-1}(\sin (c+d x))}{a d}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)}+\frac{B \tan (c+d x)}{a d}",1,"(2*Cos[(c + d*x)/2]*(A + B*Sec[c + d*x])*((-A + B)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*(-((A - B)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])) + (B*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])))))/(a*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))","B",1
85,1,109,43,0.271828,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+B \cos \left(\frac{1}{2} (c+d x)\right) \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a d}",1,"(2*Cos[(c + d*x)/2]*(B*Cos[(c + d*x)/2]*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (A - B)*Sec[c/2]*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
86,1,72,35,0.1586526,"\int \frac{A+B \sec (c+d x)}{a+a \sec (c+d x)} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(2 (B-A) \sin \left(\frac{d x}{2}\right)+A d x \cos \left(c+\frac{d x}{2}\right)+A d x \cos \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{A x}{a}-\frac{(A-B) \tan (c+d x)}{d (a \sec (c+d x)+a)}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(A*d*x*Cos[(d*x)/2] + A*d*x*Cos[c + (d*x)/2] + 2*(-A + B)*Sin[(d*x)/2]))/(a*d*(1 + Cos[c + d*x]))","B",1
87,1,76,60,0.3778411,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right) (d x (B-A)+A \sin (c+d x))+(A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)\right)}{a d (\cos (c+d x)+1)}","\frac{(2 A-B) \sin (c+d x)}{a d}-\frac{(A-B) \sin (c+d x)}{d (a \sec (c+d x)+a)}-\frac{x (A-B)}{a}",1,"(2*Cos[(c + d*x)/2]*((A - B)*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]*((-A + B)*d*x + A*Sin[c + d*x])))/(a*d*(1 + Cos[c + d*x]))","A",1
88,1,197,98,0.4624031,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(4 d x (3 A-2 B) \cos \left(c+\frac{d x}{2}\right)+4 d x (3 A-2 B) \cos \left(\frac{d x}{2}\right)-4 A \sin \left(c+\frac{d x}{2}\right)-3 A \sin \left(c+\frac{3 d x}{2}\right)-3 A \sin \left(2 c+\frac{3 d x}{2}\right)+A \sin \left(2 c+\frac{5 d x}{2}\right)+A \sin \left(3 c+\frac{5 d x}{2}\right)-20 A \sin \left(\frac{d x}{2}\right)+4 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+4 B \sin \left(2 c+\frac{3 d x}{2}\right)+20 B \sin \left(\frac{d x}{2}\right)\right)}{8 a d (\cos (c+d x)+1)}","-\frac{2 (A-B) \sin (c+d x)}{a d}+\frac{(3 A-2 B) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A-B) \sin (c+d x) \cos (c+d x)}{d (a \sec (c+d x)+a)}+\frac{x (3 A-2 B)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(4*(3*A - 2*B)*d*x*Cos[(d*x)/2] + 4*(3*A - 2*B)*d*x*Cos[c + (d*x)/2] - 20*A*Sin[(d*x)/2] + 20*B*Sin[(d*x)/2] - 4*A*Sin[c + (d*x)/2] + 4*B*Sin[c + (d*x)/2] - 3*A*Sin[c + (3*d*x)/2] + 4*B*Sin[c + (3*d*x)/2] - 3*A*Sin[2*c + (3*d*x)/2] + 4*B*Sin[2*c + (3*d*x)/2] + A*Sin[2*c + (5*d*x)/2] + A*Sin[3*c + (5*d*x)/2]))/(8*a*d*(1 + Cos[c + d*x]))","B",1
89,1,249,122,0.7476554,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-36 d x (A-B) \cos \left(c+\frac{d x}{2}\right)-36 d x (A-B) \cos \left(\frac{d x}{2}\right)+21 A \sin \left(c+\frac{d x}{2}\right)+18 A \sin \left(c+\frac{3 d x}{2}\right)+18 A \sin \left(2 c+\frac{3 d x}{2}\right)-2 A \sin \left(2 c+\frac{5 d x}{2}\right)-2 A \sin \left(3 c+\frac{5 d x}{2}\right)+A \sin \left(3 c+\frac{7 d x}{2}\right)+A \sin \left(4 c+\frac{7 d x}{2}\right)+69 A \sin \left(\frac{d x}{2}\right)-12 B \sin \left(c+\frac{d x}{2}\right)-9 B \sin \left(c+\frac{3 d x}{2}\right)-9 B \sin \left(2 c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{5 d x}{2}\right)-60 B \sin \left(\frac{d x}{2}\right)\right)}{24 a d (\cos (c+d x)+1)}","-\frac{(4 A-3 B) \sin ^3(c+d x)}{3 a d}+\frac{(4 A-3 B) \sin (c+d x)}{a d}-\frac{3 (A-B) \sin (c+d x) \cos (c+d x)}{2 a d}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{d (a \sec (c+d x)+a)}-\frac{3 x (A-B)}{2 a}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-36*(A - B)*d*x*Cos[(d*x)/2] - 36*(A - B)*d*x*Cos[c + (d*x)/2] + 69*A*Sin[(d*x)/2] - 60*B*Sin[(d*x)/2] + 21*A*Sin[c + (d*x)/2] - 12*B*Sin[c + (d*x)/2] + 18*A*Sin[c + (3*d*x)/2] - 9*B*Sin[c + (3*d*x)/2] + 18*A*Sin[2*c + (3*d*x)/2] - 9*B*Sin[2*c + (3*d*x)/2] - 2*A*Sin[2*c + (5*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2] - 2*A*Sin[3*c + (5*d*x)/2] + 3*B*Sin[3*c + (5*d*x)/2] + A*Sin[3*c + (7*d*x)/2] + A*Sin[4*c + (7*d*x)/2]))/(24*a*d*(1 + Cos[c + d*x]))","B",1
90,1,764,179,6.4338788,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4(c+d x) \left(195 A \sin \left(c-\frac{d x}{2}\right)-51 A \sin \left(c+\frac{d x}{2}\right)+189 A \sin \left(2 c+\frac{d x}{2}\right)-A \sin \left(c+\frac{3 d x}{2}\right)-81 A \sin \left(2 c+\frac{3 d x}{2}\right)+119 A \sin \left(3 c+\frac{3 d x}{2}\right)-129 A \sin \left(c+\frac{5 d x}{2}\right)-9 A \sin \left(2 c+\frac{5 d x}{2}\right)-57 A \sin \left(3 c+\frac{5 d x}{2}\right)+63 A \sin \left(4 c+\frac{5 d x}{2}\right)-75 A \sin \left(2 c+\frac{7 d x}{2}\right)-15 A \sin \left(3 c+\frac{7 d x}{2}\right)-39 A \sin \left(4 c+\frac{7 d x}{2}\right)+21 A \sin \left(5 c+\frac{7 d x}{2}\right)-32 A \sin \left(3 c+\frac{9 d x}{2}\right)-12 A \sin \left(4 c+\frac{9 d x}{2}\right)-20 A \sin \left(5 c+\frac{9 d x}{2}\right)+45 A \sin \left(\frac{d x}{2}\right)-201 A \sin \left(\frac{3 d x}{2}\right)-306 B \sin \left(c-\frac{d x}{2}\right)+42 B \sin \left(c+\frac{d x}{2}\right)-270 B \sin \left(2 c+\frac{d x}{2}\right)+50 B \sin \left(c+\frac{3 d x}{2}\right)+90 B \sin \left(2 c+\frac{3 d x}{2}\right)-170 B \sin \left(3 c+\frac{3 d x}{2}\right)+198 B \sin \left(c+\frac{5 d x}{2}\right)+42 B \sin \left(2 c+\frac{5 d x}{2}\right)+66 B \sin \left(3 c+\frac{5 d x}{2}\right)-90 B \sin \left(4 c+\frac{5 d x}{2}\right)+114 B \sin \left(2 c+\frac{7 d x}{2}\right)+36 B \sin \left(3 c+\frac{7 d x}{2}\right)+48 B \sin \left(4 c+\frac{7 d x}{2}\right)-30 B \sin \left(5 c+\frac{7 d x}{2}\right)+48 B \sin \left(3 c+\frac{9 d x}{2}\right)+22 B \sin \left(4 c+\frac{9 d x}{2}\right)+26 B \sin \left(5 c+\frac{9 d x}{2}\right)-6 B \sin \left(\frac{d x}{2}\right)+310 B \sin \left(\frac{3 d x}{2}\right)\right) (A+B \sec (c+d x))}{96 d (a \sec (c+d x)+a)^2 (A \cos (c+d x)+B)}+\frac{2 (10 B-7 A) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^2 (A \cos (c+d x)+B)}-\frac{2 (10 B-7 A) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^2 (A \cos (c+d x)+B)}","-\frac{4 (2 A-3 B) \tan ^3(c+d x)}{3 a^2 d}-\frac{4 (2 A-3 B) \tan (c+d x)}{a^2 d}+\frac{(7 A-10 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(7 A-10 B) \tan (c+d x) \sec ^3(c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{(7 A-10 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{(A-B) \tan (c+d x) \sec ^4(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(2*(-7*A + 10*B)*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (2*(-7*A + 10*B)*Cos[c/2 + (d*x)/2]^4*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^4*(A + B*Sec[c + d*x])*(45*A*Sin[(d*x)/2] - 6*B*Sin[(d*x)/2] - 201*A*Sin[(3*d*x)/2] + 310*B*Sin[(3*d*x)/2] + 195*A*Sin[c - (d*x)/2] - 306*B*Sin[c - (d*x)/2] - 51*A*Sin[c + (d*x)/2] + 42*B*Sin[c + (d*x)/2] + 189*A*Sin[2*c + (d*x)/2] - 270*B*Sin[2*c + (d*x)/2] - A*Sin[c + (3*d*x)/2] + 50*B*Sin[c + (3*d*x)/2] - 81*A*Sin[2*c + (3*d*x)/2] + 90*B*Sin[2*c + (3*d*x)/2] + 119*A*Sin[3*c + (3*d*x)/2] - 170*B*Sin[3*c + (3*d*x)/2] - 129*A*Sin[c + (5*d*x)/2] + 198*B*Sin[c + (5*d*x)/2] - 9*A*Sin[2*c + (5*d*x)/2] + 42*B*Sin[2*c + (5*d*x)/2] - 57*A*Sin[3*c + (5*d*x)/2] + 66*B*Sin[3*c + (5*d*x)/2] + 63*A*Sin[4*c + (5*d*x)/2] - 90*B*Sin[4*c + (5*d*x)/2] - 75*A*Sin[2*c + (7*d*x)/2] + 114*B*Sin[2*c + (7*d*x)/2] - 15*A*Sin[3*c + (7*d*x)/2] + 36*B*Sin[3*c + (7*d*x)/2] - 39*A*Sin[4*c + (7*d*x)/2] + 48*B*Sin[4*c + (7*d*x)/2] + 21*A*Sin[5*c + (7*d*x)/2] - 30*B*Sin[5*c + (7*d*x)/2] - 32*A*Sin[3*c + (9*d*x)/2] + 48*B*Sin[3*c + (9*d*x)/2] - 12*A*Sin[4*c + (9*d*x)/2] + 22*B*Sin[4*c + (9*d*x)/2] - 20*A*Sin[5*c + (9*d*x)/2] + 26*B*Sin[5*c + (9*d*x)/2]))/(96*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","B",1
91,1,496,156,4.187117,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{96 (4 A-7 B) \cos ^4\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \left(-14 (A-B) \sin \left(\frac{d x}{2}\right)+(64 A-97 B) \sin \left(\frac{3 d x}{2}\right)-84 A \sin \left(c-\frac{d x}{2}\right)+42 A \sin \left(c+\frac{d x}{2}\right)-56 A \sin \left(2 c+\frac{d x}{2}\right)-6 A \sin \left(c+\frac{3 d x}{2}\right)+34 A \sin \left(2 c+\frac{3 d x}{2}\right)-36 A \sin \left(3 c+\frac{3 d x}{2}\right)+48 A \sin \left(c+\frac{5 d x}{2}\right)+6 A \sin \left(2 c+\frac{5 d x}{2}\right)+30 A \sin \left(3 c+\frac{5 d x}{2}\right)-12 A \sin \left(4 c+\frac{5 d x}{2}\right)+20 A \sin \left(2 c+\frac{7 d x}{2}\right)+6 A \sin \left(3 c+\frac{7 d x}{2}\right)+14 A \sin \left(4 c+\frac{7 d x}{2}\right)+126 B \sin \left(c-\frac{d x}{2}\right)-42 B \sin \left(c+\frac{d x}{2}\right)+98 B \sin \left(2 c+\frac{d x}{2}\right)+3 B \sin \left(c+\frac{3 d x}{2}\right)-37 B \sin \left(2 c+\frac{3 d x}{2}\right)+63 B \sin \left(3 c+\frac{3 d x}{2}\right)-75 B \sin \left(c+\frac{5 d x}{2}\right)-15 B \sin \left(2 c+\frac{5 d x}{2}\right)-39 B \sin \left(3 c+\frac{5 d x}{2}\right)+21 B \sin \left(4 c+\frac{5 d x}{2}\right)-32 B \sin \left(2 c+\frac{7 d x}{2}\right)-12 B \sin \left(3 c+\frac{7 d x}{2}\right)-20 B \sin \left(4 c+\frac{7 d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","\frac{2 (5 A-8 B) \tan (c+d x)}{3 a^2 d}-\frac{(4 A-7 B) \tanh ^{-1}(\sin (c+d x))}{2 a^2 d}+\frac{(5 A-8 B) \tan (c+d x) \sec ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{(4 A-7 B) \tan (c+d x) \sec (c+d x)}{2 a^2 d}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(96*(4*A - 7*B)*Cos[(c + d*x)/2]^4*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^2*(-14*(A - B)*Sin[(d*x)/2] + (64*A - 97*B)*Sin[(3*d*x)/2] - 84*A*Sin[c - (d*x)/2] + 126*B*Sin[c - (d*x)/2] + 42*A*Sin[c + (d*x)/2] - 42*B*Sin[c + (d*x)/2] - 56*A*Sin[2*c + (d*x)/2] + 98*B*Sin[2*c + (d*x)/2] - 6*A*Sin[c + (3*d*x)/2] + 3*B*Sin[c + (3*d*x)/2] + 34*A*Sin[2*c + (3*d*x)/2] - 37*B*Sin[2*c + (3*d*x)/2] - 36*A*Sin[3*c + (3*d*x)/2] + 63*B*Sin[3*c + (3*d*x)/2] + 48*A*Sin[c + (5*d*x)/2] - 75*B*Sin[c + (5*d*x)/2] + 6*A*Sin[2*c + (5*d*x)/2] - 15*B*Sin[2*c + (5*d*x)/2] + 30*A*Sin[3*c + (5*d*x)/2] - 39*B*Sin[3*c + (5*d*x)/2] - 12*A*Sin[4*c + (5*d*x)/2] + 21*B*Sin[4*c + (5*d*x)/2] + 20*A*Sin[2*c + (7*d*x)/2] - 32*B*Sin[2*c + (7*d*x)/2] + 6*A*Sin[3*c + (7*d*x)/2] - 12*B*Sin[3*c + (7*d*x)/2] + 14*A*Sin[4*c + (7*d*x)/2] - 20*B*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
92,1,292,108,1.939814,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (A+B \sec (c+d x)) \left(-(A-B) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(B-A) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)+\cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\frac{6 B \sin (d x)}{\left(\cos \left(\frac{c}{2}\right)-\sin \left(\frac{c}{2}\right)\right) \left(\sin \left(\frac{c}{2}\right)+\cos \left(\frac{c}{2}\right)\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}-6 (A-2 B) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)-2 (4 A-7 B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2 (A \cos (c+d x)+B)}","-\frac{(A-4 B) \tan (c+d x)}{3 a^2 d}+\frac{(A-2 B) \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A-2 B) \tan (c+d x)}{a^2 d (\sec (c+d x)+1)}+\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(2*Cos[(c + d*x)/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((-A + B)*Sec[c/2]*Sin[(d*x)/2] - 2*(4*A - 7*B)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] + Cos[(c + d*x)/2]^3*(-6*(A - 2*B)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (6*B*Sin[d*x])/((Cos[c/2] - Sin[c/2])*(Cos[c/2] + Sin[c/2])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))) - (A - B)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^2)","B",1
93,1,169,79,0.5636474,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left(-(A-B) \tan \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right)+(B-A) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)-2 (A-4 B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \cos ^2\left(\frac{1}{2} (c+d x)\right)+6 B \cos ^3\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(2 A-5 B) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^2 d}-\frac{(A-B) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(-2*Cos[(c + d*x)/2]*(6*B*Cos[(c + d*x)/2]^3*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (-A + B)*Sec[c/2]*Sin[(d*x)/2] - 2*(A - 4*B)*Cos[(c + d*x)/2]^2*Sec[c/2]*Sin[(d*x)/2] - (A - B)*Cos[(c + d*x)/2]*Tan[c/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","B",1
94,1,76,65,0.2115799,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left((2 A+B) \sin \left(c+\frac{3 d x}{2}\right)+3 (A+B) \sin \left(\frac{d x}{2}\right)-3 A \sin \left(c+\frac{d x}{2}\right)\right)}{3 a^2 d (\cos (c+d x)+1)^2}","\frac{(A+2 B) \tan (c+d x)}{3 d \left(a^2 \sec (c+d x)+a^2\right)}+\frac{(A-B) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(3*(A + B)*Sin[(d*x)/2] - 3*A*Sin[c + (d*x)/2] + (2*A + B)*Sin[c + (3*d*x)/2]))/(3*a^2*d*(1 + Cos[c + d*x])^2)","A",1
95,1,153,70,0.3849111,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^3\left(\frac{1}{2} (c+d x)\right) \left(12 A \sin \left(c+\frac{d x}{2}\right)-10 A \sin \left(c+\frac{3 d x}{2}\right)+9 A d x \cos \left(c+\frac{d x}{2}\right)+3 A d x \cos \left(c+\frac{3 d x}{2}\right)+3 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-18 A \sin \left(\frac{d x}{2}\right)+9 A d x \cos \left(\frac{d x}{2}\right)-6 B \sin \left(c+\frac{d x}{2}\right)+4 B \sin \left(c+\frac{3 d x}{2}\right)+6 B \sin \left(\frac{d x}{2}\right)\right)}{24 a^2 d}","-\frac{(4 A-B) \tan (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{A x}{a^2}-\frac{(A-B) \tan (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^3*(9*A*d*x*Cos[(d*x)/2] + 9*A*d*x*Cos[c + (d*x)/2] + 3*A*d*x*Cos[c + (3*d*x)/2] + 3*A*d*x*Cos[2*c + (3*d*x)/2] - 18*A*Sin[(d*x)/2] + 6*B*Sin[(d*x)/2] + 12*A*Sin[c + (d*x)/2] - 6*B*Sin[c + (d*x)/2] - 10*A*Sin[c + (3*d*x)/2] + 4*B*Sin[c + (3*d*x)/2]))/(24*a^2*d)","B",1
96,1,245,98,0.637471,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-18 d x (2 A-B) \cos \left(c+\frac{d x}{2}\right)-18 d x (2 A-B) \cos \left(\frac{d x}{2}\right)-30 A \sin \left(c+\frac{d x}{2}\right)+41 A \sin \left(c+\frac{3 d x}{2}\right)+9 A \sin \left(2 c+\frac{3 d x}{2}\right)+3 A \sin \left(2 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{5 d x}{2}\right)-12 A d x \cos \left(c+\frac{3 d x}{2}\right)-12 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+66 A \sin \left(\frac{d x}{2}\right)+24 B \sin \left(c+\frac{d x}{2}\right)-20 B \sin \left(c+\frac{3 d x}{2}\right)+6 B d x \cos \left(c+\frac{3 d x}{2}\right)+6 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-36 B \sin \left(\frac{d x}{2}\right)\right)}{12 a^2 d (\cos (c+d x)+1)^2}","\frac{2 (5 A-2 B) \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) \sin (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-18*(2*A - B)*d*x*Cos[(d*x)/2] - 18*(2*A - B)*d*x*Cos[c + (d*x)/2] - 12*A*d*x*Cos[c + (3*d*x)/2] + 6*B*d*x*Cos[c + (3*d*x)/2] - 12*A*d*x*Cos[2*c + (3*d*x)/2] + 6*B*d*x*Cos[2*c + (3*d*x)/2] + 66*A*Sin[(d*x)/2] - 36*B*Sin[(d*x)/2] - 30*A*Sin[c + (d*x)/2] + 24*B*Sin[c + (d*x)/2] + 41*A*Sin[c + (3*d*x)/2] - 20*B*Sin[c + (3*d*x)/2] + 9*A*Sin[2*c + (3*d*x)/2] + 3*A*Sin[2*c + (5*d*x)/2] + 3*A*Sin[3*c + (5*d*x)/2]))/(12*a^2*d*(1 + Cos[c + d*x])^2)","B",1
97,1,315,143,0.7773223,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(36 d x (7 A-4 B) \cos \left(c+\frac{d x}{2}\right)+36 d x (7 A-4 B) \cos \left(\frac{d x}{2}\right)+147 A \sin \left(c+\frac{d x}{2}\right)-239 A \sin \left(c+\frac{3 d x}{2}\right)-63 A \sin \left(2 c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{5 d x}{2}\right)-15 A \sin \left(3 c+\frac{5 d x}{2}\right)+3 A \sin \left(3 c+\frac{7 d x}{2}\right)+3 A \sin \left(4 c+\frac{7 d x}{2}\right)+84 A d x \cos \left(c+\frac{3 d x}{2}\right)+84 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 A \sin \left(\frac{d x}{2}\right)-120 B \sin \left(c+\frac{d x}{2}\right)+164 B \sin \left(c+\frac{3 d x}{2}\right)+36 B \sin \left(2 c+\frac{3 d x}{2}\right)+12 B \sin \left(2 c+\frac{5 d x}{2}\right)+12 B \sin \left(3 c+\frac{5 d x}{2}\right)-48 B d x \cos \left(c+\frac{3 d x}{2}\right)-48 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+264 B \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{2 (8 A-5 B) \sin (c+d x)}{3 a^2 d}+\frac{(7 A-4 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(8 A-5 B) \sin (c+d x) \cos (c+d x)}{3 a^2 d (\sec (c+d x)+1)}+\frac{x (7 A-4 B)}{2 a^2}-\frac{(A-B) \sin (c+d x) \cos (c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(36*(7*A - 4*B)*d*x*Cos[(d*x)/2] + 36*(7*A - 4*B)*d*x*Cos[c + (d*x)/2] + 84*A*d*x*Cos[c + (3*d*x)/2] - 48*B*d*x*Cos[c + (3*d*x)/2] + 84*A*d*x*Cos[2*c + (3*d*x)/2] - 48*B*d*x*Cos[2*c + (3*d*x)/2] - 381*A*Sin[(d*x)/2] + 264*B*Sin[(d*x)/2] + 147*A*Sin[c + (d*x)/2] - 120*B*Sin[c + (d*x)/2] - 239*A*Sin[c + (3*d*x)/2] + 164*B*Sin[c + (3*d*x)/2] - 63*A*Sin[2*c + (3*d*x)/2] + 36*B*Sin[2*c + (3*d*x)/2] - 15*A*Sin[2*c + (5*d*x)/2] + 12*B*Sin[2*c + (5*d*x)/2] - 15*A*Sin[3*c + (5*d*x)/2] + 12*B*Sin[3*c + (5*d*x)/2] + 3*A*Sin[3*c + (7*d*x)/2] + 3*A*Sin[4*c + (7*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
98,1,369,170,0.7706715,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-36 d x (10 A-7 B) \cos \left(c+\frac{d x}{2}\right)-36 d x (10 A-7 B) \cos \left(\frac{d x}{2}\right)-156 A \sin \left(c+\frac{d x}{2}\right)+342 A \sin \left(c+\frac{3 d x}{2}\right)+118 A \sin \left(2 c+\frac{3 d x}{2}\right)+30 A \sin \left(2 c+\frac{5 d x}{2}\right)+30 A \sin \left(3 c+\frac{5 d x}{2}\right)-3 A \sin \left(3 c+\frac{7 d x}{2}\right)-3 A \sin \left(4 c+\frac{7 d x}{2}\right)+A \sin \left(4 c+\frac{9 d x}{2}\right)+A \sin \left(5 c+\frac{9 d x}{2}\right)-120 A d x \cos \left(c+\frac{3 d x}{2}\right)-120 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+516 A \sin \left(\frac{d x}{2}\right)+147 B \sin \left(c+\frac{d x}{2}\right)-239 B \sin \left(c+\frac{3 d x}{2}\right)-63 B \sin \left(2 c+\frac{3 d x}{2}\right)-15 B \sin \left(2 c+\frac{5 d x}{2}\right)-15 B \sin \left(3 c+\frac{5 d x}{2}\right)+3 B \sin \left(3 c+\frac{7 d x}{2}\right)+3 B \sin \left(4 c+\frac{7 d x}{2}\right)+84 B d x \cos \left(c+\frac{3 d x}{2}\right)+84 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-381 B \sin \left(\frac{d x}{2}\right)\right)}{48 a^2 d (\cos (c+d x)+1)^2}","-\frac{4 (3 A-2 B) \sin ^3(c+d x)}{3 a^2 d}+\frac{4 (3 A-2 B) \sin (c+d x)}{a^2 d}-\frac{(10 A-7 B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{(10 A-7 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{x (10 A-7 B)}{2 a^2}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-36*(10*A - 7*B)*d*x*Cos[(d*x)/2] - 36*(10*A - 7*B)*d*x*Cos[c + (d*x)/2] - 120*A*d*x*Cos[c + (3*d*x)/2] + 84*B*d*x*Cos[c + (3*d*x)/2] - 120*A*d*x*Cos[2*c + (3*d*x)/2] + 84*B*d*x*Cos[2*c + (3*d*x)/2] + 516*A*Sin[(d*x)/2] - 381*B*Sin[(d*x)/2] - 156*A*Sin[c + (d*x)/2] + 147*B*Sin[c + (d*x)/2] + 342*A*Sin[c + (3*d*x)/2] - 239*B*Sin[c + (3*d*x)/2] + 118*A*Sin[2*c + (3*d*x)/2] - 63*B*Sin[2*c + (3*d*x)/2] + 30*A*Sin[2*c + (5*d*x)/2] - 15*B*Sin[2*c + (5*d*x)/2] + 30*A*Sin[3*c + (5*d*x)/2] - 15*B*Sin[3*c + (5*d*x)/2] - 3*A*Sin[3*c + (7*d*x)/2] + 3*B*Sin[3*c + (7*d*x)/2] - 3*A*Sin[4*c + (7*d*x)/2] + 3*B*Sin[4*c + (7*d*x)/2] + A*Sin[4*c + (9*d*x)/2] + A*Sin[5*c + (9*d*x)/2]))/(48*a^2*d*(1 + Cos[c + d*x])^2)","B",1
99,1,768,202,6.4017273,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4(c+d x) \left(-2094 A \sin \left(c-\frac{d x}{2}\right)+1314 A \sin \left(c+\frac{d x}{2}\right)-1650 A \sin \left(2 c+\frac{d x}{2}\right)-450 A \sin \left(c+\frac{3 d x}{2}\right)+1230 A \sin \left(2 c+\frac{3 d x}{2}\right)-1050 A \sin \left(3 c+\frac{3 d x}{2}\right)+1278 A \sin \left(c+\frac{5 d x}{2}\right)-90 A \sin \left(2 c+\frac{5 d x}{2}\right)+918 A \sin \left(3 c+\frac{5 d x}{2}\right)-450 A \sin \left(4 c+\frac{5 d x}{2}\right)+630 A \sin \left(2 c+\frac{7 d x}{2}\right)+60 A \sin \left(3 c+\frac{7 d x}{2}\right)+480 A \sin \left(4 c+\frac{7 d x}{2}\right)-90 A \sin \left(5 c+\frac{7 d x}{2}\right)+144 A \sin \left(3 c+\frac{9 d x}{2}\right)+30 A \sin \left(4 c+\frac{9 d x}{2}\right)+114 A \sin \left(5 c+\frac{9 d x}{2}\right)-870 A \sin \left(\frac{d x}{2}\right)+1830 A \sin \left(\frac{3 d x}{2}\right)+4329 B \sin \left(c-\frac{d x}{2}\right)-1989 B \sin \left(c+\frac{d x}{2}\right)+3575 B \sin \left(2 c+\frac{d x}{2}\right)+475 B \sin \left(c+\frac{3 d x}{2}\right)-2005 B \sin \left(2 c+\frac{3 d x}{2}\right)+2275 B \sin \left(3 c+\frac{3 d x}{2}\right)-2673 B \sin \left(c+\frac{5 d x}{2}\right)-105 B \sin \left(2 c+\frac{5 d x}{2}\right)-1593 B \sin \left(3 c+\frac{5 d x}{2}\right)+975 B \sin \left(4 c+\frac{5 d x}{2}\right)-1325 B \sin \left(2 c+\frac{7 d x}{2}\right)-255 B \sin \left(3 c+\frac{7 d x}{2}\right)-875 B \sin \left(4 c+\frac{7 d x}{2}\right)+195 B \sin \left(5 c+\frac{7 d x}{2}\right)-304 B \sin \left(3 c+\frac{9 d x}{2}\right)-90 B \sin \left(4 c+\frac{9 d x}{2}\right)-214 B \sin \left(5 c+\frac{9 d x}{2}\right)+1235 B \sin \left(\frac{d x}{2}\right)-3805 B \sin \left(\frac{3 d x}{2}\right)\right) (A+B \sec (c+d x))}{480 d (a \sec (c+d x)+a)^3 (A \cos (c+d x)+B)}-\frac{4 (13 B-6 A) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^3 (A \cos (c+d x)+B)}+\frac{4 (13 B-6 A) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^3 (A \cos (c+d x)+B)}","\frac{8 (9 A-19 B) \tan (c+d x)}{15 a^3 d}-\frac{(6 A-13 B) \tanh ^{-1}(\sin (c+d x))}{2 a^3 d}+\frac{4 (9 A-19 B) \tan (c+d x) \sec ^2(c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(6 A-13 B) \tan (c+d x) \sec (c+d x)}{2 a^3 d}+\frac{(A-B) \tan (c+d x) \sec ^4(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(6 A-11 B) \tan (c+d x) \sec ^3(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-4*(-6*A + 13*B)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (4*(-6*A + 13*B)*Cos[c/2 + (d*x)/2]^6*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^4*(A + B*Sec[c + d*x])*(-870*A*Sin[(d*x)/2] + 1235*B*Sin[(d*x)/2] + 1830*A*Sin[(3*d*x)/2] - 3805*B*Sin[(3*d*x)/2] - 2094*A*Sin[c - (d*x)/2] + 4329*B*Sin[c - (d*x)/2] + 1314*A*Sin[c + (d*x)/2] - 1989*B*Sin[c + (d*x)/2] - 1650*A*Sin[2*c + (d*x)/2] + 3575*B*Sin[2*c + (d*x)/2] - 450*A*Sin[c + (3*d*x)/2] + 475*B*Sin[c + (3*d*x)/2] + 1230*A*Sin[2*c + (3*d*x)/2] - 2005*B*Sin[2*c + (3*d*x)/2] - 1050*A*Sin[3*c + (3*d*x)/2] + 2275*B*Sin[3*c + (3*d*x)/2] + 1278*A*Sin[c + (5*d*x)/2] - 2673*B*Sin[c + (5*d*x)/2] - 90*A*Sin[2*c + (5*d*x)/2] - 105*B*Sin[2*c + (5*d*x)/2] + 918*A*Sin[3*c + (5*d*x)/2] - 1593*B*Sin[3*c + (5*d*x)/2] - 450*A*Sin[4*c + (5*d*x)/2] + 975*B*Sin[4*c + (5*d*x)/2] + 630*A*Sin[2*c + (7*d*x)/2] - 1325*B*Sin[2*c + (7*d*x)/2] + 60*A*Sin[3*c + (7*d*x)/2] - 255*B*Sin[3*c + (7*d*x)/2] + 480*A*Sin[4*c + (7*d*x)/2] - 875*B*Sin[4*c + (7*d*x)/2] - 90*A*Sin[5*c + (7*d*x)/2] + 195*B*Sin[5*c + (7*d*x)/2] + 144*A*Sin[3*c + (9*d*x)/2] - 304*B*Sin[3*c + (9*d*x)/2] + 30*A*Sin[4*c + (9*d*x)/2] - 90*B*Sin[4*c + (9*d*x)/2] + 114*A*Sin[5*c + (9*d*x)/2] - 214*B*Sin[5*c + (9*d*x)/2]))/(480*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","B",1
100,1,480,156,4.278575,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(5 (32 A-51 B) \sin \left(\frac{d x}{2}\right)+(567 B-167 A) \sin \left(\frac{3 d x}{2}\right)+170 A \sin \left(c-\frac{d x}{2}\right)-170 A \sin \left(c+\frac{d x}{2}\right)+160 A \sin \left(2 c+\frac{d x}{2}\right)+75 A \sin \left(c+\frac{3 d x}{2}\right)-167 A \sin \left(2 c+\frac{3 d x}{2}\right)+75 A \sin \left(3 c+\frac{3 d x}{2}\right)-95 A \sin \left(c+\frac{5 d x}{2}\right)+15 A \sin \left(2 c+\frac{5 d x}{2}\right)-95 A \sin \left(3 c+\frac{5 d x}{2}\right)+15 A \sin \left(4 c+\frac{5 d x}{2}\right)-22 A \sin \left(2 c+\frac{7 d x}{2}\right)-22 A \sin \left(4 c+\frac{7 d x}{2}\right)-600 B \sin \left(c-\frac{d x}{2}\right)+375 B \sin \left(c+\frac{d x}{2}\right)-480 B \sin \left(2 c+\frac{d x}{2}\right)-60 B \sin \left(c+\frac{3 d x}{2}\right)+402 B \sin \left(2 c+\frac{3 d x}{2}\right)-225 B \sin \left(3 c+\frac{3 d x}{2}\right)+315 B \sin \left(c+\frac{5 d x}{2}\right)+30 B \sin \left(2 c+\frac{5 d x}{2}\right)+240 B \sin \left(3 c+\frac{5 d x}{2}\right)-45 B \sin \left(4 c+\frac{5 d x}{2}\right)+72 B \sin \left(2 c+\frac{7 d x}{2}\right)+15 B \sin \left(3 c+\frac{7 d x}{2}\right)+57 B \sin \left(4 c+\frac{7 d x}{2}\right)\right)-960 (A-3 B) \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","-\frac{(7 A-27 B) \tan (c+d x)}{15 a^3 d}+\frac{(A-3 B) \tanh ^{-1}(\sin (c+d x))}{a^3 d}-\frac{(A-3 B) \tan (c+d x)}{d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(4 A-9 B) \tan (c+d x) \sec ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-960*(A - 3*B)*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]*(5*(32*A - 51*B)*Sin[(d*x)/2] + (-167*A + 567*B)*Sin[(3*d*x)/2] + 170*A*Sin[c - (d*x)/2] - 600*B*Sin[c - (d*x)/2] - 170*A*Sin[c + (d*x)/2] + 375*B*Sin[c + (d*x)/2] + 160*A*Sin[2*c + (d*x)/2] - 480*B*Sin[2*c + (d*x)/2] + 75*A*Sin[c + (3*d*x)/2] - 60*B*Sin[c + (3*d*x)/2] - 167*A*Sin[2*c + (3*d*x)/2] + 402*B*Sin[2*c + (3*d*x)/2] + 75*A*Sin[3*c + (3*d*x)/2] - 225*B*Sin[3*c + (3*d*x)/2] - 95*A*Sin[c + (5*d*x)/2] + 315*B*Sin[c + (5*d*x)/2] + 15*A*Sin[2*c + (5*d*x)/2] + 30*B*Sin[2*c + (5*d*x)/2] - 95*A*Sin[3*c + (5*d*x)/2] + 240*B*Sin[3*c + (5*d*x)/2] + 15*A*Sin[4*c + (5*d*x)/2] - 45*B*Sin[4*c + (5*d*x)/2] - 22*A*Sin[2*c + (7*d*x)/2] + 72*B*Sin[2*c + (7*d*x)/2] + 15*B*Sin[3*c + (7*d*x)/2] - 22*A*Sin[4*c + (7*d*x)/2] + 57*B*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
101,1,197,125,0.9279548,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(5 (4 A-29 B) \sin \left(\frac{d x}{2}\right)+10 A \sin \left(c+\frac{3 d x}{2}\right)+2 A \sin \left(2 c+\frac{5 d x}{2}\right)+75 B \sin \left(c+\frac{d x}{2}\right)-95 B \sin \left(c+\frac{3 d x}{2}\right)+15 B \sin \left(2 c+\frac{3 d x}{2}\right)-22 B \sin \left(2 c+\frac{5 d x}{2}\right)\right)-240 B \cos ^6\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(4 A-29 B) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^3 d}+\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(2 A-7 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-240*B*Cos[(c + d*x)/2]^6*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*(5*(4*A - 29*B)*Sin[(d*x)/2] + 75*B*Sin[c + (d*x)/2] + 10*A*Sin[c + (3*d*x)/2] - 95*B*Sin[c + (3*d*x)/2] + 15*B*Sin[2*c + (3*d*x)/2] + 2*A*Sin[2*c + (5*d*x)/2] - 22*B*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
102,1,96,102,0.3071053,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left((3 A+2 B) \left(5 \sin \left(c+\frac{3 d x}{2}\right)+\sin \left(2 c+\frac{5 d x}{2}\right)\right)+5 (3 A+4 B) \sin \left(\frac{d x}{2}\right)-15 A \sin \left(c+\frac{d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(3 A+7 B) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(3 A-8 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(5*(3*A + 4*B)*Sin[(d*x)/2] - 15*A*Sin[c + (d*x)/2] + (3*A + 2*B)*(5*Sin[c + (3*d*x)/2] + Sin[2*c + (5*d*x)/2])))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
103,1,135,102,0.3457745,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-15 (2 A+B) \sin \left(c+\frac{d x}{2}\right)+5 (8 A+3 B) \sin \left(\frac{d x}{2}\right)+20 A \sin \left(c+\frac{3 d x}{2}\right)-15 A \sin \left(2 c+\frac{3 d x}{2}\right)+7 A \sin \left(2 c+\frac{5 d x}{2}\right)+15 B \sin \left(c+\frac{3 d x}{2}\right)+3 B \sin \left(2 c+\frac{5 d x}{2}\right)\right)}{30 a^3 d (\cos (c+d x)+1)^3}","\frac{(2 A+3 B) \tan (c+d x)}{15 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(2 A+3 B) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(5*(8*A + 3*B)*Sin[(d*x)/2] - 15*(2*A + B)*Sin[c + (d*x)/2] + 20*A*Sin[c + (3*d*x)/2] + 15*B*Sin[c + (3*d*x)/2] - 15*A*Sin[2*c + (3*d*x)/2] + 7*A*Sin[2*c + (5*d*x)/2] + 3*B*Sin[2*c + (5*d*x)/2]))/(30*a^3*d*(1 + Cos[c + d*x])^3)","A",1
104,1,241,108,0.5959101,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^5\left(\frac{1}{2} (c+d x)\right) \left(270 A \sin \left(c+\frac{d x}{2}\right)-230 A \sin \left(c+\frac{3 d x}{2}\right)+90 A \sin \left(2 c+\frac{3 d x}{2}\right)-64 A \sin \left(2 c+\frac{5 d x}{2}\right)+150 A d x \cos \left(c+\frac{d x}{2}\right)+75 A d x \cos \left(c+\frac{3 d x}{2}\right)+75 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+15 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+15 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-370 A \sin \left(\frac{d x}{2}\right)+150 A d x \cos \left(\frac{d x}{2}\right)-60 B \sin \left(c+\frac{d x}{2}\right)+40 B \sin \left(c+\frac{3 d x}{2}\right)-30 B \sin \left(2 c+\frac{3 d x}{2}\right)+14 B \sin \left(2 c+\frac{5 d x}{2}\right)+80 B \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d}","-\frac{2 (11 A-B) \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) \tan (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \tan (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^5*(150*A*d*x*Cos[(d*x)/2] + 150*A*d*x*Cos[c + (d*x)/2] + 75*A*d*x*Cos[c + (3*d*x)/2] + 75*A*d*x*Cos[2*c + (3*d*x)/2] + 15*A*d*x*Cos[2*c + (5*d*x)/2] + 15*A*d*x*Cos[3*c + (5*d*x)/2] - 370*A*Sin[(d*x)/2] + 80*B*Sin[(d*x)/2] + 270*A*Sin[c + (d*x)/2] - 60*B*Sin[c + (d*x)/2] - 230*A*Sin[c + (3*d*x)/2] + 40*B*Sin[c + (3*d*x)/2] + 90*A*Sin[2*c + (3*d*x)/2] - 30*B*Sin[2*c + (3*d*x)/2] - 64*A*Sin[2*c + (5*d*x)/2] + 14*B*Sin[2*c + (5*d*x)/2]))/(480*a^3*d)","B",1
105,1,365,136,1.0839971,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-300 d x (3 A-B) \cos \left(c+\frac{d x}{2}\right)-300 d x (3 A-B) \cos \left(\frac{d x}{2}\right)-1125 A \sin \left(c+\frac{d x}{2}\right)+1215 A \sin \left(c+\frac{3 d x}{2}\right)-225 A \sin \left(2 c+\frac{3 d x}{2}\right)+363 A \sin \left(2 c+\frac{5 d x}{2}\right)+75 A \sin \left(3 c+\frac{5 d x}{2}\right)+15 A \sin \left(3 c+\frac{7 d x}{2}\right)+15 A \sin \left(4 c+\frac{7 d x}{2}\right)-450 A d x \cos \left(c+\frac{3 d x}{2}\right)-450 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-90 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-90 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+1755 A \sin \left(\frac{d x}{2}\right)+540 B \sin \left(c+\frac{d x}{2}\right)-460 B \sin \left(c+\frac{3 d x}{2}\right)+180 B \sin \left(2 c+\frac{3 d x}{2}\right)-128 B \sin \left(2 c+\frac{5 d x}{2}\right)+150 B d x \cos \left(c+\frac{3 d x}{2}\right)+150 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+30 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+30 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-740 B \sin \left(\frac{d x}{2}\right)\right)}{120 a^3 d (\cos (c+d x)+1)^3}","\frac{2 (36 A-11 B) \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) \sin (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-300*(3*A - B)*d*x*Cos[(d*x)/2] - 300*(3*A - B)*d*x*Cos[c + (d*x)/2] - 450*A*d*x*Cos[c + (3*d*x)/2] + 150*B*d*x*Cos[c + (3*d*x)/2] - 450*A*d*x*Cos[2*c + (3*d*x)/2] + 150*B*d*x*Cos[2*c + (3*d*x)/2] - 90*A*d*x*Cos[2*c + (5*d*x)/2] + 30*B*d*x*Cos[2*c + (5*d*x)/2] - 90*A*d*x*Cos[3*c + (5*d*x)/2] + 30*B*d*x*Cos[3*c + (5*d*x)/2] + 1755*A*Sin[(d*x)/2] - 740*B*Sin[(d*x)/2] - 1125*A*Sin[c + (d*x)/2] + 540*B*Sin[c + (d*x)/2] + 1215*A*Sin[c + (3*d*x)/2] - 460*B*Sin[c + (3*d*x)/2] - 225*A*Sin[2*c + (3*d*x)/2] + 180*B*Sin[2*c + (3*d*x)/2] + 363*A*Sin[2*c + (5*d*x)/2] - 128*B*Sin[2*c + (5*d*x)/2] + 75*A*Sin[3*c + (5*d*x)/2] + 15*A*Sin[3*c + (7*d*x)/2] + 15*A*Sin[4*c + (7*d*x)/2]))/(120*a^3*d*(1 + Cos[c + d*x])^3)","B",1
106,1,435,187,0.8296437,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(600 d x (13 A-6 B) \cos \left(c+\frac{d x}{2}\right)+600 d x (13 A-6 B) \cos \left(\frac{d x}{2}\right)+7560 A \sin \left(c+\frac{d x}{2}\right)-9230 A \sin \left(c+\frac{3 d x}{2}\right)+930 A \sin \left(2 c+\frac{3 d x}{2}\right)-2782 A \sin \left(2 c+\frac{5 d x}{2}\right)-750 A \sin \left(3 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{7 d x}{2}\right)-105 A \sin \left(4 c+\frac{7 d x}{2}\right)+15 A \sin \left(4 c+\frac{9 d x}{2}\right)+15 A \sin \left(5 c+\frac{9 d x}{2}\right)+3900 A d x \cos \left(c+\frac{3 d x}{2}\right)+3900 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 A \sin \left(\frac{d x}{2}\right)-4500 B \sin \left(c+\frac{d x}{2}\right)+4860 B \sin \left(c+\frac{3 d x}{2}\right)-900 B \sin \left(2 c+\frac{3 d x}{2}\right)+1452 B \sin \left(2 c+\frac{5 d x}{2}\right)+300 B \sin \left(3 c+\frac{5 d x}{2}\right)+60 B \sin \left(3 c+\frac{7 d x}{2}\right)+60 B \sin \left(4 c+\frac{7 d x}{2}\right)-1800 B d x \cos \left(c+\frac{3 d x}{2}\right)-1800 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-360 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-360 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+7020 B \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{8 (19 A-9 B) \sin (c+d x)}{15 a^3 d}+\frac{(13 A-6 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{4 (19 A-9 B) \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 a^3}-\frac{(11 A-6 B) \sin (c+d x) \cos (c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \cos (c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(600*(13*A - 6*B)*d*x*Cos[(d*x)/2] + 600*(13*A - 6*B)*d*x*Cos[c + (d*x)/2] + 3900*A*d*x*Cos[c + (3*d*x)/2] - 1800*B*d*x*Cos[c + (3*d*x)/2] + 3900*A*d*x*Cos[2*c + (3*d*x)/2] - 1800*B*d*x*Cos[2*c + (3*d*x)/2] + 780*A*d*x*Cos[2*c + (5*d*x)/2] - 360*B*d*x*Cos[2*c + (5*d*x)/2] + 780*A*d*x*Cos[3*c + (5*d*x)/2] - 360*B*d*x*Cos[3*c + (5*d*x)/2] - 12760*A*Sin[(d*x)/2] + 7020*B*Sin[(d*x)/2] + 7560*A*Sin[c + (d*x)/2] - 4500*B*Sin[c + (d*x)/2] - 9230*A*Sin[c + (3*d*x)/2] + 4860*B*Sin[c + (3*d*x)/2] + 930*A*Sin[2*c + (3*d*x)/2] - 900*B*Sin[2*c + (3*d*x)/2] - 2782*A*Sin[2*c + (5*d*x)/2] + 1452*B*Sin[2*c + (5*d*x)/2] - 750*A*Sin[3*c + (5*d*x)/2] + 300*B*Sin[3*c + (5*d*x)/2] - 105*A*Sin[3*c + (7*d*x)/2] + 60*B*Sin[3*c + (7*d*x)/2] - 105*A*Sin[4*c + (7*d*x)/2] + 60*B*Sin[4*c + (7*d*x)/2] + 15*A*Sin[4*c + (9*d*x)/2] + 15*A*Sin[5*c + (9*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
107,1,491,218,1.2324742,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-600 d x (23 A-13 B) \cos \left(c+\frac{d x}{2}\right)-600 d x (23 A-13 B) \cos \left(\frac{d x}{2}\right)-11110 A \sin \left(c+\frac{d x}{2}\right)+15380 A \sin \left(c+\frac{3 d x}{2}\right)-380 A \sin \left(2 c+\frac{3 d x}{2}\right)+4777 A \sin \left(2 c+\frac{5 d x}{2}\right)+1625 A \sin \left(3 c+\frac{5 d x}{2}\right)+230 A \sin \left(3 c+\frac{7 d x}{2}\right)+230 A \sin \left(4 c+\frac{7 d x}{2}\right)-20 A \sin \left(4 c+\frac{9 d x}{2}\right)-20 A \sin \left(5 c+\frac{9 d x}{2}\right)+5 A \sin \left(5 c+\frac{11 d x}{2}\right)+5 A \sin \left(6 c+\frac{11 d x}{2}\right)-6900 A d x \cos \left(c+\frac{3 d x}{2}\right)-6900 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-1380 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-1380 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+20410 A \sin \left(\frac{d x}{2}\right)+7560 B \sin \left(c+\frac{d x}{2}\right)-9230 B \sin \left(c+\frac{3 d x}{2}\right)+930 B \sin \left(2 c+\frac{3 d x}{2}\right)-2782 B \sin \left(2 c+\frac{5 d x}{2}\right)-750 B \sin \left(3 c+\frac{5 d x}{2}\right)-105 B \sin \left(3 c+\frac{7 d x}{2}\right)-105 B \sin \left(4 c+\frac{7 d x}{2}\right)+15 B \sin \left(4 c+\frac{9 d x}{2}\right)+15 B \sin \left(5 c+\frac{9 d x}{2}\right)+3900 B d x \cos \left(c+\frac{3 d x}{2}\right)+3900 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+780 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+780 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-12760 B \sin \left(\frac{d x}{2}\right)\right)}{480 a^3 d (\cos (c+d x)+1)^3}","-\frac{4 (34 A-19 B) \sin ^3(c+d x)}{15 a^3 d}+\frac{4 (34 A-19 B) \sin (c+d x)}{5 a^3 d}-\frac{(23 A-13 B) \sin (c+d x) \cos (c+d x)}{2 a^3 d}-\frac{(23 A-13 B) \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)}{2 a^3}-\frac{(13 A-8 B) \sin (c+d x) \cos ^2(c+d x)}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{5 d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-600*(23*A - 13*B)*d*x*Cos[(d*x)/2] - 600*(23*A - 13*B)*d*x*Cos[c + (d*x)/2] - 6900*A*d*x*Cos[c + (3*d*x)/2] + 3900*B*d*x*Cos[c + (3*d*x)/2] - 6900*A*d*x*Cos[2*c + (3*d*x)/2] + 3900*B*d*x*Cos[2*c + (3*d*x)/2] - 1380*A*d*x*Cos[2*c + (5*d*x)/2] + 780*B*d*x*Cos[2*c + (5*d*x)/2] - 1380*A*d*x*Cos[3*c + (5*d*x)/2] + 780*B*d*x*Cos[3*c + (5*d*x)/2] + 20410*A*Sin[(d*x)/2] - 12760*B*Sin[(d*x)/2] - 11110*A*Sin[c + (d*x)/2] + 7560*B*Sin[c + (d*x)/2] + 15380*A*Sin[c + (3*d*x)/2] - 9230*B*Sin[c + (3*d*x)/2] - 380*A*Sin[2*c + (3*d*x)/2] + 930*B*Sin[2*c + (3*d*x)/2] + 4777*A*Sin[2*c + (5*d*x)/2] - 2782*B*Sin[2*c + (5*d*x)/2] + 1625*A*Sin[3*c + (5*d*x)/2] - 750*B*Sin[3*c + (5*d*x)/2] + 230*A*Sin[3*c + (7*d*x)/2] - 105*B*Sin[3*c + (7*d*x)/2] + 230*A*Sin[4*c + (7*d*x)/2] - 105*B*Sin[4*c + (7*d*x)/2] - 20*A*Sin[4*c + (9*d*x)/2] + 15*B*Sin[4*c + (9*d*x)/2] - 20*A*Sin[5*c + (9*d*x)/2] + 15*B*Sin[5*c + (9*d*x)/2] + 5*A*Sin[5*c + (11*d*x)/2] + 5*A*Sin[6*c + (11*d*x)/2]))/(480*a^3*d*(1 + Cos[c + d*x])^3)","B",1
108,1,880,238,6.5159885,"\int \frac{\sec ^6(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^6*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","-\frac{8 (21 B-8 A) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^3(c+d x) (A+B \sec (c+d x)) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{8 (21 B-8 A) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)+\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right) \sec ^3(c+d x) (A+B \sec (c+d x)) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^4}+\frac{\sec \left(\frac{c}{2}\right) \sec (c) \sec ^5(c+d x) (A+B \sec (c+d x)) \left(-38668 A \sin \left(\frac{d x}{2}\right)+73206 B \sin \left(\frac{d x}{2}\right)+64384 A \sin \left(\frac{3 d x}{2}\right)-166668 B \sin \left(\frac{3 d x}{2}\right)-70896 A \sin \left(c-\frac{d x}{2}\right)+183162 B \sin \left(c-\frac{d x}{2}\right)+50316 A \sin \left(c+\frac{d x}{2}\right)-100842 B \sin \left(c+\frac{d x}{2}\right)-59248 A \sin \left(2 c+\frac{d x}{2}\right)+155526 B \sin \left(2 c+\frac{d x}{2}\right)-22820 A \sin \left(c+\frac{3 d x}{2}\right)+37380 B \sin \left(c+\frac{3 d x}{2}\right)+48004 A \sin \left(2 c+\frac{3 d x}{2}\right)-101148 B \sin \left(2 c+\frac{3 d x}{2}\right)-39200 A \sin \left(3 c+\frac{3 d x}{2}\right)+102900 B \sin \left(3 c+\frac{3 d x}{2}\right)+46032 A \sin \left(c+\frac{5 d x}{2}\right)-119364 B \sin \left(c+\frac{5 d x}{2}\right)-8750 A \sin \left(2 c+\frac{5 d x}{2}\right)+8820 B \sin \left(2 c+\frac{5 d x}{2}\right)+35742 A \sin \left(3 c+\frac{5 d x}{2}\right)-78204 B \sin \left(3 c+\frac{5 d x}{2}\right)-19040 A \sin \left(4 c+\frac{5 d x}{2}\right)+49980 B \sin \left(4 c+\frac{5 d x}{2}\right)+24664 A \sin \left(2 c+\frac{7 d x}{2}\right)-64053 B \sin \left(2 c+\frac{7 d x}{2}\right)-1050 A \sin \left(3 c+\frac{7 d x}{2}\right)-3885 B \sin \left(3 c+\frac{7 d x}{2}\right)+19834 A \sin \left(4 c+\frac{7 d x}{2}\right)-44733 B \sin \left(4 c+\frac{7 d x}{2}\right)-5880 A \sin \left(5 c+\frac{7 d x}{2}\right)+15435 B \sin \left(5 c+\frac{7 d x}{2}\right)+8456 A \sin \left(3 c+\frac{9 d x}{2}\right)-21987 B \sin \left(3 c+\frac{9 d x}{2}\right)+630 A \sin \left(4 c+\frac{9 d x}{2}\right)-3675 B \sin \left(4 c+\frac{9 d x}{2}\right)+6986 A \sin \left(5 c+\frac{9 d x}{2}\right)-16107 B \sin \left(5 c+\frac{9 d x}{2}\right)-840 A \sin \left(6 c+\frac{9 d x}{2}\right)+2205 B \sin \left(6 c+\frac{9 d x}{2}\right)+1328 A \sin \left(4 c+\frac{11 d x}{2}\right)-3456 B \sin \left(4 c+\frac{11 d x}{2}\right)+210 A \sin \left(5 c+\frac{11 d x}{2}\right)-840 B \sin \left(5 c+\frac{11 d x}{2}\right)+1118 A \sin \left(6 c+\frac{11 d x}{2}\right)-2616 B \sin \left(6 c+\frac{11 d x}{2}\right)\right) \cos \left(\frac{c}{2}+\frac{d x}{2}\right)}{6720 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^4}","\frac{8 (83 A-216 B) \tan (c+d x)}{105 a^4 d}-\frac{(8 A-21 B) \tanh ^{-1}(\sin (c+d x))}{2 a^4 d}+\frac{(52 A-129 B) \tan (c+d x) \sec ^3(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{4 (83 A-216 B) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(8 A-21 B) \tan (c+d x) \sec (c+d x)}{2 a^4 d}+\frac{(A-B) \tan (c+d x) \sec ^5(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{(A-2 B) \tan (c+d x) \sec ^4(c+d x)}{5 a d (a \sec (c+d x)+a)^3}",1,"(-8*(-8*A + 21*B)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^4) + (8*(-8*A + 21*B)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^5*(A + B*Sec[c + d*x])*(-38668*A*Sin[(d*x)/2] + 73206*B*Sin[(d*x)/2] + 64384*A*Sin[(3*d*x)/2] - 166668*B*Sin[(3*d*x)/2] - 70896*A*Sin[c - (d*x)/2] + 183162*B*Sin[c - (d*x)/2] + 50316*A*Sin[c + (d*x)/2] - 100842*B*Sin[c + (d*x)/2] - 59248*A*Sin[2*c + (d*x)/2] + 155526*B*Sin[2*c + (d*x)/2] - 22820*A*Sin[c + (3*d*x)/2] + 37380*B*Sin[c + (3*d*x)/2] + 48004*A*Sin[2*c + (3*d*x)/2] - 101148*B*Sin[2*c + (3*d*x)/2] - 39200*A*Sin[3*c + (3*d*x)/2] + 102900*B*Sin[3*c + (3*d*x)/2] + 46032*A*Sin[c + (5*d*x)/2] - 119364*B*Sin[c + (5*d*x)/2] - 8750*A*Sin[2*c + (5*d*x)/2] + 8820*B*Sin[2*c + (5*d*x)/2] + 35742*A*Sin[3*c + (5*d*x)/2] - 78204*B*Sin[3*c + (5*d*x)/2] - 19040*A*Sin[4*c + (5*d*x)/2] + 49980*B*Sin[4*c + (5*d*x)/2] + 24664*A*Sin[2*c + (7*d*x)/2] - 64053*B*Sin[2*c + (7*d*x)/2] - 1050*A*Sin[3*c + (7*d*x)/2] - 3885*B*Sin[3*c + (7*d*x)/2] + 19834*A*Sin[4*c + (7*d*x)/2] - 44733*B*Sin[4*c + (7*d*x)/2] - 5880*A*Sin[5*c + (7*d*x)/2] + 15435*B*Sin[5*c + (7*d*x)/2] + 8456*A*Sin[3*c + (9*d*x)/2] - 21987*B*Sin[3*c + (9*d*x)/2] + 630*A*Sin[4*c + (9*d*x)/2] - 3675*B*Sin[4*c + (9*d*x)/2] + 6986*A*Sin[5*c + (9*d*x)/2] - 16107*B*Sin[5*c + (9*d*x)/2] - 840*A*Sin[6*c + (9*d*x)/2] + 2205*B*Sin[6*c + (9*d*x)/2] + 1328*A*Sin[4*c + (11*d*x)/2] - 3456*B*Sin[4*c + (11*d*x)/2] + 210*A*Sin[5*c + (11*d*x)/2] - 840*B*Sin[5*c + (11*d*x)/2] + 1118*A*Sin[6*c + (11*d*x)/2] - 2616*B*Sin[6*c + (11*d*x)/2]))/(6720*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^4)","B",1
109,1,754,194,6.4459169,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec (c) \cos \left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^4(c+d x) \left(4795 A \sin \left(c-\frac{d x}{2}\right)-4795 A \sin \left(c+\frac{d x}{2}\right)+4165 A \sin \left(2 c+\frac{d x}{2}\right)+2275 A \sin \left(c+\frac{3 d x}{2}\right)-4445 A \sin \left(2 c+\frac{3 d x}{2}\right)+2275 A \sin \left(3 c+\frac{3 d x}{2}\right)-2785 A \sin \left(c+\frac{5 d x}{2}\right)+735 A \sin \left(2 c+\frac{5 d x}{2}\right)-2785 A \sin \left(3 c+\frac{5 d x}{2}\right)+735 A \sin \left(4 c+\frac{5 d x}{2}\right)-1015 A \sin \left(2 c+\frac{7 d x}{2}\right)+105 A \sin \left(3 c+\frac{7 d x}{2}\right)-1015 A \sin \left(4 c+\frac{7 d x}{2}\right)+105 A \sin \left(5 c+\frac{7 d x}{2}\right)-160 A \sin \left(3 c+\frac{9 d x}{2}\right)-160 A \sin \left(5 c+\frac{9 d x}{2}\right)+4165 A \sin \left(\frac{d x}{2}\right)-4445 A \sin \left(\frac{3 d x}{2}\right)-20524 B \sin \left(c-\frac{d x}{2}\right)+14644 B \sin \left(c+\frac{d x}{2}\right)-16660 B \sin \left(2 c+\frac{d x}{2}\right)-4690 B \sin \left(c+\frac{3 d x}{2}\right)+14378 B \sin \left(2 c+\frac{3 d x}{2}\right)-9100 B \sin \left(3 c+\frac{3 d x}{2}\right)+11668 B \sin \left(c+\frac{5 d x}{2}\right)-630 B \sin \left(2 c+\frac{5 d x}{2}\right)+9358 B \sin \left(3 c+\frac{5 d x}{2}\right)-2940 B \sin \left(4 c+\frac{5 d x}{2}\right)+4228 B \sin \left(2 c+\frac{7 d x}{2}\right)+315 B \sin \left(3 c+\frac{7 d x}{2}\right)+3493 B \sin \left(4 c+\frac{7 d x}{2}\right)-420 B \sin \left(5 c+\frac{7 d x}{2}\right)+664 B \sin \left(3 c+\frac{9 d x}{2}\right)+105 B \sin \left(4 c+\frac{9 d x}{2}\right)+559 B \sin \left(5 c+\frac{9 d x}{2}\right)-10780 B \sin \left(\frac{d x}{2}\right)+18788 B \sin \left(\frac{3 d x}{2}\right)\right) (A+B \sec (c+d x))}{1680 d (a \sec (c+d x)+a)^4 (A \cos (c+d x)+B)}+\frac{16 (4 B-A) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) (A+B \sec (c+d x)) \log \left(\cos \left(\frac{c}{2}+\frac{d x}{2}\right)-\sin \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^4 (A \cos (c+d x)+B)}-\frac{16 (4 B-A) \cos ^8\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) (A+B \sec (c+d x)) \log \left(\sin \left(\frac{c}{2}+\frac{d x}{2}\right)+\cos \left(\frac{c}{2}+\frac{d x}{2}\right)\right)}{d (a \sec (c+d x)+a)^4 (A \cos (c+d x)+B)}","-\frac{(55 A-244 B) \tan (c+d x)}{105 a^4 d}+\frac{(A-4 B) \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{(25 A-88 B) \tan (c+d x) \sec ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{(A-4 B) \tan (c+d x)}{a^4 d (\sec (c+d x)+1)}+\frac{(A-B) \tan (c+d x) \sec ^4(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{(5 A-12 B) \tan (c+d x) \sec ^3(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(16*(-A + 4*B)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^4) - (16*(-A + 4*B)*Cos[c/2 + (d*x)/2]^8*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^4) + (Cos[c/2 + (d*x)/2]*Sec[c/2]*Sec[c]*Sec[c + d*x]^4*(A + B*Sec[c + d*x])*(4165*A*Sin[(d*x)/2] - 10780*B*Sin[(d*x)/2] - 4445*A*Sin[(3*d*x)/2] + 18788*B*Sin[(3*d*x)/2] + 4795*A*Sin[c - (d*x)/2] - 20524*B*Sin[c - (d*x)/2] - 4795*A*Sin[c + (d*x)/2] + 14644*B*Sin[c + (d*x)/2] + 4165*A*Sin[2*c + (d*x)/2] - 16660*B*Sin[2*c + (d*x)/2] + 2275*A*Sin[c + (3*d*x)/2] - 4690*B*Sin[c + (3*d*x)/2] - 4445*A*Sin[2*c + (3*d*x)/2] + 14378*B*Sin[2*c + (3*d*x)/2] + 2275*A*Sin[3*c + (3*d*x)/2] - 9100*B*Sin[3*c + (3*d*x)/2] - 2785*A*Sin[c + (5*d*x)/2] + 11668*B*Sin[c + (5*d*x)/2] + 735*A*Sin[2*c + (5*d*x)/2] - 630*B*Sin[2*c + (5*d*x)/2] - 2785*A*Sin[3*c + (5*d*x)/2] + 9358*B*Sin[3*c + (5*d*x)/2] + 735*A*Sin[4*c + (5*d*x)/2] - 2940*B*Sin[4*c + (5*d*x)/2] - 1015*A*Sin[2*c + (7*d*x)/2] + 4228*B*Sin[2*c + (7*d*x)/2] + 105*A*Sin[3*c + (7*d*x)/2] + 315*B*Sin[3*c + (7*d*x)/2] - 1015*A*Sin[4*c + (7*d*x)/2] + 3493*B*Sin[4*c + (7*d*x)/2] + 105*A*Sin[5*c + (7*d*x)/2] - 420*B*Sin[5*c + (7*d*x)/2] - 160*A*Sin[3*c + (9*d*x)/2] + 664*B*Sin[3*c + (9*d*x)/2] + 105*B*Sin[4*c + (9*d*x)/2] - 160*A*Sin[5*c + (9*d*x)/2] + 559*B*Sin[5*c + (9*d*x)/2]))/(1680*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^4)","B",1
110,1,239,163,1.5703457,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(70 (3 A-49 B) \sin \left(\frac{d x}{2}\right)+126 A \sin \left(c+\frac{3 d x}{2}\right)+42 A \sin \left(2 c+\frac{5 d x}{2}\right)+6 A \sin \left(3 c+\frac{7 d x}{2}\right)+2170 B \sin \left(c+\frac{d x}{2}\right)-2625 B \sin \left(c+\frac{3 d x}{2}\right)+735 B \sin \left(2 c+\frac{3 d x}{2}\right)-1015 B \sin \left(2 c+\frac{5 d x}{2}\right)+105 B \sin \left(3 c+\frac{5 d x}{2}\right)-160 B \sin \left(3 c+\frac{7 d x}{2}\right)\right)-6720 B \cos ^8\left(\frac{1}{2} (c+d x)\right) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{(12 A-215 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(6 A-55 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{B \tanh ^{-1}(\sin (c+d x))}{a^4 d}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{(3 A-10 B) \tan (c+d x) \sec ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(-6720*B*Cos[(c + d*x)/2]^8*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + Cos[(c + d*x)/2]*Sec[c/2]*(70*(3*A - 49*B)*Sin[(d*x)/2] + 2170*B*Sin[c + (d*x)/2] + 126*A*Sin[c + (3*d*x)/2] - 2625*B*Sin[c + (3*d*x)/2] + 735*B*Sin[2*c + (3*d*x)/2] + 42*A*Sin[2*c + (5*d*x)/2] - 1015*B*Sin[2*c + (5*d*x)/2] + 105*B*Sin[3*c + (5*d*x)/2] + 6*A*Sin[3*c + (7*d*x)/2] - 160*B*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
111,1,109,146,0.3757262,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left((4 A+3 B) \left(21 \sin \left(c+\frac{3 d x}{2}\right)+7 \sin \left(2 c+\frac{5 d x}{2}\right)+\sin \left(3 c+\frac{7 d x}{2}\right)\right)+35 (2 A+3 B) \sin \left(\frac{d x}{2}\right)-70 A \sin \left(c+\frac{d x}{2}\right)\right)}{210 a^4 d (\cos (c+d x)+1)^4}","\frac{(4 A+3 B) \tan (c+d x)}{15 d \left(a^4 \sec (c+d x)+a^4\right)}-\frac{8 (4 A+3 B) \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}-\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{7 d (a \sec (c+d x)+a)^4}+\frac{(4 A+3 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(35*(2*A + 3*B)*Sin[(d*x)/2] - 70*A*Sin[c + (d*x)/2] + (4*A + 3*B)*(21*Sin[c + (3*d*x)/2] + 7*Sin[2*c + (5*d*x)/2] + Sin[3*c + (7*d*x)/2])))/(210*a^4*d*(1 + Cos[c + d*x])^4)","A",1
112,1,163,138,0.4160619,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-35 (5 A+4 B) \sin \left(c+\frac{d x}{2}\right)+140 (2 A+B) \sin \left(\frac{d x}{2}\right)+168 A \sin \left(c+\frac{3 d x}{2}\right)-105 A \sin \left(2 c+\frac{3 d x}{2}\right)+91 A \sin \left(2 c+\frac{5 d x}{2}\right)+13 A \sin \left(3 c+\frac{7 d x}{2}\right)+168 B \sin \left(c+\frac{3 d x}{2}\right)+56 B \sin \left(2 c+\frac{5 d x}{2}\right)+8 B \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{(8 A+13 B) \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{(8 A+13 B) \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{(4 A-11 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(140*(2*A + B)*Sin[(d*x)/2] - 35*(5*A + 4*B)*Sin[c + (d*x)/2] + 168*A*Sin[c + (3*d*x)/2] + 168*B*Sin[c + (3*d*x)/2] - 105*A*Sin[2*c + (3*d*x)/2] + 91*A*Sin[2*c + (5*d*x)/2] + 56*B*Sin[2*c + (5*d*x)/2] + 13*A*Sin[3*c + (7*d*x)/2] + 8*B*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
113,1,193,138,0.4757712,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-35 (18 A+5 B) \sin \left(c+\frac{d x}{2}\right)+70 (9 A+4 B) \sin \left(\frac{d x}{2}\right)+441 A \sin \left(c+\frac{3 d x}{2}\right)-315 A \sin \left(2 c+\frac{3 d x}{2}\right)+147 A \sin \left(2 c+\frac{5 d x}{2}\right)-105 A \sin \left(3 c+\frac{5 d x}{2}\right)+36 A \sin \left(3 c+\frac{7 d x}{2}\right)+168 B \sin \left(c+\frac{3 d x}{2}\right)-105 B \sin \left(2 c+\frac{3 d x}{2}\right)+91 B \sin \left(2 c+\frac{5 d x}{2}\right)+13 B \sin \left(3 c+\frac{7 d x}{2}\right)\right)}{420 a^4 d (\cos (c+d x)+1)^4}","\frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left(a^4 \sec (c+d x)+a^4\right)}+\frac{2 (3 A+4 B) \tan (c+d x)}{105 d \left(a^2 \sec (c+d x)+a^2\right)^2}+\frac{(3 A+4 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}+\frac{(A-B) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(70*(9*A + 4*B)*Sin[(d*x)/2] - 35*(18*A + 5*B)*Sin[c + (d*x)/2] + 441*A*Sin[c + (3*d*x)/2] + 168*B*Sin[c + (3*d*x)/2] - 315*A*Sin[2*c + (3*d*x)/2] - 105*B*Sin[2*c + (3*d*x)/2] + 147*A*Sin[2*c + (5*d*x)/2] + 91*B*Sin[2*c + (5*d*x)/2] - 105*A*Sin[3*c + (5*d*x)/2] + 36*A*Sin[3*c + (7*d*x)/2] + 13*B*Sin[3*c + (7*d*x)/2]))/(420*a^4*d*(1 + Cos[c + d*x])^4)","A",1
114,1,329,138,0.7926862,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \sec ^7\left(\frac{1}{2} (c+d x)\right) \left(8260 A \sin \left(c+\frac{d x}{2}\right)-7140 A \sin \left(c+\frac{3 d x}{2}\right)+3780 A \sin \left(2 c+\frac{3 d x}{2}\right)-2800 A \sin \left(2 c+\frac{5 d x}{2}\right)+840 A \sin \left(3 c+\frac{5 d x}{2}\right)-520 A \sin \left(3 c+\frac{7 d x}{2}\right)+3675 A d x \cos \left(c+\frac{d x}{2}\right)+2205 A d x \cos \left(c+\frac{3 d x}{2}\right)+2205 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+735 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+735 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+105 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+105 A d x \cos \left(4 c+\frac{7 d x}{2}\right)-9940 A \sin \left(\frac{d x}{2}\right)+3675 A d x \cos \left(\frac{d x}{2}\right)-1260 B \sin \left(c+\frac{d x}{2}\right)+882 B \sin \left(c+\frac{3 d x}{2}\right)-630 B \sin \left(2 c+\frac{3 d x}{2}\right)+294 B \sin \left(2 c+\frac{5 d x}{2}\right)-210 B \sin \left(3 c+\frac{5 d x}{2}\right)+72 B \sin \left(3 c+\frac{7 d x}{2}\right)+1260 B \sin \left(\frac{d x}{2}\right)\right)}{13440 a^4 d}","-\frac{2 (80 A-3 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(55 A-6 B) \tan (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}+\frac{A x}{a^4}-\frac{(10 A-3 B) \tan (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B) \tan (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Sec[c/2]*Sec[(c + d*x)/2]^7*(3675*A*d*x*Cos[(d*x)/2] + 3675*A*d*x*Cos[c + (d*x)/2] + 2205*A*d*x*Cos[c + (3*d*x)/2] + 2205*A*d*x*Cos[2*c + (3*d*x)/2] + 735*A*d*x*Cos[2*c + (5*d*x)/2] + 735*A*d*x*Cos[3*c + (5*d*x)/2] + 105*A*d*x*Cos[3*c + (7*d*x)/2] + 105*A*d*x*Cos[4*c + (7*d*x)/2] - 9940*A*Sin[(d*x)/2] + 1260*B*Sin[(d*x)/2] + 8260*A*Sin[c + (d*x)/2] - 1260*B*Sin[c + (d*x)/2] - 7140*A*Sin[c + (3*d*x)/2] + 882*B*Sin[c + (3*d*x)/2] + 3780*A*Sin[2*c + (3*d*x)/2] - 630*B*Sin[2*c + (3*d*x)/2] - 2800*A*Sin[2*c + (5*d*x)/2] + 294*B*Sin[2*c + (5*d*x)/2] + 840*A*Sin[3*c + (5*d*x)/2] - 210*B*Sin[3*c + (5*d*x)/2] - 520*A*Sin[3*c + (7*d*x)/2] + 72*B*Sin[3*c + (7*d*x)/2]))/(13440*a^4*d)","B",1
115,1,485,166,1.108248,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-7350 d x (4 A-B) \cos \left(c+\frac{d x}{2}\right)-7350 d x (4 A-B) \cos \left(\frac{d x}{2}\right)-46130 A \sin \left(c+\frac{d x}{2}\right)+46116 A \sin \left(c+\frac{3 d x}{2}\right)-18060 A \sin \left(2 c+\frac{3 d x}{2}\right)+19292 A \sin \left(2 c+\frac{5 d x}{2}\right)-2100 A \sin \left(3 c+\frac{5 d x}{2}\right)+3791 A \sin \left(3 c+\frac{7 d x}{2}\right)+735 A \sin \left(4 c+\frac{7 d x}{2}\right)+105 A \sin \left(4 c+\frac{9 d x}{2}\right)+105 A \sin \left(5 c+\frac{9 d x}{2}\right)-17640 A d x \cos \left(c+\frac{3 d x}{2}\right)-17640 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-5880 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-5880 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-840 A d x \cos \left(3 c+\frac{7 d x}{2}\right)-840 A d x \cos \left(4 c+\frac{7 d x}{2}\right)+60830 A \sin \left(\frac{d x}{2}\right)+16520 B \sin \left(c+\frac{d x}{2}\right)-14280 B \sin \left(c+\frac{3 d x}{2}\right)+7560 B \sin \left(2 c+\frac{3 d x}{2}\right)-5600 B \sin \left(2 c+\frac{5 d x}{2}\right)+1680 B \sin \left(3 c+\frac{5 d x}{2}\right)-1040 B \sin \left(3 c+\frac{7 d x}{2}\right)+4410 B d x \cos \left(c+\frac{3 d x}{2}\right)+4410 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+1470 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+1470 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+210 B d x \cos \left(3 c+\frac{7 d x}{2}\right)+210 B d x \cos \left(4 c+\frac{7 d x}{2}\right)-19880 B \sin \left(\frac{d x}{2}\right)\right)}{1680 a^4 d (\cos (c+d x)+1)^4}","\frac{8 (83 A-20 B) \sin (c+d x)}{105 a^4 d}-\frac{(4 A-B) \sin (c+d x)}{a^4 d (\sec (c+d x)+1)}-\frac{(88 A-25 B) \sin (c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{x (4 A-B)}{a^4}-\frac{(12 A-5 B) \sin (c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-7350*(4*A - B)*d*x*Cos[(d*x)/2] - 7350*(4*A - B)*d*x*Cos[c + (d*x)/2] - 17640*A*d*x*Cos[c + (3*d*x)/2] + 4410*B*d*x*Cos[c + (3*d*x)/2] - 17640*A*d*x*Cos[2*c + (3*d*x)/2] + 4410*B*d*x*Cos[2*c + (3*d*x)/2] - 5880*A*d*x*Cos[2*c + (5*d*x)/2] + 1470*B*d*x*Cos[2*c + (5*d*x)/2] - 5880*A*d*x*Cos[3*c + (5*d*x)/2] + 1470*B*d*x*Cos[3*c + (5*d*x)/2] - 840*A*d*x*Cos[3*c + (7*d*x)/2] + 210*B*d*x*Cos[3*c + (7*d*x)/2] - 840*A*d*x*Cos[4*c + (7*d*x)/2] + 210*B*d*x*Cos[4*c + (7*d*x)/2] + 60830*A*Sin[(d*x)/2] - 19880*B*Sin[(d*x)/2] - 46130*A*Sin[c + (d*x)/2] + 16520*B*Sin[c + (d*x)/2] + 46116*A*Sin[c + (3*d*x)/2] - 14280*B*Sin[c + (3*d*x)/2] - 18060*A*Sin[2*c + (3*d*x)/2] + 7560*B*Sin[2*c + (3*d*x)/2] + 19292*A*Sin[2*c + (5*d*x)/2] - 5600*B*Sin[2*c + (5*d*x)/2] - 2100*A*Sin[3*c + (5*d*x)/2] + 1680*B*Sin[3*c + (5*d*x)/2] + 3791*A*Sin[3*c + (7*d*x)/2] - 1040*B*Sin[3*c + (7*d*x)/2] + 735*A*Sin[4*c + (7*d*x)/2] + 105*A*Sin[4*c + (9*d*x)/2] + 105*A*Sin[5*c + (9*d*x)/2]))/(1680*a^4*d*(1 + Cos[c + d*x])^4)","B",1
116,1,555,223,1.2045044,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(14700 d x (21 A-8 B) \cos \left(c+\frac{d x}{2}\right)+14700 d x (21 A-8 B) \cos \left(\frac{d x}{2}\right)+386190 A \sin \left(c+\frac{d x}{2}\right)-422478 A \sin \left(c+\frac{3 d x}{2}\right)+132930 A \sin \left(2 c+\frac{3 d x}{2}\right)-181461 A \sin \left(2 c+\frac{5 d x}{2}\right)+3675 A \sin \left(3 c+\frac{5 d x}{2}\right)-36003 A \sin \left(3 c+\frac{7 d x}{2}\right)-9555 A \sin \left(4 c+\frac{7 d x}{2}\right)-945 A \sin \left(4 c+\frac{9 d x}{2}\right)-945 A \sin \left(5 c+\frac{9 d x}{2}\right)+105 A \sin \left(5 c+\frac{11 d x}{2}\right)+105 A \sin \left(6 c+\frac{11 d x}{2}\right)+185220 A d x \cos \left(c+\frac{3 d x}{2}\right)+185220 A d x \cos \left(2 c+\frac{3 d x}{2}\right)+61740 A d x \cos \left(2 c+\frac{5 d x}{2}\right)+61740 A d x \cos \left(3 c+\frac{5 d x}{2}\right)+8820 A d x \cos \left(3 c+\frac{7 d x}{2}\right)+8820 A d x \cos \left(4 c+\frac{7 d x}{2}\right)-539490 A \sin \left(\frac{d x}{2}\right)-184520 B \sin \left(c+\frac{d x}{2}\right)+184464 B \sin \left(c+\frac{3 d x}{2}\right)-72240 B \sin \left(2 c+\frac{3 d x}{2}\right)+77168 B \sin \left(2 c+\frac{5 d x}{2}\right)-8400 B \sin \left(3 c+\frac{5 d x}{2}\right)+15164 B \sin \left(3 c+\frac{7 d x}{2}\right)+2940 B \sin \left(4 c+\frac{7 d x}{2}\right)+420 B \sin \left(4 c+\frac{9 d x}{2}\right)+420 B \sin \left(5 c+\frac{9 d x}{2}\right)-70560 B d x \cos \left(c+\frac{3 d x}{2}\right)-70560 B d x \cos \left(2 c+\frac{3 d x}{2}\right)-23520 B d x \cos \left(2 c+\frac{5 d x}{2}\right)-23520 B d x \cos \left(3 c+\frac{5 d x}{2}\right)-3360 B d x \cos \left(3 c+\frac{7 d x}{2}\right)-3360 B d x \cos \left(4 c+\frac{7 d x}{2}\right)+243320 B \sin \left(\frac{d x}{2}\right)\right)}{6720 a^4 d (\cos (c+d x)+1)^4}","-\frac{8 (216 A-83 B) \sin (c+d x)}{105 a^4 d}+\frac{(21 A-8 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{4 (216 A-83 B) \sin (c+d x) \cos (c+d x)}{105 a^4 d (\sec (c+d x)+1)}-\frac{(129 A-52 B) \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 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{(A-B) \sin (c+d x) \cos (c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(14700*(21*A - 8*B)*d*x*Cos[(d*x)/2] + 14700*(21*A - 8*B)*d*x*Cos[c + (d*x)/2] + 185220*A*d*x*Cos[c + (3*d*x)/2] - 70560*B*d*x*Cos[c + (3*d*x)/2] + 185220*A*d*x*Cos[2*c + (3*d*x)/2] - 70560*B*d*x*Cos[2*c + (3*d*x)/2] + 61740*A*d*x*Cos[2*c + (5*d*x)/2] - 23520*B*d*x*Cos[2*c + (5*d*x)/2] + 61740*A*d*x*Cos[3*c + (5*d*x)/2] - 23520*B*d*x*Cos[3*c + (5*d*x)/2] + 8820*A*d*x*Cos[3*c + (7*d*x)/2] - 3360*B*d*x*Cos[3*c + (7*d*x)/2] + 8820*A*d*x*Cos[4*c + (7*d*x)/2] - 3360*B*d*x*Cos[4*c + (7*d*x)/2] - 539490*A*Sin[(d*x)/2] + 243320*B*Sin[(d*x)/2] + 386190*A*Sin[c + (d*x)/2] - 184520*B*Sin[c + (d*x)/2] - 422478*A*Sin[c + (3*d*x)/2] + 184464*B*Sin[c + (3*d*x)/2] + 132930*A*Sin[2*c + (3*d*x)/2] - 72240*B*Sin[2*c + (3*d*x)/2] - 181461*A*Sin[2*c + (5*d*x)/2] + 77168*B*Sin[2*c + (5*d*x)/2] + 3675*A*Sin[3*c + (5*d*x)/2] - 8400*B*Sin[3*c + (5*d*x)/2] - 36003*A*Sin[3*c + (7*d*x)/2] + 15164*B*Sin[3*c + (7*d*x)/2] - 9555*A*Sin[4*c + (7*d*x)/2] + 2940*B*Sin[4*c + (7*d*x)/2] - 945*A*Sin[4*c + (9*d*x)/2] + 420*B*Sin[4*c + (9*d*x)/2] - 945*A*Sin[5*c + (9*d*x)/2] + 420*B*Sin[5*c + (9*d*x)/2] + 105*A*Sin[5*c + (11*d*x)/2] + 105*A*Sin[6*c + (11*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x])^4)","B",1
117,1,611,256,1.8012792,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^4,x]","\frac{\sec \left(\frac{c}{2}\right) \cos \left(\frac{1}{2} (c+d x)\right) \left(-14700 d x (44 A-21 B) \cos \left(c+\frac{d x}{2}\right)-14700 d x (44 A-21 B) \cos \left(\frac{d x}{2}\right)-687260 A \sin \left(c+\frac{d x}{2}\right)+814107 A \sin \left(c+\frac{3 d x}{2}\right)-204645 A \sin \left(2 c+\frac{3 d x}{2}\right)+357609 A \sin \left(2 c+\frac{5 d x}{2}\right)+18025 A \sin \left(3 c+\frac{5 d x}{2}\right)+72522 A \sin \left(3 c+\frac{7 d x}{2}\right)+24010 A \sin \left(4 c+\frac{7 d x}{2}\right)+2310 A \sin \left(4 c+\frac{9 d x}{2}\right)+2310 A \sin \left(5 c+\frac{9 d x}{2}\right)-175 A \sin \left(5 c+\frac{11 d x}{2}\right)-175 A \sin \left(6 c+\frac{11 d x}{2}\right)+35 A \sin \left(6 c+\frac{13 d x}{2}\right)+35 A \sin \left(7 c+\frac{13 d x}{2}\right)-388080 A d x \cos \left(c+\frac{3 d x}{2}\right)-388080 A d x \cos \left(2 c+\frac{3 d x}{2}\right)-129360 A d x \cos \left(2 c+\frac{5 d x}{2}\right)-129360 A d x \cos \left(3 c+\frac{5 d x}{2}\right)-18480 A d x \cos \left(3 c+\frac{7 d x}{2}\right)-18480 A d x \cos \left(4 c+\frac{7 d x}{2}\right)+1010660 A \sin \left(\frac{d x}{2}\right)+386190 B \sin \left(c+\frac{d x}{2}\right)-422478 B \sin \left(c+\frac{3 d x}{2}\right)+132930 B \sin \left(2 c+\frac{3 d x}{2}\right)-181461 B \sin \left(2 c+\frac{5 d x}{2}\right)+3675 B \sin \left(3 c+\frac{5 d x}{2}\right)-36003 B \sin \left(3 c+\frac{7 d x}{2}\right)-9555 B \sin \left(4 c+\frac{7 d x}{2}\right)-945 B \sin \left(4 c+\frac{9 d x}{2}\right)-945 B \sin \left(5 c+\frac{9 d x}{2}\right)+105 B \sin \left(5 c+\frac{11 d x}{2}\right)+105 B \sin \left(6 c+\frac{11 d x}{2}\right)+185220 B d x \cos \left(c+\frac{3 d x}{2}\right)+185220 B d x \cos \left(2 c+\frac{3 d x}{2}\right)+61740 B d x \cos \left(2 c+\frac{5 d x}{2}\right)+61740 B d x \cos \left(3 c+\frac{5 d x}{2}\right)+8820 B d x \cos \left(3 c+\frac{7 d x}{2}\right)+8820 B d x \cos \left(4 c+\frac{7 d x}{2}\right)-539490 B \sin \left(\frac{d x}{2}\right)\right)}{6720 a^4 d (\cos (c+d x)+1)^4}","-\frac{8 (227 A-108 B) \sin ^3(c+d x)}{105 a^4 d}+\frac{8 (227 A-108 B) \sin (c+d x)}{35 a^4 d}-\frac{(44 A-21 B) \sin (c+d x) \cos (c+d x)}{2 a^4 d}-\frac{(44 A-21 B) \sin (c+d x) \cos ^2(c+d x)}{3 a^4 d (\sec (c+d x)+1)}-\frac{(178 A-87 B) \sin (c+d x) \cos ^2(c+d x)}{105 a^4 d (\sec (c+d x)+1)^2}-\frac{x (44 A-21 B)}{2 a^4}-\frac{(16 A-9 B) \sin (c+d x) \cos ^2(c+d x)}{35 a d (a \sec (c+d x)+a)^3}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{7 d (a \sec (c+d x)+a)^4}",1,"(Cos[(c + d*x)/2]*Sec[c/2]*(-14700*(44*A - 21*B)*d*x*Cos[(d*x)/2] - 14700*(44*A - 21*B)*d*x*Cos[c + (d*x)/2] - 388080*A*d*x*Cos[c + (3*d*x)/2] + 185220*B*d*x*Cos[c + (3*d*x)/2] - 388080*A*d*x*Cos[2*c + (3*d*x)/2] + 185220*B*d*x*Cos[2*c + (3*d*x)/2] - 129360*A*d*x*Cos[2*c + (5*d*x)/2] + 61740*B*d*x*Cos[2*c + (5*d*x)/2] - 129360*A*d*x*Cos[3*c + (5*d*x)/2] + 61740*B*d*x*Cos[3*c + (5*d*x)/2] - 18480*A*d*x*Cos[3*c + (7*d*x)/2] + 8820*B*d*x*Cos[3*c + (7*d*x)/2] - 18480*A*d*x*Cos[4*c + (7*d*x)/2] + 8820*B*d*x*Cos[4*c + (7*d*x)/2] + 1010660*A*Sin[(d*x)/2] - 539490*B*Sin[(d*x)/2] - 687260*A*Sin[c + (d*x)/2] + 386190*B*Sin[c + (d*x)/2] + 814107*A*Sin[c + (3*d*x)/2] - 422478*B*Sin[c + (3*d*x)/2] - 204645*A*Sin[2*c + (3*d*x)/2] + 132930*B*Sin[2*c + (3*d*x)/2] + 357609*A*Sin[2*c + (5*d*x)/2] - 181461*B*Sin[2*c + (5*d*x)/2] + 18025*A*Sin[3*c + (5*d*x)/2] + 3675*B*Sin[3*c + (5*d*x)/2] + 72522*A*Sin[3*c + (7*d*x)/2] - 36003*B*Sin[3*c + (7*d*x)/2] + 24010*A*Sin[4*c + (7*d*x)/2] - 9555*B*Sin[4*c + (7*d*x)/2] + 2310*A*Sin[4*c + (9*d*x)/2] - 945*B*Sin[4*c + (9*d*x)/2] + 2310*A*Sin[5*c + (9*d*x)/2] - 945*B*Sin[5*c + (9*d*x)/2] - 175*A*Sin[5*c + (11*d*x)/2] + 105*B*Sin[5*c + (11*d*x)/2] - 175*A*Sin[6*c + (11*d*x)/2] + 105*B*Sin[6*c + (11*d*x)/2] + 35*A*Sin[6*c + (13*d*x)/2] + 35*A*Sin[7*c + (13*d*x)/2]))/(6720*a^4*d*(1 + Cos[c + d*x])^4)","B",1
118,1,98,187,0.60119,"\int \sec ^4(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 a \tan (c+d x) \left(5 (9 A+8 B) \sec ^3(c+d x)+6 (9 A+8 B) \sec ^2(c+d x)+8 (9 A+8 B) \sec (c+d x)+16 (9 A+8 B)+35 B \sec ^4(c+d x)\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a (9 A+8 B) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{4 (9 A+8 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 a d}-\frac{8 (9 A+8 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{4 a (9 A+8 B) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a B \tan (c+d x) \sec ^4(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(16*(9*A + 8*B) + 8*(9*A + 8*B)*Sec[c + d*x] + 6*(9*A + 8*B)*Sec[c + d*x]^2 + 5*(9*A + 8*B)*Sec[c + d*x]^3 + 35*B*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
119,1,81,144,0.2903776,"\int \sec ^3(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 a \tan (c+d x) \left(3 (7 A+6 B) \sec ^2(c+d x)+4 (7 A+6 B) \sec (c+d x)+8 (7 A+6 B)+15 B \sec ^3(c+d x)\right)}{105 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (7 A+6 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 a d}-\frac{4 (7 A+6 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (7 A+6 B) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a B \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(8*(7*A + 6*B) + 4*(7*A + 6*B)*Sec[c + d*x] + 3*(7*A + 6*B)*Sec[c + d*x]^2 + 15*B*Sec[c + d*x]^3)*Tan[c + d*x])/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
120,1,80,101,0.3353888,"\int \sec ^2(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \tan (c+d x) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} ((5 A+4 B) \cos (c+d x)+(5 A+4 B) \cos (2 (c+d x))+5 A+7 B)}{15 d (\cos (c+d x)+1)}","\frac{2 (5 A-2 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a (5 A+7 B) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 a d}",1,"(2*(5*A + 7*B + (5*A + 4*B)*Cos[c + d*x] + (5*A + 4*B)*Cos[2*(c + d*x)])*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(15*d*(1 + Cos[c + d*x]))","A",1
121,1,53,62,0.1708941,"\int \sec (c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \tan (c+d x) \sqrt{a (\sec (c+d x)+1)} ((3 A+2 B) \cos (c+d x)+B)}{3 d (\cos (c+d x)+1)}","\frac{2 a (3 A+B) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*(B + (3*A + 2*B)*Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[c + d*x])/(3*d*(1 + Cos[c + d*x]))","A",1
122,1,76,66,0.3220661,"\int \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{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 B \tan (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*B*Sin[(c + d*x)/2]))/d","A",1
123,1,93,68,0.2518403,"\int \cos (c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (A+2 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 A \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)}\right)}{2 d}","\frac{\sqrt{a} (A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a A \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(A + 2*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*A*Sqrt[Cos[c + d*x]]*Sin[(c + d*x)/2]))/(2*d)","A",1
124,1,117,117,0.4124806,"\int \cos ^2(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(2 A \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)+B \left(\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{a (3 A+4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (3 A+4 B) \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) \cos (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"((B*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] + Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]]) + 2*A*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","C",1
125,1,70,160,0.1869438,"\int \cos ^3(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(A \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+B \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)\right)}{d}","\frac{a (5 A+6 B) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (5 A+6 B) \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)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(2*(B*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]] + A*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
126,1,70,203,0.1824171,"\int \cos ^4(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(A \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+B \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)\right)}{d}","\frac{5 a (7 A+8 B) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{5 \sqrt{a} (7 A+8 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a (7 A+8 B) \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{5 a (7 A+8 B) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}",1,"(2*(B*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]] + A*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/d","C",1
127,1,100,189,0.7873424,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 \tan (c+d x) \left(5 (9 A+17 B) \sec ^3(c+d x)+3 (39 A+34 B) \sec ^2(c+d x)+4 (39 A+34 B) \sec (c+d x)+8 (39 A+34 B)+35 B \sec ^4(c+d x)\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^2 (9 A+10 B) \tan (c+d x) \sec ^3(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (39 A+34 B) \tan (c+d x)}{45 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (39 A+34 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}-\frac{4 a (39 A+34 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 a B \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}",1,"(2*a^2*(8*(39*A + 34*B) + 4*(39*A + 34*B)*Sec[c + d*x] + 3*(39*A + 34*B)*Sec[c + d*x]^2 + 5*(9*A + 17*B)*Sec[c + d*x]^3 + 35*B*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
128,1,82,138,0.4205754,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 \tan (c+d x) \left(3 (7 A+13 B) \sec ^2(c+d x)+(63 A+52 B) \sec (c+d x)+2 (63 A+52 B)+15 B \sec ^3(c+d x)\right)}{105 d \sqrt{a (\sec (c+d x)+1)}}","\frac{8 a^2 (21 A+19 B) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (7 A-2 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 a (21 A+19 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 B \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 a d}",1,"(2*a^2*(2*(63*A + 52*B) + (63*A + 52*B)*Sec[c + d*x] + 3*(7*A + 13*B)*Sec[c + d*x]^2 + 15*B*Sec[c + d*x]^3)*Tan[c + d*x])/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
129,1,70,101,0.3330347,"\int \sec (c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a \sqrt{a (\sec (c+d x)+1)} ((25 A+18 B) \sin (c+d x)+\tan (c+d x) (5 A+3 B \sec (c+d x)+9 B))}{15 d (\cos (c+d x)+1)}","\frac{8 a^2 (5 A+3 B) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (5 A+3 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(2*a*Sqrt[a*(1 + Sec[c + d*x])]*((25*A + 18*B)*Sin[c + d*x] + (5*A + 9*B + 3*B*Sec[c + d*x])*Tan[c + d*x]))/(15*d*(1 + Cos[c + d*x]))","A",1
130,1,102,105,0.6224529,"\int (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((3 A+5 B) \cos (c+d x)+B)+3 \sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)\right)}{3 d}","\frac{2 a^{3/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (3 A+4 B) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a B \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(a*Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + 2*(B + (3*A + 5*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(3*d)","A",1
131,1,97,103,0.461557,"\int \cos (c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (3 A+2 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) (A \cos (c+d x)+2 B)\right)}{2 d}","\frac{a^{3/2} (3 A+2 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 (A-2 B) \sin (c+d x)}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 a B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(3*A + 2*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(2*B + A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(2*d)","A",1
132,1,111,119,0.8130132,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (7 A+12 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} (2 A \cos (c+d x)+7 A+4 B)\right)}{8 d}","\frac{a^{3/2} (7 A+12 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (5 A+4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a A \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(a*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(7*A + 12*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sqrt[Cos[c + d*x]]*(7*A + 4*B + 2*A*Cos[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
133,1,137,164,1.0130573,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\sin (c+d x) \sqrt{1-\sec (c+d x)} (2 (11 A+6 B) \cos (c+d x)+4 A \cos (2 (c+d x))+37 A+42 B)+3 (11 A+14 B) \tan (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{24 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{a^{3/2} (11 A+14 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (11 A+14 B) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (7 A+6 B) \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{a A \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(a*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*((37*A + 42*B + 2*(11*A + 6*B)*Cos[c + d*x] + 4*A*Cos[2*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 3*(11*A + 14*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x]))/(24*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","A",1
134,1,154,209,1.4778064,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(\sin (c+d x) \sqrt{1-\sec (c+d x)} (2 (93 A+88 B) \cos (c+d x)+4 (15 A+8 B) \cos (2 (c+d x))+12 A \cos (3 (c+d x))+285 A+296 B)+3 (75 A+88 B) \tan (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{192 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{a^{3/2} (75 A+88 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (75 A+88 B) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (9 A+8 B) \sin (c+d x) \cos ^2(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (75 A+88 B) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a A \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"(a*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*((285*A + 296*B + 2*(93*A + 88*B)*Cos[c + d*x] + 4*(15*A + 8*B)*Cos[2*(c + d*x)] + 12*A*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 3*(75*A + 88*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x]))/(192*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","A",1
135,1,487,237,6.189808,"\int \sec ^3(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 A \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{5/2}}{9 d (\sec (c+d x)+1)^2}+\frac{38 A \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{5/2}}{63 d (\sec (c+d x)+1)^3}+\frac{146 A \tan (c+d x) \sec ^2(c+d x) (a (\sec (c+d x)+1))^{5/2}}{105 d (\sec (c+d x)+1)^3}+\frac{584 A \tan (c+d x) \sec (c+d x) (a (\sec (c+d x)+1))^{5/2}}{315 d (\sec (c+d x)+1)^3}+\frac{1168 A \tan (c+d x) (a (\sec (c+d x)+1))^{5/2}}{315 d (\sec (c+d x)+1)^3}+\frac{2 B \tan (c+d x) \sec ^4(c+d x) (a (\sec (c+d x)+1))^{5/2}}{11 d (\sec (c+d x)+1)^2}+\frac{46 B \tan (c+d x) \sec ^4(c+d x) (a (\sec (c+d x)+1))^{5/2}}{99 d (\sec (c+d x)+1)^3}+\frac{710 B \tan (c+d x) \sec ^3(c+d x) (a (\sec (c+d x)+1))^{5/2}}{693 d (\sec (c+d x)+1)^3}+\frac{284 B \tan (c+d x) \sec ^2(c+d x) (a (\sec (c+d x)+1))^{5/2}}{231 d (\sec (c+d x)+1)^3}+\frac{1136 B \tan (c+d x) \sec (c+d x) (a (\sec (c+d x)+1))^{5/2}}{693 d (\sec (c+d x)+1)^3}+\frac{2272 B \tan (c+d x) (a (\sec (c+d x)+1))^{5/2}}{693 d (\sec (c+d x)+1)^3}","\frac{2 a^3 (209 A+194 B) \tan (c+d x) \sec ^3(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (803 A+710 B) \tan (c+d x)}{495 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (11 A+14 B) \tan (c+d x) \sec ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{99 d}-\frac{4 a^2 (803 A+710 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3465 d}+\frac{2 a (803 A+710 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{1155 d}+\frac{2 a B \tan (c+d x) \sec ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}",1,"(1168*A*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(315*d*(1 + Sec[c + d*x])^3) + (2272*B*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(693*d*(1 + Sec[c + d*x])^3) + (584*A*Sec[c + d*x]*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(315*d*(1 + Sec[c + d*x])^3) + (1136*B*Sec[c + d*x]*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(693*d*(1 + Sec[c + d*x])^3) + (146*A*Sec[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(105*d*(1 + Sec[c + d*x])^3) + (284*B*Sec[c + d*x]^2*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(231*d*(1 + Sec[c + d*x])^3) + (38*A*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(63*d*(1 + Sec[c + d*x])^3) + (710*B*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(693*d*(1 + Sec[c + d*x])^3) + (46*B*Sec[c + d*x]^4*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(99*d*(1 + Sec[c + d*x])^3) + (2*A*Sec[c + d*x]^3*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(9*d*(1 + Sec[c + d*x])^2) + (2*B*Sec[c + d*x]^4*(a*(1 + Sec[c + d*x]))^(5/2)*Tan[c + d*x])/(11*d*(1 + Sec[c + d*x])^2)","B",1
136,1,96,175,0.6351023,"\int \sec ^2(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^3 \tan (c+d x) \left(5 (9 A+26 B) \sec ^3(c+d x)+3 (60 A+73 B) \sec ^2(c+d x)+(345 A+292 B) \sec (c+d x)+690 A+35 B \sec ^4(c+d x)+584 B\right)}{315 d \sqrt{a (\sec (c+d x)+1)}}","\frac{64 a^3 (15 A+13 B) \tan (c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (15 A+13 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{315 d}+\frac{2 (9 A-2 B) \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{63 d}+\frac{2 a (15 A+13 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{105 d}+\frac{2 B \tan (c+d x) (a \sec (c+d x)+a)^{7/2}}{9 a d}",1,"(2*a^3*(690*A + 584*B + (345*A + 292*B)*Sec[c + d*x] + 3*(60*A + 73*B)*Sec[c + d*x]^2 + 5*(9*A + 26*B)*Sec[c + d*x]^3 + 35*B*Sec[c + d*x]^4)*Tan[c + d*x])/(315*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
137,1,89,138,0.513903,"\int \sec (c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 \sqrt{a (\sec (c+d x)+1)} \left((301 A+230 B) \sin (c+d x)+\tan (c+d x) \left(3 (7 A+20 B) \sec (c+d x)+98 A+15 B \sec ^2(c+d x)+115 B\right)\right)}{105 d (\cos (c+d x)+1)}","\frac{64 a^3 (7 A+5 B) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (7 A+5 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (7 A+5 B) \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 B \tan (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(2*a^2*Sqrt[a*(1 + Sec[c + d*x])]*((301*A + 230*B)*Sin[c + d*x] + (98*A + 115*B + 3*(7*A + 20*B)*Sec[c + d*x] + 15*B*Sec[c + d*x]^2)*Tan[c + d*x]))/(105*d*(1 + Cos[c + d*x]))","A",1
138,1,128,142,0.9433004,"\int (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sqrt{a (\sec (c+d x)+1)} \left(2 \sin \left(\frac{1}{2} (c+d x)\right) (2 (5 A+14 B) \cos (c+d x)+(40 A+43 B) \cos (2 (c+d x))+40 A+49 B)+30 \sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{5}{2}}(c+d x)\right)}{30 d}","\frac{2 a^{5/2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (35 A+32 B) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (5 A+8 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a B \tan (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]^2*Sqrt[a*(1 + Sec[c + d*x])]*(30*Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(5/2) + 2*(40*A + 49*B + 2*(5*A + 14*B)*Cos[c + d*x] + (40*A + 43*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(30*d)","A",1
139,1,126,143,0.8276743,"\int \cos (c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (5 A+2 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \cos ^{\frac{3}{2}}(c+d x)+\sin \left(\frac{1}{2} (c+d x)\right) (4 (3 A+8 B) \cos (c+d x)+3 A \cos (2 (c+d x))+3 A+4 B)\right)}{6 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^3 (3 A+14 B) \sin (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (A+2 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d}+\frac{2 a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sec[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(5*A + 2*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^(3/2) + (3*A + 4*B + 4*(3*A + 8*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(6*d)","A",1
140,1,116,154,0.9653875,"\int \cos ^2(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (19 A+20 B) \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right) \sqrt{\cos (c+d x)}+2 \sin \left(\frac{1}{2} (c+d x)\right) ((11 A+4 B) \cos (c+d x)+A \cos (2 (c+d x))+A+8 B)\right)}{8 d}","\frac{a^{5/2} (19 A+20 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (A-4 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}+\frac{a A \sin (c+d x) \cos (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(19*A + 20*B)*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]]*Sqrt[Cos[c + d*x]] + 2*(A + 8*B + (11*A + 4*B)*Cos[c + d*x] + A*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(8*d)","A",1
141,1,312,164,1.0394104,"\int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","-\frac{a^2 \cos (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(-192 A \tan (c+d x) \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)-165 A \sin (c+d x) \sqrt{1-\sec (c+d x)}-31 A \sin (2 (c+d x)) \sqrt{1-\sec (c+d x)}+8 A \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec (c+d x)}-165 A \tan (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-576 B \tan (c+d x) \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)+18 B \sin (c+d x) \sqrt{1-\sec (c+d x)}+54 B \sin (2 (c+d x)) \sqrt{1-\sec (c+d x)}-126 B \tan (c+d x) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{72 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{a^{5/2} (25 A+38 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (49 A+54 B) \sin (c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"-1/72*(a^2*Cos[c + d*x]*Sqrt[a*(1 + Sec[c + d*x])]*(-165*A*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 18*B*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 8*A*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] - 31*A*Sqrt[1 - Sec[c + d*x]]*Sin[2*(c + d*x)] + 54*B*Sqrt[1 - Sec[c + d*x]]*Sin[2*(c + d*x)] - 165*A*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 126*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 576*B*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]*Tan[c + d*x] - 192*A*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]*Tan[c + d*x]))/(d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
142,1,366,209,1.2962035,"\int \cos ^4(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(4608 A \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+2079 A \sqrt{1-\sec (c+d x)}+7641 A \cos (c+d x) \sqrt{1-\sec (c+d x)}+2097 A \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}+522 A \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+18 A \cos (4 (c+d x)) \sqrt{1-\sec (c+d x)}+6075 A \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+7680 B \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)+1240 B \sqrt{1-\sec (c+d x)}+6360 B \cos (c+d x) \sqrt{1-\sec (c+d x)}+1240 B \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}-80 B \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+6600 B \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{2880 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{a^{5/2} (163 A+200 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (163 A+200 B) \sin (c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (95 A+104 B) \sin (c+d x) \cos (c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (11 A+8 B) \sin (c+d x) \cos ^2(c+d x) \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a^2*(6075*A*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 6600*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 2079*A*Sqrt[1 - Sec[c + d*x]] + 1240*B*Sqrt[1 - Sec[c + d*x]] + 7641*A*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 6360*B*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 2097*A*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 1240*B*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 522*A*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] - 80*B*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 18*A*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 7680*B*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 4608*A*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(2880*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
143,1,416,254,1.8171277,"\int \cos ^5(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sin (c+d x) \sqrt{a (\sec (c+d x)+1)} \left(15360 A \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},6;\frac{3}{2};1-\sec (c+d x)\right)+11651 A \sqrt{1-\sec (c+d x)}+37029 A \cos (c+d x) \sqrt{1-\sec (c+d x)}+12653 A \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}+3818 A \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+1002 A \cos (4 (c+d x)) \sqrt{1-\sec (c+d x)}+72 A \cos (5 (c+d x)) \sqrt{1-\sec (c+d x)}+25935 A \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+21504 B \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},5;\frac{3}{2};1-\sec (c+d x)\right)+9702 B \sqrt{1-\sec (c+d x)}+35658 B \cos (c+d x) \sqrt{1-\sec (c+d x)}+9786 B \cos (2 (c+d x)) \sqrt{1-\sec (c+d x)}+2436 B \cos (3 (c+d x)) \sqrt{1-\sec (c+d x)}+84 B \cos (4 (c+d x)) \sqrt{1-\sec (c+d x)}+28350 B \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{13440 d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)}}","\frac{a^{5/2} (283 A+326 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (283 A+326 B) \sin (c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (157 A+170 B) \sin (c+d x) \cos ^2(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (283 A+326 B) \sin (c+d x) \cos (c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (13 A+10 B) \sin (c+d x) \cos ^3(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*(25935*A*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 28350*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 11651*A*Sqrt[1 - Sec[c + d*x]] + 9702*B*Sqrt[1 - Sec[c + d*x]] + 37029*A*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 35658*B*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 12653*A*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 9786*B*Cos[2*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 3818*A*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 2436*B*Cos[3*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 1002*A*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 84*B*Cos[4*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 72*A*Cos[5*(c + d*x)]*Sqrt[1 - Sec[c + d*x]] + 21504*B*Hypergeometric2F1[1/2, 5, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]] + 15360*A*Hypergeometric2F1[1/2, 6, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(13440*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])","C",1
144,1,140,202,0.5369354,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left(3 (7 A-B) \sec ^2(c+d x)+(31 B-7 A) \sec (c+d x)+91 A+15 B \sec ^3(c+d x)-43 B\right)-105 \sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{105 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (7 A-B) \tan (c+d x) \sec ^2(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (A-B) \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 A-31 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{105 a d}+\frac{4 (49 A-37 B) \tan (c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) \sec ^3(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"((-105*Sqrt[2]*(A - B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*Sqrt[1 - Sec[c + d*x]]*(91*A - 43*B + (-7*A + 31*B)*Sec[c + d*x] + 3*(7*A - B)*Sec[c + d*x]^2 + 15*B*Sec[c + d*x]^3))*Tan[c + d*x])/(105*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
145,1,123,159,0.4100007,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left((5 A-B) \sec (c+d x)-5 A+3 B \sec ^2(c+d x)+13 B\right)+15 \sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{15 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A-B) \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 A-B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{15 a d}-\frac{4 (5 A-7 B) \tan (c+d x)}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) \sec ^2(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"((15*Sqrt[2]*(A - B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*Sqrt[1 - Sec[c + d*x]]*(-5*A + 13*B + (5*A - B)*Sec[c + d*x] + 3*B*Sec[c + d*x]^2))*Tan[c + d*x])/(15*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
146,1,106,118,0.3022174,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} (3 A+B \sec (c+d x)-B)-3 \sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{3 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (A-B) \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 A-2 B) \tan (c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 B \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{3 a d}",1,"((-3*Sqrt[2]*(A - B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*Sqrt[1 - Sec[c + d*x]]*(3*A - B + B*Sec[c + d*x]))*Tan[c + d*x])/(3*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
147,1,88,78,0.173596,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)+2 B \sqrt{1-\sec (c+d x)}\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (A-B) \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 (c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"((Sqrt[2]*(A - B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + 2*B*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
148,1,92,91,0.2920709,"\int \frac{A+B \sec (c+d x)}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/Sqrt[a + a*Sec[c + d*x]],x]","\frac{2 \cos \left(\frac{1}{2} (c+d x)\right) \left((B-A) \tan ^{-1}\left(\frac{\sin \left(\frac{1}{2} (c+d x)\right)}{\sqrt{\cos (c+d x)}}\right)+\sqrt{2} A \sin ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\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{\sqrt{2} (A-B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"(2*(Sqrt[2]*A*ArcSin[Sqrt[2]*Sin[(c + d*x)/2]] + (-A + B)*ArcTan[Sin[(c + d*x)/2]/Sqrt[Cos[c + d*x]]])*Cos[(c + d*x)/2])/(d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
149,1,11162,119,26.7002539,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\text{Result too large to show}","-\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{\sqrt{2} (A-B) \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}}",1,"Result too large to show","C",0
150,1,135,165,0.4476917,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} (2 A \cos (c+d x)-A+4 B)+(7 A-4 B) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-4 \sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{(A-4 B) \sin (c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{(7 A-4 B) \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) \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 (c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(((7*A - 4*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2]*(A - B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(-A + 4*B + 2*A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
151,1,150,206,0.7071647,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\cos (c+d x) \sqrt{1-\sec (c+d x)} \left(-2 (A-6 B) \cos (c+d x)+8 A \cos ^2(c+d x)+21 A-6 B\right)+(42 B-27 A) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+24 \sqrt{2} (A-B) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{24 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(7 A-2 B) \sin (c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}-\frac{(9 A-14 B) \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) \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,"(((-27*A + 42*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]] + 24*Sqrt[2]*(A - B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] + Cos[c + d*x]*(21*A - 6*B - 2*(A - 6*B)*Cos[c + d*x] + 8*A*Cos[c + d*x]^2)*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(24*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
152,1,160,216,2.3957596,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(4 (5 A-3 B) \sec ^2(c+d x)-12 (5 A-9 B) \sec (c+d x)-95 A+12 B \sec ^3(c+d x)+147 B\right)+15 \sqrt{2} (11 A-15 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{30 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{(11 A-15 B) \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 A-39 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{30 a^2 d}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(5 A-9 B) \tan (c+d x) \sec ^2(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{(65 A-93 B) \tan (c+d x)}{15 a d \sqrt{a \sec (c+d x)+a}}",1,"((15*Sqrt[2]*(11*A - 15*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(-95*A + 147*B - 12*(5*A - 9*B)*Sec[c + d*x] + 4*(5*A - 3*B)*Sec[c + d*x]^2 + 12*B*Sec[c + d*x]^3))*Tan[c + d*x])/(30*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
153,1,141,171,1.3559832,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(12 (A-B) \sec (c+d x)+15 A+4 B \sec ^2(c+d x)-19 B\right)-3 \sqrt{2} (7 A-11 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{6 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{(7 A-11 B) \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 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{6 a^2 d}+\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{(9 A-13 B) \tan (c+d x)}{3 a d \sqrt{a \sec (c+d x)+a}}",1,"((-3*Sqrt[2]*(7*A - 11*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(15*A - 19*B + 12*(A - B)*Sec[c + d*x] + 4*B*Sec[c + d*x]^2))*Tan[c + d*x])/(6*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
154,1,125,118,0.8715807,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} (-A+4 B \sec (c+d x)+5 B)+\sqrt{2} (3 A-7 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{2 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{(3 A-7 B) \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) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}+\frac{2 B \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}}",1,"((Sqrt[2]*(3*A - 7*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Sec[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(-A + 5*B + 4*B*Sec[c + d*x]))*Tan[c + d*x])/(2*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
155,1,127,87,0.8167437,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 (A-B) \sin (c+d x) \sqrt{1-\sec (c+d x)}+2 \sqrt{2} (A+3 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tan (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)}{4 a d (\cos (c+d x)+1) \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(A+3 B) \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) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*(A - B)*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 2*Sqrt[2]*(A + 3*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^2*Tan[c + d*x])/(4*a*d*(1 + Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
156,1,11183,127,26.7865576,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(5 A-B) \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) \tan (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
157,1,11954,170,27.2713185,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\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{(9 A-5 B) \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-B) \sin (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"Result too large to show","C",0
158,1,395,221,2.4440187,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sec (c+d x) \left((91 A-48 B) (\sin (c+d x)+\tan (c+d x)) \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-40 A \sqrt{1-\sec (c+d x)} \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right) (\sin (c+d x)+\tan (c+d x))-13 A \sin (c+d x) \sqrt{1-\sec (c+d x)}+\frac{13}{2} A \sin (2 (c+d x)) \sqrt{1-\sec (c+d x)}+18 A \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec (c+d x)}-52 \sqrt{2} A \sin (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)-52 \sqrt{2} A \tan (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)+24 B \sin (c+d x) \sqrt{1-\sec (c+d x)}+8 B \sin (2 (c+d x)) \sqrt{1-\sec (c+d x)}+36 \sqrt{2} B \sin (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)+36 \sqrt{2} B \tan (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{(19 A-12 B) \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) \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) \sin (c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}+\frac{(2 A-B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x) \cos (c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Sec[c + d*x]*(-52*Sqrt[2]*A*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Sin[c + d*x] + 36*Sqrt[2]*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Sin[c + d*x] - 13*A*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 24*B*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + 18*A*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]]*Sin[c + d*x] + (13*A*Sqrt[1 - Sec[c + d*x]]*Sin[2*(c + d*x)])/2 + 8*B*Sqrt[1 - Sec[c + d*x]]*Sin[2*(c + d*x)] - 52*Sqrt[2]*A*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Tan[c + d*x] + 36*Sqrt[2]*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Tan[c + d*x] + (91*A - 48*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]]*(Sin[c + d*x] + Tan[c + d*x]) - 40*A*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]]*(Sin[c + d*x] + Tan[c + d*x])))/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
159,1,502,268,6.1529038,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","-\frac{A (\sec (c+d x)+1)^{3/2} \left(\frac{336 \tan (c+d x) \, _2F_1\left(\frac{1}{2},4;\frac{3}{2};1-\sec (c+d x)\right)}{d \sqrt{\sec (c+d x)+1}}+\frac{17 \tan (c+d x) \left(-8 \cos ^3(c+d x) \sqrt{1-\sec (c+d x)}+2 \cos ^2(c+d x) \sqrt{1-\sec (c+d x)}+3 \left(-7 \cos (c+d x) \sqrt{1-\sec (c+d x)}+9 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-8 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1}}\right)}{96 (a (\sec (c+d x)+1))^{3/2}}-\frac{A \sin (c+d x) \cos ^2(c+d x)}{2 d (a (\sec (c+d x)+1))^{3/2}}-\frac{B (\sec (c+d x)+1)^{3/2} \left(\frac{40 \tan (c+d x) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)}{d \sqrt{\sec (c+d x)+1}}-\frac{13 \tan (c+d x) \left(2 \cos ^2(c+d x) \sqrt{1-\sec (c+d x)}-\cos (c+d x) \sqrt{1-\sec (c+d x)}+7 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1}}\right)}{16 (a (\sec (c+d x)+1))^{3/2}}-\frac{B \sin (c+d x) \cos (c+d x)}{2 d (a (\sec (c+d x)+1))^{3/2}}","-\frac{(47 A-38 B) \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) \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 (3 A-2 B) \sin (c+d x)}{8 a d \sqrt{a \sec (c+d x)+a}}+\frac{(5 A-3 B) \sin (c+d x) \cos ^2(c+d x)}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(13 A-12 B) \sin (c+d x) \cos (c+d x)}{12 a d \sqrt{a \sec (c+d x)+a}}",1,"-1/2*(B*Cos[c + d*x]*Sin[c + d*x])/(d*(a*(1 + Sec[c + d*x]))^(3/2)) - (A*Cos[c + d*x]^2*Sin[c + d*x])/(2*d*(a*(1 + Sec[c + d*x]))^(3/2)) - (B*(1 + Sec[c + d*x])^(3/2)*((40*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) - (13*(7*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 2*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(16*(a*(1 + Sec[c + d*x]))^(3/2)) - (A*(1 + Sec[c + d*x])^(3/2)*((336*Hypergeometric2F1[1/2, 4, 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (17*(3*(9*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 8*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - 7*Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]]) + 2*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]] - 8*Cos[c + d*x]^3*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(96*(a*(1 + Sec[c + d*x]))^(3/2))","C",1
160,1,161,216,2.7076134,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left(32 (3 A-5 B) \sec ^2(c+d x)+(255 A-503 B) \sec (c+d x)+147 A+32 B \sec ^3(c+d x)-299 B\right)-6 \sqrt{2} (75 A-163 B) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{48 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{(75 A-163 B) \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 A-95 B) \tan (c+d x) \sqrt{a \sec (c+d x)+a}}{48 a^3 d}+\frac{(93 A-197 B) \tan (c+d x)}{24 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(A-B) \tan (c+d x) \sec ^3(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(9 A-17 B) \tan (c+d x) \sec ^2(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((-6*Sqrt[2]*(75*A - 163*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(147*A - 299*B + (255*A - 503*B)*Sec[c + d*x] + 32*(3*A - 5*B)*Sec[c + d*x]^2 + 32*B*Sec[c + d*x]^3))*Tan[c + d*x])/(48*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
161,1,144,169,1.5768441,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} \left((85 B-13 A) \sec (c+d x)-9 A+32 B \sec ^2(c+d x)+49 B\right)+2 \sqrt{2} (19 A-75 B) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{(19 A-75 B) \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 B) \tan (c+d x)}{4 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(A-B) \tan (c+d x) \sec ^2(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}-\frac{(5 A-13 B) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"((2*Sqrt[2]*(19*A - 75*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(-9*A + 49*B + (-13*A + 85*B)*Sec[c + d*x] + 32*B*Sec[c + d*x]^2))*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
162,1,131,126,1.6395606,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \left(\sqrt{1-\sec (c+d x)} ((5 A-13 B) \sec (c+d x)+A-9 B)+2 \sqrt{2} (5 A+19 B) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{(5 A+19 B) \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-13 B) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"((2*Sqrt[2]*(5*A + 19*B)*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^2 + Sqrt[1 - Sec[c + d*x]]*(A - 9*B + (5*A - 13*B)*Sec[c + d*x]))*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
163,1,206,126,1.5929394,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{64 A \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sqrt{1-\sec (c+d x)} \sec (c+d x) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x))\right)+B (10 \sin (c+d x)+\sin (2 (c+d x))) \sqrt{1-\sec (c+d x)}+40 \sqrt{2} B \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)}{32 a^2 d (\cos (c+d x)+1)^2 \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(3 A+5 B) \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 A+5 B) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}+\frac{(A-B) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(40*Sqrt[2]*B*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]]*Cos[(c + d*x)/2]^5*Sec[c + d*x]*Sin[(c + d*x)/2] + 64*A*Cos[(c + d*x)/2]^5*Hypergeometric2F1[1/2, 3, 3/2, (1 - Sec[c + d*x])/2]*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]*Sin[(c + d*x)/2] + B*Sqrt[1 - Sec[c + d*x]]*(10*Sin[c + d*x] + Sin[2*(c + d*x)]))/(32*a^2*d*(1 + Cos[c + d*x])^2*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","C",1
164,1,11243,164,27.0402498,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{(43 A-3 B) \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) \tan (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \tan (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
165,1,12012,207,27.3097754,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\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{(115 A-43 B) \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a \sec (c+d x)+a}}\right)}{16 \sqrt{2} a^{5/2} d}+\frac{(35 A-11 B) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(15 A-7 B) \sin (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"Result too large to show","C",0
166,1,512,264,6.1690269,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","-\frac{A (\sec (c+d x)+1)^{5/2} \left(\frac{760 \tan (c+d x) \, _2F_1\left(\frac{1}{2},3;\frac{3}{2};1-\sec (c+d x)\right)}{d \sqrt{\sec (c+d x)+1}}+\frac{152 \sin (c+d x) \cos (c+d x)}{d (\sec (c+d x)+1)^{3/2}}-\frac{219 \tan (c+d x) \left(2 \cos ^2(c+d x) \sqrt{1-\sec (c+d x)}-\cos (c+d x) \sqrt{1-\sec (c+d x)}+7 \tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-4 \sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1}}\right)}{128 (a (\sec (c+d x)+1))^{5/2}}-\frac{A \sin (c+d x) \cos (c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}-\frac{B \sin (c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}-\frac{5 B (\sec (c+d x)+1)^{5/2} \left(\frac{6 \sin (c+d x)}{d (\sec (c+d x)+1)^{3/2}}+\frac{9 \tan (c+d x) \left(\cos (c+d x)+\frac{\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)}{\sqrt{1-\sec (c+d x)}}\right)}{d \sqrt{\sec (c+d x)+1}}+\frac{23 \tan (c+d x) \left(-\cos (c+d x) \sqrt{1-\sec (c+d x)}+\tanh ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-\sqrt{2} \tanh ^{-1}\left(\frac{\sqrt{1-\sec (c+d x)}}{\sqrt{2}}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{\sec (c+d x)+1}}\right)}{32 (a (\sec (c+d x)+1))^{5/2}}","\frac{(39 A-20 B) \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) \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 (9 A-5 B) \sin (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(31 A-15 B) \sin (c+d x) \cos (c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(19 A-11 B) \sin (c+d x) \cos (c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \cos (c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"-1/4*(B*Sin[c + d*x])/(d*(a*(1 + Sec[c + d*x]))^(5/2)) - (A*Cos[c + d*x]*Sin[c + d*x])/(4*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (5*B*(1 + Sec[c + d*x])^(5/2)*((6*Sin[c + d*x])/(d*(1 + Sec[c + d*x])^(3/2)) + (9*(Cos[c + d*x] + ArcTanh[Sqrt[1 - Sec[c + d*x]]]/Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) + (23*(ArcTanh[Sqrt[1 - Sec[c + d*x]]] - Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(32*(a*(1 + Sec[c + d*x]))^(5/2)) - (A*(1 + Sec[c + d*x])^(5/2)*((152*Cos[c + d*x]*Sin[c + d*x])/(d*(1 + Sec[c + d*x])^(3/2)) + (760*Hypergeometric2F1[1/2, 3, 3/2, 1 - Sec[c + d*x]]*Tan[c + d*x])/(d*Sqrt[1 + Sec[c + d*x]]) - (219*(7*ArcTanh[Sqrt[1 - Sec[c + d*x]]] - 4*Sqrt[2]*ArcTanh[Sqrt[1 - Sec[c + d*x]]/Sqrt[2]] - Cos[c + d*x]*Sqrt[1 - Sec[c + d*x]] + 2*Cos[c + d*x]^2*Sqrt[1 - Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[1 + Sec[c + d*x]])))/(128*(a*(1 + Sec[c + d*x]))^(5/2))","C",1
167,1,140,89,0.6436278,"\int \frac{A+A \sec (c+d x)}{\sqrt{a-a \sec (c+d x)}} \, dx","Integrate[(A + A*Sec[c + d*x])/Sqrt[a - a*Sec[c + d*x]],x]","-\frac{i A \left(-1+e^{i (c+d x)}\right) \left(\sqrt{2} \sinh ^{-1}\left(e^{i (c+d x)}\right)-4 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+\sqrt{2} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{2} d \sqrt{1+e^{2 i (c+d x)}} \sqrt{a-a \sec (c+d x)}}","\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}-\frac{2 \sqrt{2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}",1,"((-I)*A*(-1 + E^(I*(c + d*x)))*(Sqrt[2]*ArcSinh[E^(I*(c + d*x))] - 4*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + Sqrt[2]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]]))/(Sqrt[2]*d*Sqrt[1 + E^((2*I)*(c + d*x))]*Sqrt[a - a*Sec[c + d*x]])","C",1
168,1,269,115,1.4953475,"\int \frac{\cos (c+d x) (A+A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],x]","\frac{A e^{-\frac{1}{2} i (c+d x)} \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos \left(\frac{1}{2} (c+d x)\right)+i \sin \left(\frac{1}{2} (c+d x)\right)\right) \left(3 e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+e^{-\frac{1}{2} i (c+d x)} \left(e^{-i (c+d x)}+e^{i (c+d x)}+e^{2 i (c+d x)}-4 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+3 \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+1\right)\right)}{2 d \sqrt{a-a \sec (c+d x)}}","\frac{A \sin (c+d x)}{d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}-\frac{2 \sqrt{2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}",1,"(A*((3*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/E^((I/2)*(c + d*x)) + (1 + E^((-I)*(c + d*x)) + E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) - 4*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + 3*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/E^((I/2)*(c + d*x)))*Sec[c + d*x]*(Cos[(c + d*x)/2] + I*Sin[(c + d*x)/2])*Sin[(c + d*x)/2])/(2*d*E^((I/2)*(c + d*x))*Sqrt[a - a*Sec[c + d*x]])","C",1
169,1,297,155,1.7886241,"\int \frac{\cos ^2(c+d x) (A+A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],x]","\frac{A e^{-\frac{1}{2} i (c+d x)} \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos \left(\frac{1}{2} (c+d x)\right)+i \sin \left(\frac{1}{2} (c+d x)\right)\right) \left(11 e^{-\frac{1}{2} i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)+e^{-\frac{1}{2} i (c+d x)} \left(6 e^{-i (c+d x)}+7 e^{i (c+d x)}+e^{-2 i (c+d x)}+6 e^{2 i (c+d x)}+e^{3 i (c+d x)}-16 \sqrt{2} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+11 \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+7\right)\right)}{8 d \sqrt{a-a \sec (c+d x)}}","\frac{5 A \sin (c+d x)}{4 d \sqrt{a-a \sec (c+d x)}}+\frac{11 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{4 \sqrt{a} d}-\frac{2 \sqrt{2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}+\frac{A \sin (c+d x) \cos (c+d x)}{2 d \sqrt{a-a \sec (c+d x)}}",1,"(A*((11*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))])/E^((I/2)*(c + d*x)) + (7 + 6/E^(I*(c + d*x)) + 7*E^(I*(c + d*x)) + E^((-2*I)*(c + d*x)) + 6*E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x)) - 16*Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + 11*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])/E^((I/2)*(c + d*x)))*Sec[c + d*x]*(Cos[(c + d*x)/2] + I*Sin[(c + d*x)/2])*Sin[(c + d*x)/2])/(8*d*E^((I/2)*(c + d*x))*Sqrt[a - a*Sec[c + d*x]])","C",1
170,1,330,192,1.9498319,"\int \frac{\cos ^3(c+d x) (A+A \sec (c+d x))}{\sqrt{a-a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/Sqrt[a - a*Sec[c + d*x]],x]","\frac{A e^{-4 i (c+d x)} \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \left(\cos \left(\frac{1}{2} (c+d x)\right)+i \sin \left(\frac{1}{2} (c+d x)\right)\right) \left(9 e^{i (c+d x)}+40 e^{2 i (c+d x)}+47 e^{3 i (c+d x)}+47 e^{4 i (c+d x)}+40 e^{5 i (c+d x)}+9 e^{6 i (c+d x)}+2 e^{7 i (c+d x)}+69 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \sinh ^{-1}\left(e^{i (c+d x)}\right)-96 \sqrt{2} e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+69 e^{3 i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)+2\right)}{48 d \sqrt{a-a \sec (c+d x)}}","\frac{9 A \sin (c+d x)}{8 d \sqrt{a-a \sec (c+d x)}}+\frac{23 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{8 \sqrt{a} d}-\frac{2 \sqrt{2} A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{a} d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 d \sqrt{a-a \sec (c+d x)}}+\frac{7 A \sin (c+d x) \cos (c+d x)}{12 d \sqrt{a-a \sec (c+d x)}}",1,"(A*(2 + 9*E^(I*(c + d*x)) + 40*E^((2*I)*(c + d*x)) + 47*E^((3*I)*(c + d*x)) + 47*E^((4*I)*(c + d*x)) + 40*E^((5*I)*(c + d*x)) + 9*E^((6*I)*(c + d*x)) + 2*E^((7*I)*(c + d*x)) + 69*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcSinh[E^(I*(c + d*x))] - 96*Sqrt[2]*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + 69*E^((3*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]*(Cos[(c + d*x)/2] + I*Sin[(c + d*x)/2])*Sin[(c + d*x)/2])/(48*d*E^((4*I)*(c + d*x))*Sqrt[a - a*Sec[c + d*x]])","C",1
171,1,322,116,6.6933162,"\int \frac{A+A \sec (c+d x)}{(a-a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(3/2),x]","A \left(\frac{\sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \left(-\frac{4 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}+\frac{4 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}-\frac{2 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{2 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}\right)}{(a-a \sec (c+d x))^{3/2}}-\frac{2 \sqrt{2} e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(\sinh ^{-1}\left(e^{i (c+d x)}\right)-\frac{3 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}+\tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{d (a-a \sec (c+d x))^{3/2}}\right)","\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{a^{3/2} d}-\frac{3 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{2} a^{3/2} d}-\frac{A \tan (c+d x)}{d (a-a \sec (c+d x))^{3/2}}",1,"A*((-2*Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(ArcSinh[E^(I*(c + d*x))] - (3*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] + ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sin[c/2 + (d*x)/2]^3)/(d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(3/2)) + (Sec[c + d*x]^2*((4*Cos[c/2]*Cos[(d*x)/2])/d - (2*Cot[c/2]*Csc[c/2 + (d*x)/2])/d + (2*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/d - (4*Sin[c/2]*Sin[(d*x)/2])/d)*Sin[c/2 + (d*x)/2]^3)/(a - a*Sec[c + d*x])^(3/2))","C",1
172,1,361,146,6.6279023,"\int \frac{\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(3/2),x]","A \left(\frac{\sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \left(-\frac{2 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}+\frac{2 \sin \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}+\frac{2 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}-\frac{2 \cos \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)}{d}-\frac{2 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{2 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}\right)}{(a-a \sec (c+d x))^{3/2}}+\frac{\sqrt{2} e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(-5 \sinh ^{-1}\left(e^{i (c+d x)}\right)+7 \sqrt{2} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)-5 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{d (a-a \sec (c+d x))^{3/2}}\right)","\frac{5 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{a^{3/2} d}-\frac{7 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{2} a^{3/2} d}+\frac{2 A \sin (c+d x)}{a d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x)}{d (a-a \sec (c+d x))^{3/2}}",1,"A*((Sqrt[2]*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(-5*ArcSinh[E^(I*(c + d*x))] + 7*Sqrt[2]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] - 5*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sin[c/2 + (d*x)/2]^3)/(d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(3/2)) + (Sec[c + d*x]^2*((2*Cos[c/2]*Cos[(d*x)/2])/d - (2*Cos[(3*c)/2]*Cos[(3*d*x)/2])/d - (2*Cot[c/2]*Csc[c/2 + (d*x)/2])/d + (2*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/d - (2*Sin[c/2]*Sin[(d*x)/2])/d + (2*Sin[(3*c)/2]*Sin[(3*d*x)/2])/d)*Sin[c/2 + (d*x)/2]^3)/(a - a*Sec[c + d*x])^(3/2))","C",1
173,1,408,194,6.7656988,"\int \frac{\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(3/2),x]","A \left(\frac{\sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \left(\frac{3 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{2 d}+\frac{5 \sin \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}+\frac{\sin \left(\frac{5 c}{2}\right) \sin \left(\frac{5 d x}{2}\right)}{2 d}-\frac{3 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{2 d}-\frac{5 \cos \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)}{d}-\frac{\cos \left(\frac{5 c}{2}\right) \cos \left(\frac{5 d x}{2}\right)}{2 d}-\frac{2 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{2 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}\right)}{(a-a \sec (c+d x))^{3/2}}-\frac{e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(31 \sinh ^{-1}\left(e^{i (c+d x)}\right)-44 \sqrt{2} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+31 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{2 \sqrt{2} d (a-a \sec (c+d x))^{3/2}}\right)","\frac{31 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{4 a^{3/2} d}-\frac{11 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{2} a^{3/2} d}+\frac{13 A \sin (c+d x)}{4 a d \sqrt{a-a \sec (c+d x)}}+\frac{3 A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x) \cos (c+d x)}{d (a-a \sec (c+d x))^{3/2}}",1,"A*(-1/2*(Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(31*ArcSinh[E^(I*(c + d*x))] - 44*Sqrt[2]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + 31*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sin[c/2 + (d*x)/2]^3)/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(3/2)) + (Sec[c + d*x]^2*((-3*Cos[c/2]*Cos[(d*x)/2])/(2*d) - (5*Cos[(3*c)/2]*Cos[(3*d*x)/2])/d - (Cos[(5*c)/2]*Cos[(5*d*x)/2])/(2*d) - (2*Cot[c/2]*Csc[c/2 + (d*x)/2])/d + (2*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/d + (3*Sin[c/2]*Sin[(d*x)/2])/(2*d) + (5*Sin[(3*c)/2]*Sin[(3*d*x)/2])/d + (Sin[(5*c)/2]*Sin[(5*d*x)/2])/(2*d))*Sin[c/2 + (d*x)/2]^3)/(a - a*Sec[c + d*x])^(3/2))","C",1
174,1,452,236,6.732779,"\int \frac{\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(3/2),x]","A \left(\frac{\sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^2(c+d x) \left(\frac{65 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{12 d}+\frac{25 \sin \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{3 d}+\frac{5 \sin \left(\frac{5 c}{2}\right) \sin \left(\frac{5 d x}{2}\right)}{4 d}+\frac{\sin \left(\frac{7 c}{2}\right) \sin \left(\frac{7 d x}{2}\right)}{6 d}-\frac{65 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{12 d}-\frac{25 \cos \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)}{3 d}-\frac{5 \cos \left(\frac{5 c}{2}\right) \cos \left(\frac{5 d x}{2}\right)}{4 d}-\frac{\cos \left(\frac{7 c}{2}\right) \cos \left(\frac{7 d x}{2}\right)}{6 d}-\frac{2 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{2 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}\right)}{(a-a \sec (c+d x))^{3/2}}-\frac{5 e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^3\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{3}{2}}(c+d x) \left(17 \sinh ^{-1}\left(e^{i (c+d x)}\right)-24 \sqrt{2} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+17 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{4 \sqrt{2} d (a-a \sec (c+d x))^{3/2}}\right)","\frac{85 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{8 a^{3/2} d}-\frac{15 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{\sqrt{2} a^{3/2} d}+\frac{35 A \sin (c+d x)}{8 a d \sqrt{a-a \sec (c+d x)}}+\frac{4 A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt{a-a \sec (c+d x)}}-\frac{A \sin (c+d x) \cos ^2(c+d x)}{d (a-a \sec (c+d x))^{3/2}}+\frac{25 A \sin (c+d x) \cos (c+d x)}{12 a d \sqrt{a-a \sec (c+d x)}}",1,"A*((-5*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(17*ArcSinh[E^(I*(c + d*x))] - 24*Sqrt[2]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + 17*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(3/2)*Sin[c/2 + (d*x)/2]^3)/(4*Sqrt[2]*d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(3/2)) + (Sec[c + d*x]^2*((-65*Cos[c/2]*Cos[(d*x)/2])/(12*d) - (25*Cos[(3*c)/2]*Cos[(3*d*x)/2])/(3*d) - (5*Cos[(5*c)/2]*Cos[(5*d*x)/2])/(4*d) - (Cos[(7*c)/2]*Cos[(7*d*x)/2])/(6*d) - (2*Cot[c/2]*Csc[c/2 + (d*x)/2])/d + (2*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/d + (65*Sin[c/2]*Sin[(d*x)/2])/(12*d) + (25*Sin[(3*c)/2]*Sin[(3*d*x)/2])/(3*d) + (5*Sin[(5*c)/2]*Sin[(5*d*x)/2])/(4*d) + (Sin[(7*c)/2]*Sin[(7*d*x)/2])/(6*d))*Sin[c/2 + (d*x)/2]^3)/(a - a*Sec[c + d*x])^(3/2))","C",1
175,1,387,152,6.8140485,"\int \frac{A+A \sec (c+d x)}{(a-a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + A*Sec[c + d*x])/(a - a*Sec[c + d*x])^(5/2),x]","A \left(\frac{\sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(\frac{11 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}-\frac{11 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}-\frac{\cot \left(\frac{c}{2}\right) \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{15 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{15 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}\right)}{(a-a \sec (c+d x))^{5/2}}+\frac{e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{5}{2}}(c+d x) \left(8 \sinh ^{-1}\left(e^{i (c+d x)}\right)-\frac{23 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}+8 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{2} d (a-a \sec (c+d x))^{5/2}}\right)","\frac{2 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{a^{5/2} d}-\frac{23 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{8 \sqrt{2} a^{5/2} d}-\frac{7 A \tan (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac{A \tan (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}",1,"A*((Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(8*ArcSinh[E^(I*(c + d*x))] - (23*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] + 8*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(5/2)*Sin[c/2 + (d*x)/2]^5)/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(5/2)) + (Sec[c + d*x]^3*((-11*Cos[c/2]*Cos[(d*x)/2])/d + (15*Cot[c/2]*Csc[c/2 + (d*x)/2])/(2*d) - (Cot[c/2]*Csc[c/2 + (d*x)/2]^3)/d - (15*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(2*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/d + (11*Sin[c/2]*Sin[(d*x)/2])/d)*Sin[c/2 + (d*x)/2]^5)/(a - a*Sec[c + d*x])^(5/2))","C",1
176,1,423,184,6.8010095,"\int \frac{\cos (c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]","A \left(\frac{\sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(\frac{15 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}-\frac{4 \sin \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}-\frac{15 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}+\frac{4 \cos \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)}{d}-\frac{\cot \left(\frac{c}{2}\right) \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{23 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{23 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}\right)}{(a-a \sec (c+d x))^{5/2}}+\frac{e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{5}{2}}(c+d x) \left(28 \sinh ^{-1}\left(e^{i (c+d x)}\right)-\frac{79 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}+28 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{2} d (a-a \sec (c+d x))^{5/2}}\right)","\frac{7 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{a^{5/2} d}-\frac{79 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{8 \sqrt{2} a^{5/2} d}+\frac{23 A \sin (c+d x)}{8 a^2 d \sqrt{a-a \sec (c+d x)}}-\frac{11 A \sin (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac{A \sin (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}",1,"A*((Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(28*ArcSinh[E^(I*(c + d*x))] - (79*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] + 28*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(5/2)*Sin[c/2 + (d*x)/2]^5)/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(5/2)) + (Sec[c + d*x]^3*((-15*Cos[c/2]*Cos[(d*x)/2])/d + (4*Cos[(3*c)/2]*Cos[(3*d*x)/2])/d + (23*Cot[c/2]*Csc[c/2 + (d*x)/2])/(2*d) - (Cot[c/2]*Csc[c/2 + (d*x)/2]^3)/d - (23*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(2*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/d + (15*Sin[c/2]*Sin[(d*x)/2])/d - (4*Sin[(3*c)/2]*Sin[(3*d*x)/2])/d)*Sin[c/2 + (d*x)/2]^5)/(a - a*Sec[c + d*x])^(5/2))","C",1
177,1,458,236,6.8210214,"\int \frac{\cos ^2(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]","A \left(\frac{\sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(\frac{12 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{d}-\frac{14 \sin \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{d}-\frac{\sin \left(\frac{5 c}{2}\right) \sin \left(\frac{5 d x}{2}\right)}{d}-\frac{12 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{d}+\frac{14 \cos \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)}{d}+\frac{\cos \left(\frac{5 c}{2}\right) \cos \left(\frac{5 d x}{2}\right)}{d}-\frac{\cot \left(\frac{c}{2}\right) \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{31 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{31 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}\right)}{(a-a \sec (c+d x))^{5/2}}+\frac{e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{5}{2}}(c+d x) \left(59 \sinh ^{-1}\left(e^{i (c+d x)}\right)-\frac{167 \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)}{\sqrt{2}}+59 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{\sqrt{2} d (a-a \sec (c+d x))^{5/2}}\right)","\frac{59 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{4 a^{5/2} d}-\frac{167 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{8 \sqrt{2} a^{5/2} d}+\frac{49 A \sin (c+d x)}{8 a^2 d \sqrt{a-a \sec (c+d x)}}+\frac{23 A \sin (c+d x) \cos (c+d x)}{8 a^2 d \sqrt{a-a \sec (c+d x)}}-\frac{15 A \sin (c+d x) \cos (c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac{A \sin (c+d x) \cos (c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}",1,"A*((Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(59*ArcSinh[E^(I*(c + d*x))] - (167*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])])/Sqrt[2] + 59*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(5/2)*Sin[c/2 + (d*x)/2]^5)/(Sqrt[2]*d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(5/2)) + (Sec[c + d*x]^3*((-12*Cos[c/2]*Cos[(d*x)/2])/d + (14*Cos[(3*c)/2]*Cos[(3*d*x)/2])/d + (Cos[(5*c)/2]*Cos[(5*d*x)/2])/d + (31*Cot[c/2]*Csc[c/2 + (d*x)/2])/(2*d) - (Cot[c/2]*Csc[c/2 + (d*x)/2]^3)/d - (31*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(2*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/d + (12*Sin[c/2]*Sin[(d*x)/2])/d - (14*Sin[(3*c)/2]*Sin[(3*d*x)/2])/d - (Sin[(5*c)/2]*Sin[(5*d*x)/2])/d)*Sin[c/2 + (d*x)/2]^5)/(a - a*Sec[c + d*x])^(5/2))","C",1
178,1,514,280,6.8412809,"\int \frac{\cos ^3(c+d x) (A+A \sec (c+d x))}{(a-a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + A*Sec[c + d*x]))/(a - a*Sec[c + d*x])^(5/2),x]","A \left(\frac{\sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^3(c+d x) \left(\frac{7 \sin \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right)}{6 d}-\frac{92 \sin \left(\frac{3 c}{2}\right) \sin \left(\frac{3 d x}{2}\right)}{3 d}-\frac{7 \sin \left(\frac{5 c}{2}\right) \sin \left(\frac{5 d x}{2}\right)}{2 d}-\frac{\sin \left(\frac{7 c}{2}\right) \sin \left(\frac{7 d x}{2}\right)}{3 d}-\frac{7 \cos \left(\frac{c}{2}\right) \cos \left(\frac{d x}{2}\right)}{6 d}+\frac{92 \cos \left(\frac{3 c}{2}\right) \cos \left(\frac{3 d x}{2}\right)}{3 d}+\frac{7 \cos \left(\frac{5 c}{2}\right) \cos \left(\frac{5 d x}{2}\right)}{2 d}+\frac{\cos \left(\frac{7 c}{2}\right) \cos \left(\frac{7 d x}{2}\right)}{3 d}-\frac{\cot \left(\frac{c}{2}\right) \csc ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{39 \cot \left(\frac{c}{2}\right) \csc \left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{\csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{39 \csc \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \csc ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}\right)}{(a-a \sec (c+d x))^{5/2}}+\frac{7 e^{-\frac{1}{2} i (c+d x)} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \sin ^5\left(\frac{c}{2}+\frac{d x}{2}\right) \sec ^{\frac{5}{2}}(c+d x) \left(29 \sinh ^{-1}\left(e^{i (c+d x)}\right)-41 \sqrt{2} \tanh ^{-1}\left(\frac{1+e^{i (c+d x)}}{\sqrt{2} \sqrt{1+e^{2 i (c+d x)}}}\right)+29 \tanh ^{-1}\left(\sqrt{1+e^{2 i (c+d x)}}\right)\right)}{2 \sqrt{2} d (a-a \sec (c+d x))^{5/2}}\right)","\frac{203 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a-a \sec (c+d x)}}\right)}{8 a^{5/2} d}-\frac{287 A \tan ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{2} \sqrt{a-a \sec (c+d x)}}\right)}{8 \sqrt{2} a^{5/2} d}+\frac{21 A \sin (c+d x)}{2 a^2 d \sqrt{a-a \sec (c+d x)}}+\frac{77 A \sin (c+d x) \cos ^2(c+d x)}{24 a^2 d \sqrt{a-a \sec (c+d x)}}+\frac{119 A \sin (c+d x) \cos (c+d x)}{24 a^2 d \sqrt{a-a \sec (c+d x)}}-\frac{19 A \sin (c+d x) \cos ^2(c+d x)}{8 a d (a-a \sec (c+d x))^{3/2}}-\frac{A \sin (c+d x) \cos ^2(c+d x)}{2 d (a-a \sec (c+d x))^{5/2}}",1,"A*((7*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*(29*ArcSinh[E^(I*(c + d*x))] - 41*Sqrt[2]*ArcTanh[(1 + E^(I*(c + d*x)))/(Sqrt[2]*Sqrt[1 + E^((2*I)*(c + d*x))])] + 29*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]])*Sec[c + d*x]^(5/2)*Sin[c/2 + (d*x)/2]^5)/(2*Sqrt[2]*d*E^((I/2)*(c + d*x))*(a - a*Sec[c + d*x])^(5/2)) + (Sec[c + d*x]^3*((-7*Cos[c/2]*Cos[(d*x)/2])/(6*d) + (92*Cos[(3*c)/2]*Cos[(3*d*x)/2])/(3*d) + (7*Cos[(5*c)/2]*Cos[(5*d*x)/2])/(2*d) + (Cos[(7*c)/2]*Cos[(7*d*x)/2])/(3*d) + (39*Cot[c/2]*Csc[c/2 + (d*x)/2])/(2*d) - (Cot[c/2]*Csc[c/2 + (d*x)/2]^3)/d - (39*Csc[c/2]*Csc[c/2 + (d*x)/2]^2*Sin[(d*x)/2])/(2*d) + (Csc[c/2]*Csc[c/2 + (d*x)/2]^4*Sin[(d*x)/2])/d + (7*Sin[c/2]*Sin[(d*x)/2])/(6*d) - (92*Sin[(3*c)/2]*Sin[(3*d*x)/2])/(3*d) - (7*Sin[(5*c)/2]*Sin[(5*d*x)/2])/(2*d) - (Sin[(7*c)/2]*Sin[(7*d*x)/2])/(3*d))*Sin[c/2 + (d*x)/2]^5)/(a - a*Sec[c + d*x])^(5/2))","C",1
179,1,200,199,0.7802796,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1) (A+B \sec (c+d x)) \left(5 (7 A+5 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-63 (A+B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+63 A \sin (c+d x)+35 A \tan (c+d x)+21 A \tan (c+d x) \sec (c+d x)+63 B \sin (c+d x)+25 B \tan (c+d x)+15 B \tan (c+d x) \sec ^2(c+d x)+21 B \tan (c+d x) \sec (c+d x)\right)}{105 d \sec ^{\frac{3}{2}}(c+d x) (A \cos (c+d x)+B)}","\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}+\frac{2 a (7 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{6 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (7 A+5 B) \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{6 a (A+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 B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{7 d}",1,"(a*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])*(A + B*Sec[c + d*x])*(-63*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(7*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 63*A*Sin[c + d*x] + 63*B*Sin[c + d*x] + 35*A*Tan[c + d*x] + 25*B*Tan[c + d*x] + 21*A*Sec[c + d*x]*Tan[c + d*x] + 21*B*Sec[c + d*x]*Tan[c + d*x] + 15*B*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d*(B + A*Cos[c + d*x])*Sec[c + d*x]^(3/2))","A",1
180,1,168,172,0.6735665,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \sec ^2\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1) (A+B \sec (c+d x)) \left(5 (A+B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 (5 A+3 B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 A \sin (c+d x)+5 A \tan (c+d x)+9 B \sin (c+d x)+5 B \tan (c+d x)+3 B \tan (c+d x) \sec (c+d x)\right)}{15 d \sec ^{\frac{3}{2}}(c+d x) (A \cos (c+d x)+B)}","\frac{2 a (A+B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a (5 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a (A+B) \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 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 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(a*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])*(A + B*Sec[c + d*x])*(-3*(5*A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 15*A*Sin[c + d*x] + 9*B*Sin[c + d*x] + 5*A*Tan[c + d*x] + 5*B*Tan[c + d*x] + 3*B*Sec[c + d*x]*Tan[c + d*x]))/(15*d*(B + A*Cos[c + d*x])*Sec[c + d*x]^(3/2))","A",1
181,1,94,135,0.5303982,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a \sec ^{\frac{3}{2}}(c+d x) \left(2 (3 A+B) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 (A+B) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (3 (A+B) \cos (c+d x)+B)\right)}{3 d}","\frac{2 a (A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 a (3 A+B) \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) \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 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(a*Sec[c + d*x]^(3/2)*(-6*(A + B)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 2*(3*A + B)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 2*(B + 3*(A + B)*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
182,1,77,106,0.3072789,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 a \sqrt{\sec (c+d x)} \left((A+B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+(A-B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+B \sin (c+d x)\right)}{d}","\frac{2 a (A+B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A-B) \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(2*a*Sqrt[Sec[c + d*x]]*((A - B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + B*Sin[c + d*x]))/d","A",1
183,1,83,110,0.3180513,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(2 (A+3 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (A+B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \sin (2 (c+d x))\right)}{3 d}","\frac{2 a (A+3 B) \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) \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)}}",1,"(a*Sqrt[Sec[c + d*x]]*(6*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(A + 3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + A*Sin[2*(c + d*x)]))/(3*d)","A",1
184,1,99,141,0.5881349,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (5 (A+B)+3 A \cos (c+d x))+10 (A+B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (3 A+5 B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a (A+B) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\frac{2 a (A+B) \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) \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)}",1,"(a*Sqrt[Sec[c + d*x]]*(6*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(A + B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*(A + B) + 3*A*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
185,1,113,172,0.9865574,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{a \sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (42 (A+B) \cos (c+d x)+15 A \cos (2 (c+d x))+65 A+70 B)+20 (5 A+7 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 (A+B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a (A+B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 A+7 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+7 B) \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{6 a (A+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 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a*Sqrt[Sec[c + d*x]]*(252*(A + B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(5*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*A + 70*B + 42*(A + B)*Cos[c + d*x] + 15*A*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
186,1,463,234,4.6656241,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 \csc (c) e^{-i d x} \cos ^3(c+d x) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) (4 A+3 B) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left(5 i (7 A+6 B) \left(1+e^{2 i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+7 A \left(9 e^{i (c+d x)}-5 e^{2 i (c+d x)}+36 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+39 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+12 e^{7 i (c+d x)}-5\right)+3 B \left(7 e^{i (c+d x)}-20 e^{2 i (c+d x)}+63 e^{3 i (c+d x)}+20 e^{4 i (c+d x)}+77 e^{5 i (c+d x)}+10 e^{6 i (c+d x)}+21 e^{7 i (c+d x)}-10\right)\right)}{\left(1+e^{2 i (c+d x)}\right)^3}\right)}{210 d (A \cos (c+d x)+B)}","\frac{2 a^2 (7 A+9 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{4 a^2 (7 A+6 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{4 a^2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (7 A+6 B) \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 (4 A+3 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 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{7 d}",1,"(a^2*Cos[c + d*x]^3*Csc[c]*Sec[(c + d*x)/2]^4*(7*Sqrt[2]*(4*A + 3*B)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(7*A*(-5 + 9*E^(I*(c + d*x)) - 5*E^((2*I)*(c + d*x)) + 36*E^((3*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) + 39*E^((5*I)*(c + d*x)) + 5*E^((6*I)*(c + d*x)) + 12*E^((7*I)*(c + d*x))) + 3*B*(-10 + 7*E^(I*(c + d*x)) - 20*E^((2*I)*(c + d*x)) + 63*E^((3*I)*(c + d*x)) + 20*E^((4*I)*(c + d*x)) + 77*E^((5*I)*(c + d*x)) + 10*E^((6*I)*(c + d*x)) + 21*E^((7*I)*(c + d*x))) + (5*I)*(7*A + 6*B)*(1 + E^((2*I)*(c + d*x)))^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/(E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3))*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/(210*d*E^(I*d*x)*(B + A*Cos[c + d*x]))","C",1
187,1,321,199,6.5704334,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 e^{-i c} \left(-1+e^{2 i c}\right) \csc (c) \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left(2 (5 A+4 B) e^{i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-10 i (2 A+B) \left(1+e^{2 i (c+d x)}\right)^2 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-30 A e^{i (c+d x)}-60 A e^{3 i (c+d x)}-5 A e^{4 i (c+d x)}-30 A e^{5 i (c+d x)}+5 A-18 B e^{i (c+d x)}-54 B e^{3 i (c+d x)}-10 B e^{4 i (c+d x)}-24 B e^{5 i (c+d x)}+10 B\right)}{60 d \left(1+e^{2 i (c+d x)}\right)^2 \sec ^{\frac{5}{2}}(c+d x) (A \cos (c+d x)+B)}","\frac{2 a^2 (5 A+7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{4 a^2 (5 A+4 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^2 (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)}{3 d}-\frac{4 a^2 (5 A+4 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 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{5 d}",1,"(a^2*(-1 + E^((2*I)*c))*Csc[c]*(5*A + 10*B - 30*A*E^(I*(c + d*x)) - 18*B*E^(I*(c + d*x)) - 60*A*E^((3*I)*(c + d*x)) - 54*B*E^((3*I)*(c + d*x)) - 5*A*E^((4*I)*(c + d*x)) - 10*B*E^((4*I)*(c + d*x)) - 30*A*E^((5*I)*(c + d*x)) - 24*B*E^((5*I)*(c + d*x)) - (10*I)*(2*A + B)*(1 + E^((2*I)*(c + d*x)))^2*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(5*A + 4*B)*E^(I*(c + d*x))*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/(60*d*E^(I*c)*(1 + E^((2*I)*(c + d*x)))^2*(B + A*Cos[c + d*x])*Sec[c + d*x]^(5/2))","C",1
188,1,295,160,3.2111986,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left(\frac{-3 \csc (c) \cos (d x) (A \cos (2 c)-A-4 B)+6 A \cos (c) \sin (d x)+2 B \tan (c+d x)}{4 d \sec ^{\frac{5}{2}}(c+d x)}-\frac{i \sqrt{2} \cos ^3(c+d x) \left(\left(-1+e^{2 i c}\right) (3 A+2 B) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 B \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+3 B \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)}{3 (A \cos (c+d x)+B)}","\frac{2 a^2 (3 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (3 A+2 B) \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 B \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 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}",1,"(a^2*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(((-I)*Sqrt[2]*Cos[c + d*x]^3*(3*B*Sqrt[1 + E^((2*I)*(c + d*x))] + 3*B*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] + (3*A + 2*B)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (-3*(-A - 4*B + A*Cos[2*c])*Cos[d*x]*Csc[c] + 6*A*Cos[c]*Sin[d*x] + 2*B*Tan[c + d*x])/(4*d*Sec[c + d*x]^(5/2))))/(3*(B + A*Cos[c + d*x]))","C",1
189,1,299,158,3.0106501,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{a^2 \sec ^4\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^2 (A+B \sec (c+d x)) \left(\frac{-3 (2 A-B) \csc (c) \cos (d x)-3 (2 A+B) \csc (c) \cos (2 c+d x)+A \sin (2 (c+d x))}{4 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{i \sqrt{2} \cos ^3(c+d x) \left(-\left(-1+e^{2 i c}\right) (2 A+3 B) e^{i (c+d x)} \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-e^{2 i (c+d x)}\right)+3 A \left(-1+e^{2 i c}\right) \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i (c+d x)}\right)+3 A \sqrt{1+e^{2 i (c+d x)}}\right)}{\left(-1+e^{2 i c}\right) d \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}}}\right)}{3 (A \cos (c+d x)+B)}","-\frac{2 a^2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{4 a^2 (2 A+3 B) \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 \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{3 d \sqrt{\sec (c+d x)}}+\frac{4 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)}{d}",1,"(a^2*Sec[(c + d*x)/2]^4*(1 + Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((I*Sqrt[2]*Cos[c + d*x]^3*(3*A*Sqrt[1 + E^((2*I)*(c + d*x))] + 3*A*(-1 + E^((2*I)*c))*Hypergeometric2F1[-1/4, 1/2, 3/4, -E^((2*I)*(c + d*x))] - (2*A + 3*B)*E^(I*(c + d*x))*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/4, 1/2, 5/4, -E^((2*I)*(c + d*x))]))/(d*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]) + (-3*(2*A - B)*Cos[d*x]*Csc[c] - 3*(2*A + B)*Cos[2*c + d*x]*Csc[c] + A*Sin[2*(c + d*x)])/(4*d*Sec[c + d*x]^(5/2))))/(3*(B + A*Cos[c + d*x]))","C",1
190,1,153,166,1.7971893,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{a^2 \sqrt{\sec (c+d x)} \left(-4 i (4 A+5 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (10 (2 A+B) \sin (c+d x)+3 A \sin (2 (c+d x))+48 i A+60 i B)+20 (A+2 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 a^2 (7 A+5 B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (A+2 B) \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 (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) \left(a^2 \sec (c+d x)+a^2\right)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(20*(A + 2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (4*I)*(4*A + 5*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((48*I)*A + (60*I)*B + 10*(2*A + B)*Sin[c + d*x] + 3*A*Sin[2*(c + d*x)])))/(15*d)","C",1
191,1,193,201,2.3823081,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i (3 A+4 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (5 (51 A+56 B) \sin (c+d x)+42 (2 A+B) \sin (2 (c+d x))+15 A \sin (3 (c+d x))+504 i A+672 i B)+40 (6 A+7 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a^2 (9 A+7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (6 A+7 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (6 A+7 B) \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) \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) \left(a^2 \sec (c+d x)+a^2\right)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(40*(6*A + 7*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(3*A + 4*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((504*I)*A + (672*I)*B + 5*(51*A + 56*B)*Sin[c + d*x] + 42*(2*A + B)*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
192,1,217,234,2.9973493,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{a^2 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-112 i (8 A+9 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (46 A+51 B) \sin (c+d x)+14 (37 A+36 B) \sin (2 (c+d x))+180 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+2688 i A+90 B \sin (3 (c+d x))+3024 i B)+240 (5 A+6 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{4 a^2 (8 A+9 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (11 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (5 A+6 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^2 (5 A+6 B) \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) \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 \sin (c+d x) \left(a^2 \sec (c+d x)+a^2\right)}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(240*(5*A + 6*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(8*A + 9*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((2688*I)*A + (3024*I)*B + 30*(46*A + 51*B)*Sin[c + d*x] + 14*(37*A + 36*B)*Sin[2*(c + d*x)] + 180*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(1260*d*E^(I*d*x))","C",1
193,1,793,277,6.9556267,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{7 A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{30 \sqrt{2} d (A \cos (c+d x)+B)}+\frac{17 B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \cos ^4(c+d x) \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{90 \sqrt{2} d (A \cos (c+d x)+B)}+\frac{13 A \sqrt{\cos (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{42 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B)}+\frac{11 B \sqrt{\cos (c+d x)} \sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x))}{42 d \sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B)}+\frac{\sec ^6\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x)) \left(\frac{(21 A+17 B) \csc (c) \cos (d x)}{30 d}+\frac{\sec (c) \sec ^3(c+d x) (9 A \sin (d x)+7 B \sin (c)+27 B \sin (d x))}{252 d}+\frac{\sec (c) \sec ^2(c+d x) (45 A \sin (c)+189 A \sin (d x)+135 B \sin (c)+238 B \sin (d x))}{1260 d}+\frac{\sec (c) \sec (c+d x) (189 A \sin (c)+390 A \sin (d x)+238 B \sin (c)+330 B \sin (d x))}{1260 d}+\frac{(13 A+11 B) \tan (c)}{42 d}+\frac{B \sec (c) \sin (d x) \sec ^4(c+d x)}{36 d}\right)}{\sec ^{\frac{7}{2}}(c+d x) (A \cos (c+d x)+B)}","\frac{4 a^3 (24 A+23 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{105 d}+\frac{4 a^3 (13 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 (9 A+13 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{63 d}+\frac{4 a^3 (21 A+17 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{4 a^3 (13 A+11 B) \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 (21 A+17 B) \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 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^2}{9 d}",1,"(7*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(30*Sqrt[2]*d*E^(I*d*x)*(B + A*Cos[c + d*x])) + (17*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c + d*x]^4*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(90*Sqrt[2]*d*E^(I*d*x)*(B + A*Cos[c + d*x])) + (13*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(42*d*(B + A*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + (11*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(42*d*(B + A*Cos[c + d*x])*Sec[c + d*x]^(7/2)) + (Sec[c/2 + (d*x)/2]^6*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*(((21*A + 17*B)*Cos[d*x]*Csc[c])/(30*d) + (B*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(36*d) + (Sec[c]*Sec[c + d*x]^3*(7*B*Sin[c] + 9*A*Sin[d*x] + 27*B*Sin[d*x]))/(252*d) + (Sec[c]*Sec[c + d*x]^2*(45*A*Sin[c] + 135*B*Sin[c] + 189*A*Sin[d*x] + 238*B*Sin[d*x]))/(1260*d) + (Sec[c]*Sec[c + d*x]*(189*A*Sin[c] + 238*B*Sin[c] + 390*A*Sin[d*x] + 330*B*Sin[d*x]))/(1260*d) + ((13*A + 11*B)*Tan[c])/(42*d)))/((B + A*Cos[c + d*x])*Sec[c + d*x]^(7/2))","C",0
194,1,465,244,5.2309367,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 \csc (c) e^{-i d x} \cos ^4(c+d x) \sec ^6\left(\frac{1}{2} (c+d x)\right) (\sec (c+d x)+1)^3 (A+B \sec (c+d x)) \left(7 \sqrt{2} \left(-1+e^{2 i c}\right) (9 A+7 B) e^{2 i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-\frac{\left(-1+e^{2 i c}\right) e^{-i (c-d x)} \sqrt{\sec (c+d x)} \left(10 i (21 A+13 B) \left(1+e^{2 i (c+d x)}\right)^3 \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 A \left(16 e^{i (c+d x)}-5 e^{2 i (c+d x)}+54 e^{3 i (c+d x)}+5 e^{4 i (c+d x)}+56 e^{5 i (c+d x)}+5 e^{6 i (c+d x)}+18 e^{7 i (c+d x)}-5\right)+2 B \left(84 e^{i (c+d x)}-95 e^{2 i (c+d x)}+441 e^{3 i (c+d x)}+95 e^{4 i (c+d x)}+504 e^{5 i (c+d x)}+65 e^{6 i (c+d x)}+147 e^{7 i (c+d x)}-65\right)\right)}{2 \left(1+e^{2 i (c+d x)}\right)^3}\right)}{420 d (A \cos (c+d x)+B)}","\frac{4 a^3 (42 A+41 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{105 d}+\frac{2 (7 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d}+\frac{4 a^3 (9 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{4 a^3 (21 A+13 B) \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 (9 A+7 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 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}{7 d}",1,"(a^3*Cos[c + d*x]^4*Csc[c]*Sec[(c + d*x)/2]^6*(7*Sqrt[2]*(9*A + 7*B)*E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] - ((-1 + E^((2*I)*c))*(21*A*(-5 + 16*E^(I*(c + d*x)) - 5*E^((2*I)*(c + d*x)) + 54*E^((3*I)*(c + d*x)) + 5*E^((4*I)*(c + d*x)) + 56*E^((5*I)*(c + d*x)) + 5*E^((6*I)*(c + d*x)) + 18*E^((7*I)*(c + d*x))) + 2*B*(-65 + 84*E^(I*(c + d*x)) - 95*E^((2*I)*(c + d*x)) + 441*E^((3*I)*(c + d*x)) + 95*E^((4*I)*(c + d*x)) + 504*E^((5*I)*(c + d*x)) + 65*E^((6*I)*(c + d*x)) + 147*E^((7*I)*(c + d*x))) + (10*I)*(21*A + 13*B)*(1 + E^((2*I)*(c + d*x)))^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2])*Sqrt[Sec[c + d*x]])/(2*E^(I*(c - d*x))*(1 + E^((2*I)*(c + d*x)))^3))*(1 + Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(420*d*E^(I*d*x)*(B + A*Cos[c + d*x]))","C",1
195,1,244,211,2.4096363,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 e^{-i d x} \sec ^{\frac{5}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(2 i (5 A+9 B) e^{-i (c+d x)} \left(1+e^{2 i (c+d x)}\right)^{5/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+40 (5 A+3 B) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+45 A \sin (c+d x)+10 A \sin (2 (c+d x))+45 A \sin (3 (c+d x))-90 i A \cos (c+d x)-30 i A \cos (3 (c+d x))+66 B \sin (c+d x)+30 B \sin (2 (c+d x))+54 B \sin (3 (c+d x))-162 i B \cos (c+d x)-54 i B \cos (3 (c+d x))\right)}{30 d}","\frac{4 a^3 (20 A+21 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (5 A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{15 d}+\frac{4 a^3 (5 A+3 B) \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 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 B \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}{5 d}",1,"(a^3*Sec[c + d*x]^(5/2)*(Cos[d*x] + I*Sin[d*x])*((-90*I)*A*Cos[c + d*x] - (162*I)*B*Cos[c + d*x] - (30*I)*A*Cos[3*(c + d*x)] - (54*I)*B*Cos[3*(c + d*x)] + 40*(5*A + 3*B)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + ((2*I)*(5*A + 9*B)*(1 + E^((2*I)*(c + d*x)))^(5/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 45*A*Sin[c + d*x] + 66*B*Sin[c + d*x] + 10*A*Sin[2*(c + d*x)] + 30*B*Sin[2*(c + d*x)] + 45*A*Sin[3*(c + d*x)] + 54*B*Sin[3*(c + d*x)]))/(30*d*E^(I*d*x))","C",1
196,1,202,199,2.0488282,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{a^3 e^{-i d x} \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(-4 i (A-B) \left(1+e^{2 i (c+d x)}\right)^{3/2} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+40 (A+B) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+A \sin (c+d x)+6 A \sin (2 (c+d x))+A \sin (3 (c+d x))+12 i A \cos (2 (c+d x))+12 i A+4 B \sin (c+d x)+18 B \sin (2 (c+d x))-12 i B \cos (2 (c+d x))-12 i B\right)}{6 d}","\frac{4 a^3 (A+4 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}-\frac{2 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)} \left(a^3 \sec (c+d x)+a^3\right)}{3 d}+\frac{20 a^3 (A+B) \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 (A-B) \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) (a \sec (c+d x)+a)^2}{3 d \sqrt{\sec (c+d x)}}",1,"(a^3*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*((12*I)*A - (12*I)*B + (12*I)*A*Cos[2*(c + d*x)] - (12*I)*B*Cos[2*(c + d*x)] + 40*(A + B)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] - (4*I)*(A - B)*(1 + E^((2*I)*(c + d*x)))^(3/2)*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + A*Sin[c + d*x] + 4*B*Sin[c + d*x] + 6*A*Sin[2*(c + d*x)] + 18*B*Sin[2*(c + d*x)] + A*Sin[3*(c + d*x)]))/(6*d*E^(I*d*x))","C",1
197,1,207,211,1.7744237,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-8 i (9 A+5 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+40 (3 A+5 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 A \sin (c+d x)+30 A \sin (2 (c+d x))+3 A \sin (3 (c+d x))+216 i A \cos (c+d x)+60 B \sin (c+d x)+10 B \sin (2 (c+d x))+120 i B \cos (c+d x)\right)}{30 d}","-\frac{4 a^3 (6 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 (9 A+5 B) \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 (3 A+5 B) \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) \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) (a \sec (c+d x)+a)^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*((216*I)*A*Cos[c + d*x] + (120*I)*B*Cos[c + d*x] + 40*(3*A + 5*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (8*I)*(9*A + 5*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 3*A*Sin[c + d*x] + 60*B*Sin[c + d*x] + 30*A*Sin[2*(c + d*x)] + 10*B*Sin[2*(c + d*x)] + 3*A*Sin[3*(c + d*x)]))/(30*d*E^(I*d*x))","C",1
198,1,194,211,2.6971801,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-56 i (7 A+9 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (5 (107 A+84 B) \sin (c+d x)+42 (3 A+B) \sin (2 (c+d x))+168 i (7 A+9 B)+15 A \sin (3 (c+d x)))+40 (13 A+21 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 (11 A+7 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{4 a^3 (41 A+42 B) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (13 A+21 B) \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) \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) (a \sec (c+d x)+a)^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(40*(13*A + 21*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (56*I)*(7*A + 9*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((168*I)*(7*A + 9*B) + 5*(107*A + 84*B)*Sin[c + d*x] + 42*(3*A + B)*Sin[2*(c + d*x)] + 15*A*Sin[3*(c + d*x)])))/(210*d*E^(I*d*x))","C",1
199,1,196,244,3.1694253,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{a^3 \sqrt{\sec (c+d x)} \left(-112 i (17 A+21 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (97 A+107 B) \sin (c+d x)+14 (73 A+54 B) \sin (2 (c+d x))+270 A \sin (3 (c+d x))+35 A \sin (4 (c+d x))+5712 i A+90 B \sin (3 (c+d x))+7056 i B)+240 (11 A+13 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{4 a^3 (23 A+24 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (13 A+9 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{4 a^3 (11 A+13 B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (11 A+13 B) \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) \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) (a \sec (c+d x)+a)^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(240*(11*A + 13*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (112*I)*(17*A + 21*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((5712*I)*A + (7056*I)*B + 30*(97*A + 107*B)*Sin[c + d*x] + 14*(73*A + 54*B)*Sin[2*(c + d*x)] + 270*A*Sin[3*(c + d*x)] + 90*B*Sin[3*(c + d*x)] + 35*A*Sin[4*(c + d*x)])))/(1260*d)","C",1
200,1,239,277,3.7374019,"\int \frac{(a+a \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]","\frac{a^3 e^{-i d x} \sqrt{\sec (c+d x)} (\cos (d x)+i \sin (d x)) \left(-2464 i (15 A+17 B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+\cos (c+d x) (30 (1953 A+2134 B) \sin (c+d x)+308 (75 A+73 B) \sin (2 (c+d x))+8505 A \sin (3 (c+d x))+2310 A \sin (4 (c+d x))+315 A \sin (5 (c+d x))+110880 i A+5940 B \sin (3 (c+d x))+770 B \sin (4 (c+d x))+125664 i B)+480 (105 A+121 B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{27720 d}","\frac{4 a^3 (15 A+17 B) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{20 a^3 (21 A+22 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 (15 A+11 B) \sin (c+d x) \left(a^3 \sec (c+d x)+a^3\right)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{4 a^3 (105 A+121 B) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{4 a^3 (105 A+121 B) \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) \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) (a \sec (c+d x)+a)^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(a^3*Sqrt[Sec[c + d*x]]*(Cos[d*x] + I*Sin[d*x])*(480*(105*A + 121*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] - (2464*I)*(15*A + 17*B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + Cos[c + d*x]*((110880*I)*A + (125664*I)*B + 30*(1953*A + 2134*B)*Sin[c + d*x] + 308*(75*A + 73*B)*Sin[2*(c + d*x)] + 8505*A*Sin[3*(c + d*x)] + 5940*B*Sin[3*(c + d*x)] + 2310*A*Sin[4*(c + d*x)] + 770*B*Sin[4*(c + d*x)] + 315*A*Sin[5*(c + d*x)])))/(27720*d*E^(I*d*x))","C",1
201,1,814,229,7.6978213,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","-\frac{A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{2} d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{7 B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 \sqrt{2} d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{5 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} (A+B \sec (c+d x)) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{5 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sqrt{\sec (c+d x)} (A+B \sec (c+d x)) \sin (c) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x)) \left(\frac{4 B \sec (c) \sin (d x) \sec ^2(c+d x)}{5 d}+\frac{4 \sec (c) (3 B \sin (c)+5 A \sin (d x)-5 B \sin (d x)) \sec (c+d x)}{15 d}+\frac{3 (7 B-5 A) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{(B-A) \sec \left(\frac{c}{2}\right) \sec (c) \left(5 \sin \left(\frac{3 c}{2}\right)-\sin \left(\frac{c}{2}\right)\right)}{3 d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)}","\frac{(A-B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{d (a \sec (c+d x)+a)}-\frac{(5 A-7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 a d}+\frac{5 (A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}-\frac{3 (5 A-7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 a d}+\frac{5 (A-B) \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 B) \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,"-((A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x]))/(Sqrt[2]*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x]))) + (7*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*(A + B*Sec[c + d*x]))/(5*Sqrt[2]*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (5*A*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - (5*B*Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x])*((3*(-5*A + 7*B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - ((-A + B)*Sec[c/2]*Sec[c]*(-Sin[c/2] + 5*Sin[(3*c)/2]))/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/d + (4*B*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(5*d) + (4*Sec[c]*Sec[c + d*x]*(3*B*Sin[c] + 5*A*Sin[d*x] - 5*B*Sin[d*x]))/(15*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x]))","C",1
202,1,372,192,3.5297793,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{e^{-\frac{1}{2} i (c+d x)} \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} (A+B \sec (c+d x)) \left(i \left(3 (A-B) e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(e^{i (c+d x)}+e^{2 i (c+d x)}+e^{3 i (c+d x)}+1\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-6 A e^{i (c+d x)}-12 A e^{2 i (c+d x)}-6 A e^{3 i (c+d x)}-9 A e^{4 i (c+d x)}-3 A+8 B e^{i (c+d x)}+10 B e^{2 i (c+d x)}+4 B e^{3 i (c+d x)}+9 B e^{4 i (c+d x)}+5 B\right)-(3 A-5 B) \left(e^{i (c+d x)}+e^{2 i (c+d x)}+e^{3 i (c+d x)}+1\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a d \left(1+e^{2 i (c+d x)}\right) (\sec (c+d x)+1) (A \cos (c+d x)+B)}","\frac{(A-B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{d (a \sec (c+d x)+a)}-\frac{(3 A-5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a d}+\frac{3 (A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}-\frac{(3 A-5 B) \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-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]*(-((3*A - 5*B)*(1 + E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]) + I*(-3*A + 5*B - 6*A*E^(I*(c + d*x)) + 8*B*E^(I*(c + d*x)) - 12*A*E^((2*I)*(c + d*x)) + 10*B*E^((2*I)*(c + d*x)) - 6*A*E^((3*I)*(c + d*x)) + 4*B*E^((3*I)*(c + d*x)) - 9*A*E^((4*I)*(c + d*x)) + 9*B*E^((4*I)*(c + d*x)) + 3*(A - B)*E^(I*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d*x))]*(1 + E^(I*(c + d*x)) + E^((2*I)*(c + d*x)) + E^((3*I)*(c + d*x)))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(3*a*d*E^((I/2)*(c + d*x))*(1 + E^((2*I)*(c + d*x)))*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))","C",1
203,1,420,153,4.8289506,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) (A+B \sec (c+d x)) \left(-6 \sqrt{\sec (c+d x)} \left(2 (B-A) \tan \left(\frac{1}{2} (c+d x)\right)+2 (A-3 B) \csc (c) \cos (d x)\right)-2 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+12 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-12 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 a d (\sec (c+d x)+1) (A \cos (c+d x)+B)}","\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d (a \sec (c+d x)+a)}-\frac{(A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d}+\frac{(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 d}+\frac{(A-3 B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]^2*(A + B*Sec[c + d*x])*((-2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (6*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 12*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 12*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 6*Sqrt[Sec[c + d*x]]*(2*(A - 3*B)*Cos[d*x]*Csc[c] + 2*(-A + B)*Tan[(c + d*x)/2])))/(6*a*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))","C",1
204,1,200,123,1.1751304,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{\left(-1+e^{2 i c}\right) e^{-\frac{1}{2} i (4 c+d x)} \left(\csc \left(\frac{c}{2}\right)+i \sec \left(\frac{c}{2}\right)\right) \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left((A-B) \left(e^{i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \left(1+e^{2 i (c+d x)}\right)\right)-3 i (A+B) \left(1+e^{i (c+d x)}\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{24 a d}","\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}+\frac{(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 d}-\frac{(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 d}",1,"((-1 + E^((2*I)*c))*((-3*I)*(A + B)*(1 + E^(I*(c + d*x)))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (A - B)*(-3*(1 + E^((2*I)*(c + d*x))) + E^(I*(c + d*x))*(1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))*(Csc[c/2] + I*Sec[c/2])*Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]])/(24*a*d*E^((I/2)*(4*c + d*x)))","C",1
205,1,445,128,2.7838149,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) (A+B \sec (c+d x)) \left(-\frac{6 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right) \left((2 A-B) \cos \left(\frac{1}{2} (c-d x)\right)+A \cos \left(\frac{1}{2} (3 c+d x)\right)\right)}{\sqrt{\sec (c+d x)}}-6 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-12 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+12 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{6 a d (\sec (c+d x)+1) (A \cos (c+d x)+B)}","-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d (a \sec (c+d x)+a)}-\frac{(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 d}+\frac{(3 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 d}",1,"(Cos[(c + d*x)/2]^2*((-6*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (2*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - (6*((2*A - B)*Cos[(c - d*x)/2] + A*Cos[(3*c + d*x)/2])*Csc[c/2]*Sec[c/2]*Sec[(c + d*x)/2])/Sqrt[Sec[c + d*x]] - 12*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 12*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(A + B*Sec[c + d*x]))/(6*a*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))","C",1
206,1,232,164,2.4248012,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) (\cos (d x)+i \sin (d x)) \left(3 i (A-B) e^{\frac{1}{2} i (c+d x)} \left(1+e^{i (c+d x)}\right) \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)+2 \cos (c+d x) \left(\sin \left(\frac{1}{2} (c+d x)\right) (2 A \cos (c+d x)+5 A-3 B)-9 i (A-B) \cos \left(\frac{1}{2} (c+d x)\right)\right)+2 (5 A-3 B) \cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a d (\sec (c+d x)+1)}","\frac{(5 A-3 B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{(A-B) \sin (c+d x)}{d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)}+\frac{(5 A-3 B) \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-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(3/2)*(Cos[d*x] + I*Sin[d*x])*(2*(5*A - 3*B)*Cos[(c + d*x)/2]*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (3*I)*(A - B)*E^((I/2)*(c + d*x))*(1 + E^(I*(c + d*x)))*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))] + 2*Cos[c + d*x]*((-9*I)*(A - B)*Cos[(c + d*x)/2] + (5*A - 3*B + 2*A*Cos[c + d*x])*Sin[(c + d*x)/2])))/(3*a*d*E^(I*d*x)*(1 + Sec[c + d*x]))","C",1
207,1,540,197,3.9778483,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) (A+B \sec (c+d x)) \left(\sqrt{\sec (c+d x)} \left(-40 (A-B) \sin (2 c) \cos (2 d x)+12 (33 A-20 B) \cos (c) \sin (d x)-40 (A-B) \cos (2 c) \sin (2 d x)+120 (A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)-3 \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x) ((33 A-20 B) \cos (2 c)+51 A-40 B)+120 (A-B) \tan \left(\frac{c}{2}\right)+12 A \sin (3 c) \cos (3 d x)+12 A \cos (3 c) \sin (3 d x)\right)-84 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-200 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+60 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+200 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{60 a d (\sec (c+d x)+1) (A \cos (c+d x)+B)}","-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)}+\frac{(7 A-5 B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (A-B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}-\frac{5 (A-B) \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) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(Cos[(c + d*x)/2]^2*(A + B*Sec[c + d*x])*((-84*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (60*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - 200*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + 200*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + Sqrt[Sec[c + d*x]]*(-3*(51*A - 40*B + (33*A - 20*B)*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2] - 40*(A - B)*Cos[2*d*x]*Sin[2*c] + 12*A*Cos[3*d*x]*Sin[3*c] + 120*(A - B)*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] + 12*(33*A - 20*B)*Cos[c]*Sin[d*x] - 40*(A - B)*Cos[2*c]*Sin[2*d*x] + 12*A*Cos[3*c]*Sin[3*d*x] + 120*(A - B)*Tan[c/2])))/(60*a*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))","C",1
208,1,568,230,4.1584725,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(7/2)*(a + a*Sec[c + d*x])),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) (A+B \sec (c+d x)) \left(\sqrt{\sec (c+d x)} \left(20 (27 A-14 B) \sin (2 c) \cos (2 d x)-84 (A-B) \sin (3 c) \cos (3 d x)-2772 (A-B) \cos (c) \sin (d x)+20 (27 A-14 B) \cos (2 c) \sin (2 d x)-84 (A-B) \cos (3 c) \sin (3 d x)-840 (A-B) \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{1}{2} (c+d x)\right)+63 (A-B) (11 \cos (2 c)+17) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \cos (d x)-840 (A-B) \tan \left(\frac{c}{2}\right)+30 A \sin (4 c) \cos (4 d x)+30 A \cos (4 c) \sin (4 d x)\right)+588 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+1800 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-588 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-1400 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{420 a d (\sec (c+d x)+1) (A \cos (c+d x)+B)}","-\frac{(A-B) \sin (c+d x)}{d \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)}-\frac{7 (A-B) \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{(9 A-7 B) \sin (c+d x)}{7 a d \sec ^{\frac{5}{2}}(c+d x)}+\frac{5 (9 A-7 B) \sin (c+d x)}{21 a d \sqrt{\sec (c+d x)}}+\frac{5 (9 A-7 B) \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{21 (A-B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}",1,"(Cos[(c + d*x)/2]^2*(A + B*Sec[c + d*x])*((588*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) - (588*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 1800*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 1400*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] + Sqrt[Sec[c + d*x]]*(63*(A - B)*(17 + 11*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2] + 20*(27*A - 14*B)*Cos[2*d*x]*Sin[2*c] - 84*(A - B)*Cos[3*d*x]*Sin[3*c] + 30*A*Cos[4*d*x]*Sin[4*c] - 840*(A - B)*Sec[c/2]*Sec[(c + d*x)/2]*Sin[(d*x)/2] - 2772*(A - B)*Cos[c]*Sin[d*x] + 20*(27*A - 14*B)*Cos[2*c]*Sin[2*d*x] - 84*(A - B)*Cos[3*c]*Sin[3*d*x] + 30*A*Cos[4*c]*Sin[4*d*x] - 840*(A - B)*Tan[c/2])))/(420*a*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x]))","C",1
209,1,865,237,8.03485,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{4 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{7 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{10 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{20 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(8 B \sin \left(\frac{d x}{2}\right)-5 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (7 B-4 A) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{8 B \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{4 (10 \cos (c) B+2 B-5 A \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}","\frac{(4 A-7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{5 (A-2 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d}+\frac{(4 A-7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}-\frac{5 (A-2 B) \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-7 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) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(4*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (7*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (10*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (20*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*((-2*(-4*A + 7*B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-5*A*Sin[(d*x)/2] + 8*B*Sin[(d*x)/2]))/(3*d) + (8*B*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (4*(2*B - 5*A*Cos[c] + 10*B*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",1
210,1,455,204,7.1338219,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{\cos ^4\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (A+B \sec (c+d x)) \left(-2 \sqrt{\sec (c+d x)} \left(6 (A-4 B) \csc (c) \cos (d x)-\tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) ((2 A-5 B) \cos (c+d x)+3 (A-2 B))\right)-2 \sqrt{2} A \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)+8 A \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+8 \sqrt{2} B \csc (c) e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \left(\left(-1+e^{2 i c}\right) e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right)-20 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 a^2 d (\sec (c+d x)+1)^2 (A \cos (c+d x)+B)}","\frac{(2 A-5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\sec (c+d x)+1)}-\frac{(A-4 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d}+\frac{(2 A-5 B) \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) \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) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]^4*Sec[c + d*x]*(A + B*Sec[c + d*x])*((-2*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + (8*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Csc[c]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))]))/E^(I*d*x) + 8*A*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 20*B*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]] - 2*Sqrt[Sec[c + d*x]]*(6*(A - 4*B)*Cos[d*x]*Csc[c] - (3*(A - 2*B) + (2*A - 5*B)*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])))/(3*a^2*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^2)","C",1
211,1,256,161,2.9467553,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(2 i \cos (c+d x) (-i (A-B) \sin (c+d x)+(A+5 B) \cos (c+d x)-A+7 B)+8 (A+2 B) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)-i B e^{-i (c+d x)} \sqrt{1+e^{2 i (c+d x)}} \left(1+e^{i (c+d x)}\right)^3 \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)\right)}{6 a^2 d (\sec (c+d x)+1)^2}","\frac{(A+2 B) \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 \sin (c+d x) \sqrt{\sec (c+d x)}}{a^2 d (\sec (c+d x)+1)}+\frac{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) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(((-I)*B*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) + 8*(A + 2*B)*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + (2*I)*Cos[c + d*x]*(-A + 7*B + (A + 5*B)*Cos[c + d*x] - I*(A - B)*Sin[c + d*x]))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
212,1,256,168,3.649014,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{e^{-i d x} \cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\cos \left(\frac{1}{2} (c+3 d x)\right)+i \sin \left(\frac{1}{2} (c+3 d x)\right)\right) \left(i \left(A e^{-i (c+d x)} \left(1+e^{i (c+d x)}\right)^3 \sqrt{1+e^{2 i (c+d x)}} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-2 \cos (c+d x) (-i (A-B) \sin (c+d x)+(7 A-B) \cos (c+d x)+5 A+B)\right)+8 (2 A+B) \sqrt{\cos (c+d x)} \cos ^3\left(\frac{1}{2} (c+d x)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \left(\cos \left(\frac{1}{2} (c+d x)\right)-i \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{6 a^2 d (\sec (c+d x)+1)^2}","\frac{(2 A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}+\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)}{3 a^2 d}-\frac{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-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*(8*(2*A + B)*Cos[(c + d*x)/2]^3*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*(Cos[(c + d*x)/2] - I*Sin[(c + d*x)/2]) + I*((A*(1 + E^(I*(c + d*x)))^3*Sqrt[1 + E^((2*I)*(c + d*x))]*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])/E^(I*(c + d*x)) - 2*Cos[c + d*x]*(5*A + B + (7*A - B)*Cos[c + d*x] - I*(A - B)*Sin[c + d*x])))*(Cos[(c + 3*d*x)/2] + I*Sin[(c + 3*d*x)/2]))/(6*a^2*d*E^(I*d*x)*(1 + Sec[c + d*x])^2)","C",1
213,1,854,177,6.8792732,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{4 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{10 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{4 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(4 B \sin \left(\frac{d x}{2}\right)-7 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (\cos (2 c) A+3 A-B) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{8 A \cos (c) \sin (d x)}{d}-\frac{4 (4 B-7 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}","-\frac{(5 A-2 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d (\sec (c+d x)+1)}-\frac{(5 A-2 B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d (a \sec (c+d x)+a)^2}",1,"(-4*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (10*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (4*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*((-2*(3*A - B + A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-7*A*Sin[(d*x)/2] + 4*B*Sin[(d*x)/2]))/(3*d) + (8*A*Cos[c]*Sin[d*x])/d - (4*(-7*A + 4*B)*Tan[c/2])/(3*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",1
214,1,899,211,6.9118693,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{7 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{4 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{20 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{10 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(7 B \sin \left(\frac{d x}{2}\right)-10 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (-2 \cos (2 c) A-5 A+3 B+B \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{d}+\frac{4 A \cos (2 d x) \sin (2 c)}{3 d}+\frac{8 (B-2 A) \cos (c) \sin (d x)}{d}+\frac{4 A \cos (2 c) \sin (2 d x)}{3 d}+\frac{4 (7 B-10 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}","\frac{5 (2 A-B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{(7 A-4 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)} (\sec (c+d x)+1)}+\frac{5 (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)}{3 a^2 d}-\frac{(7 A-4 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) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}",1,"(7*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (4*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (20*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (10*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*((-2*(-5*A + 3*B - 2*A*Cos[2*c] + B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/d + (4*A*Cos[2*d*x]*Sin[2*c])/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-10*A*Sin[(d*x)/2] + 7*B*Sin[(d*x)/2]))/(3*d) + (8*(-2*A + B)*Cos[c]*Sin[d*x])/d + (4*A*Cos[2*c]*Sin[2*d*x])/(3*d) + (4*(-10*A + 7*B)*Tan[c/2])/(3*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",1
215,1,946,244,7.1538184,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{56 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{7 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{10 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{20 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(10 B \sin \left(\frac{d x}{2}\right)-13 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{(-73 \cos (2 c) A-151 A+100 B+40 B \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{10 d}+\frac{4 (B-2 A) \cos (2 d x) \sin (2 c)}{3 d}+\frac{2 A \cos (3 d x) \sin (3 c)}{5 d}-\frac{2 (40 B-73 A) \cos (c) \sin (d x)}{5 d}+\frac{4 (B-2 A) \cos (2 c) \sin (2 d x)}{3 d}+\frac{2 A \cos (3 c) \sin (3 d x)}{5 d}-\frac{4 (10 B-13 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}","-\frac{(3 A-2 B) \sin (c+d x)}{a^2 d \sec ^{\frac{3}{2}}(c+d x) (\sec (c+d x)+1)}+\frac{7 (8 A-5 B) \sin (c+d x)}{15 a^2 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{5 (3 A-2 B) \sin (c+d x)}{3 a^2 d \sqrt{\sec (c+d x)}}-\frac{5 (3 A-2 B) \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 (8 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 a^2 d}-\frac{(A-B) \sin (c+d x)}{3 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}",1,"(-56*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (7*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x]))/(3*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (10*A*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (20*B*Cos[c/2 + (d*x)/2]^4*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*(((-151*A + 100*B - 73*A*Cos[2*c] + 40*B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(10*d) + (4*(-2*A + B)*Cos[2*d*x]*Sin[2*c])/(3*d) + (2*A*Cos[3*d*x]*Sin[3*c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-13*A*Sin[(d*x)/2] + 10*B*Sin[(d*x)/2]))/(3*d) - (2*(-73*A + 40*B)*Cos[c]*Sin[d*x])/(5*d) + (4*(-2*A + B)*Cos[2*c]*Sin[2*d*x])/(3*d) + (2*A*Cos[3*c]*Sin[3*d*x])/(5*d) - (4*(-13*A + 10*B)*Tan[c/2])/(3*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",1
216,1,953,292,8.49765,"\int \frac{\sec ^{\frac{9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(9/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{49 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{119 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{26 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{22 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(13 B \sin \left(\frac{d x}{2}\right)-8 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (13 B-8 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(29 B \sin \left(\frac{d x}{2}\right)-13 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{14 (17 B-7 A) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{16 B \sec (c) \sec (c+d x) \sin (d x)}{3 d}+\frac{4 (33 \cos (c) B+4 B-13 A \cos (c)) \sec (c) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{7 (7 A-17 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{30 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(13 A-33 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 a^3 d}+\frac{7 (7 A-17 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}-\frac{(13 A-33 B) \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 (7 A-17 B) \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) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(A-2 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 a d (a \sec (c+d x)+a)^2}",1,"(49*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (119*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (26*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (22*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*((-14*(-7*A + 17*B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-8*A*Sin[(d*x)/2] + 13*B*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-13*A*Sin[(d*x)/2] + 29*B*Sin[(d*x)/2]))/(3*d) + (16*B*Sec[c]*Sec[c + d*x]*Sin[d*x])/(3*d) + (4*(4*B - 13*A*Cos[c] + 33*B*Cos[c])*Sec[c]*Tan[c/2])/(3*d) + (4*(-8*A + 13*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",1
217,1,924,261,7.5182913,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\frac{3 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{49 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{26 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(8 B \sin \left(\frac{d x}{2}\right)-3 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (8 B-3 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(13 B \sin \left(\frac{d x}{2}\right)-3 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (49 B-9 A) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}-\frac{4 (13 B-3 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(3 A-13 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{6 d \left(a^3 \sec (c+d x)+a^3\right)}-\frac{(9 A-49 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 a^3 d}+\frac{(3 A-13 B) \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) \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) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(3 A-8 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"(-3*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(5*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (49*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (26*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*((2*(-9*A + 49*B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-3*A*Sin[(d*x)/2] + 8*B*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-3*A*Sin[(d*x)/2] + 13*B*Sin[(d*x)/2]))/(3*d) - (4*(-3*A + 13*B)*Tan[c/2])/(3*d) - (4*(-3*A + 8*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",1
218,1,919,220,7.1057804,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{3 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(2 A \sin \left(\frac{d x}{2}\right)+3 B \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (2 A+3 B) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+3 B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (A+9 B) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{4 (A+3 B) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","-\frac{(A+9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{10 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+3 B) \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) \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) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}+\frac{(A-6 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 a d (a \sec (c+d x)+a)^2}",1,"-1/15*(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (3*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(5*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*((-2*(A + 9*B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 3*B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(2*A*Sin[(d*x)/2] + 3*B*Sin[(d*x)/2]))/(15*d) + (4*(A + 3*B)*Tan[c/2])/(3*d) + (4*(2*A + 3*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",1
219,1,918,216,6.9414903,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{\sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(2 B \sin \left(\frac{d x}{2}\right)-7 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (2 B-7 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (B-A) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{4 (A+B) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(A+B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 d \left(a^3 \sec (c+d x)+a^3\right)}+\frac{(A+B) \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) \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) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{5 d (a \sec (c+d x)+a)^3}-\frac{(A+4 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}",1,"(Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*((-2*(-A + B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-7*A*Sin[(d*x)/2] + 2*B*Sin[(d*x)/2]))/(15*d) + (4*(A + B)*Tan[c/2])/(3*d) + (4*(-7*A + 2*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",1
220,1,919,222,7.0970051,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{3 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(7 B \sin \left(\frac{d x}{2}\right)-12 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (7 B-12 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-9 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (9 A+B) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{4 (B-9 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(3 A+B) \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) \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) \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{(3 A+2 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(3*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(5*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*((2*(9*A + B)*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-9*A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-12*A*Sin[(d*x)/2] + 7*B*Sin[(d*x)/2]))/(15*d) + (4*(-9*A + B)*Tan[c/2])/(3*d) - (4*(-12*A + 7*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
221,1,943,228,7.1569931,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\frac{49 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{3 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{26 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{2 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(12 B \sin \left(\frac{d x}{2}\right)-17 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (12 B-17 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(9 B \sin \left(\frac{d x}{2}\right)-23 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (10 \cos (2 c) A+39 A-9 B) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{16 A \cos (c) \sin (d x)}{d}-\frac{4 (9 B-23 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","-\frac{(13 A-3 B) \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) \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) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 a d (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d (a \sec (c+d x)+a)^3}",1,"(-49*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (3*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(5*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (26*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (2*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*((-2*(39*A - 9*B + 10*A*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-23*A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-17*A*Sin[(d*x)/2] + 12*B*Sin[(d*x)/2]))/(15*d) + (16*A*Cos[c]*Sin[d*x])/d - (4*(-23*A + 9*B)*Tan[c/2])/(3*d) + (4*(-17*A + 12*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",1
222,1,988,261,7.2756781,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","\frac{119 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{49 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{22 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{26 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(17 B \sin \left(\frac{d x}{2}\right)-22 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (17 B-22 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(23 B \sin \left(\frac{d x}{2}\right)-43 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (-30 \cos (2 c) A-89 A+39 B+10 B \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{8 A \cos (2 d x) \sin (2 c)}{3 d}+\frac{16 (B-3 A) \cos (c) \sin (d x)}{d}+\frac{8 A \cos (2 c) \sin (2 d x)}{3 d}+\frac{4 (23 B-43 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","\frac{(33 A-13 B) \sin (c+d x)}{6 a^3 d \sqrt{\sec (c+d x)}}-\frac{7 (17 A-7 B) \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) \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 (17 A-7 B) \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 A-B) \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^3}",1,"(119*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (49*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (22*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (26*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(3*d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*((-2*(-89*A + 39*B - 30*A*Cos[2*c] + 10*B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (8*A*Cos[2*d*x]*Sin[2*c])/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-22*A*Sin[(d*x)/2] + 17*B*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-43*A*Sin[(d*x)/2] + 23*B*Sin[(d*x)/2]))/(3*d) + (16*(-3*A + B)*Cos[c]*Sin[d*x])/d + (8*A*Cos[2*c]*Sin[2*d*x])/(3*d) + (4*(-43*A + 23*B)*Tan[c/2])/(3*d) - (4*(-22*A + 17*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
223,1,1032,294,7.5856846,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{77 \sqrt{2} A e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{119 \sqrt{2} B e^{-i d x} \sqrt{\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{1+e^{2 i (c+d x)}} \csc \left(\frac{c}{2}\right) \left(e^{2 i d x} \left(-1+e^{2 i c}\right) \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (c+d x)}\right)-3 \sqrt{1+e^{2 i (c+d x)}}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{42 A \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{22 B \sqrt{\cos (c+d x)} \csc \left(\frac{c}{2}\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right) \sec \left(\frac{c}{2}\right) \sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \sin (c) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(22 B \sin \left(\frac{d x}{2}\right)-27 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (22 B-27 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(43 B \sin \left(\frac{d x}{2}\right)-69 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{(-133 \cos (2 c) A-329 A+178 B+60 B \cos (2 c)) \cos (d x) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right)}{5 d}+\frac{8 (B-3 A) \cos (2 d x) \sin (2 c)}{3 d}+\frac{4 A \cos (3 d x) \sin (3 c)}{5 d}-\frac{4 (60 B-133 A) \cos (c) \sin (d x)}{5 d}+\frac{8 (B-3 A) \cos (2 c) \sin (2 d x)}{3 d}+\frac{4 A \cos (3 c) \sin (3 d x)}{5 d}-\frac{4 (43 B-69 A) \tan \left(\frac{c}{2}\right)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}","-\frac{3 (21 A-11 B) \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) \sin (c+d x)}{30 a^3 d \sec ^{\frac{3}{2}}(c+d x)}-\frac{(21 A-11 B) \sin (c+d x)}{2 a^3 d \sqrt{\sec (c+d x)}}-\frac{(21 A-11 B) \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{7 (33 A-17 B) \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) \sin (c+d x)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^3}",1,"(-77*Sqrt[2]*A*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(5*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (119*Sqrt[2]*B*Sqrt[E^(I*(c + d*x))/(1 + E^((2*I)*(c + d*x)))]*Sqrt[1 + E^((2*I)*(c + d*x))]*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*(-3*Sqrt[1 + E^((2*I)*(c + d*x))] + E^((2*I)*d*x)*(-1 + E^((2*I)*c))*Hypergeometric2F1[1/2, 3/4, 7/4, -E^((2*I)*(c + d*x))])*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(15*d*E^(I*d*x)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (42*A*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (22*B*Cos[c/2 + (d*x)/2]^6*Sqrt[Cos[c + d*x]]*Csc[c/2]*EllipticF[(c + d*x)/2, 2]*Sec[c/2]*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*Sin[c])/(d*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*(((-329*A + 178*B - 133*A*Cos[2*c] + 60*B*Cos[2*c])*Cos[d*x]*Csc[c/2]*Sec[c/2])/(5*d) + (8*(-3*A + B)*Cos[2*d*x]*Sin[2*c])/(3*d) + (4*A*Cos[3*d*x]*Sin[3*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-27*A*Sin[(d*x)/2] + 22*B*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-69*A*Sin[(d*x)/2] + 43*B*Sin[(d*x)/2]))/(3*d) - (4*(-133*A + 60*B)*Cos[c]*Sin[d*x])/(5*d) + (8*(-3*A + B)*Cos[2*c]*Sin[2*d*x])/(3*d) + (4*A*Cos[3*c]*Sin[3*d*x])/(5*d) - (4*(-69*A + 43*B)*Tan[c/2])/(3*d) + (4*(-27*A + 22*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
224,1,131,176,1.5534436,"\int \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{a (\sec (c+d x)+1)} \left(\tan \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (4 (6 A+5 B) \cos (c+d x)+3 (6 A+5 B) \cos (2 (c+d x))+18 A+31 B)+3 \sqrt{2} (6 A+5 B) \sec \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \sqrt{\sec (c+d x)}}","\frac{a (6 A+5 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{a (6 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{3 d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(6*A + 5*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Sec[(c + d*x)/2] + (18*A + 31*B + 4*(6*A + 5*B)*Cos[c + d*x] + 3*(6*A + 5*B)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Tan[(c + d*x)/2]))/(48*d*Sqrt[Sec[c + d*x]])","A",1
225,1,106,131,0.5230249,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (4 A+3 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 A+2 B \sec (c+d x)+3 B)\right)}{8 d \sqrt{\sec (c+d x)}}","\frac{a (4 A+3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (4 A+3 B) \sinh ^{-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) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(4*A + 3*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*A + 3*B + 2*B*Sec[c + d*x])*Sin[(c + d*x)/2]))/(8*d*Sqrt[Sec[c + d*x]])","A",1
226,1,89,78,0.286456,"\int \sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (2 A+B) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 d}","\frac{\sqrt{a} (2 A+B) \sinh ^{-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) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(2*A + B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] + 2*B*Sin[(c + d*x)/2]))/(2*d)","A",1
227,1,83,76,0.4707194,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 a \left(A \sin (c+d x) \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-B \tan (c+d x) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} B \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}",1,"(2*a*(A*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sin[c + d*x] - B*ArcSin[Sqrt[Sec[c + d*x]]]*Tan[c + d*x]))/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
228,1,56,82,0.2372894,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{2 \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (A \cos (c+d x)+2 A+3 B)}{3 d \sqrt{\sec (c+d x)}}","\frac{2 a (A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}",1,"(2*(2*A + 3*B + A*Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3*d*Sqrt[Sec[c + d*x]])","A",1
229,1,71,130,0.3198524,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{a \sin (c+d x) \sqrt{\sec (c+d x)} (2 (4 A+5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+19 A+20 B)}{15 d \sqrt{a (\sec (c+d x)+1)}}","\frac{4 a (4 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (4 A+5 B) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(a*(19*A + 20*B + 2*(4*A + 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
230,1,91,175,0.3536892,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{2 a \sin (c+d x) \left(8 (6 A+7 B) \sec ^3(c+d x)+4 (6 A+7 B) \sec ^2(c+d x)+3 (6 A+7 B) \sec (c+d x)+15 A\right)}{105 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a (6 A+7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a (6 A+7 B) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(2*a*(15*A + 3*(6*A + 7*B)*Sec[c + d*x] + 4*(6*A + 7*B)*Sec[c + d*x]^2 + 8*(6*A + 7*B)*Sec[c + d*x]^3)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
231,1,153,227,1.503651,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(6 \sqrt{2} (88 A+75 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) ((1048 A+1155 B) \cos (c+d x)+4 (88 A+75 B) \cos (2 (c+d x))+264 A \cos (3 (c+d x))+352 A+225 B \cos (3 (c+d x))+492 B)\right)}{768 d \sqrt{\sec (c+d x)}}","\frac{a^{3/2} (88 A+75 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^2 (8 A+9 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(88*A + 75*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (352*A + 492*B + (1048*A + 1155*B)*Cos[c + d*x] + 4*(88*A + 75*B)*Cos[2*(c + d*x)] + 264*A*Cos[3*(c + d*x)] + 225*B*Cos[3*(c + d*x)])*Sec[c + d*x]^4*Sin[(c + d*x)/2]))/(768*d*Sqrt[Sec[c + d*x]])","A",1
232,1,134,180,1.5480267,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (14 A+11 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (4 (6 A+11 B) \cos (c+d x)+(42 A+33 B) \cos (2 (c+d x))+7 (6 A+7 B))\right)}{48 d \sqrt{\sec (c+d x)}}","\frac{a^{3/2} (14 A+11 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^2 (6 A+7 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{12 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (14 A+11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{8 d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(14*A + 11*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (7*(6*A + 7*B) + 4*(6*A + 11*B)*Cos[c + d*x] + (42*A + 33*B)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]))/(48*d*Sqrt[Sec[c + d*x]])","A",1
233,1,107,133,0.7308342,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (12 A+7 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 A+2 B \sec (c+d x)+7 B)\right)}{8 d \sqrt{\sec (c+d x)}}","\frac{a^{3/2} (12 A+7 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{2 d}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(12*A + 7*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*A + 7*B + 2*B*Sec[c + d*x])*Sin[(c + d*x)/2]))/(8*d*Sqrt[Sec[c + d*x]])","A",1
234,1,107,124,1.7370069,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a^2 \tan (c+d x) \left(\sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)} (2 A \cos (c+d x)+B)+2 A \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-3 B \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{a^{3/2} (2 A+3 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^2 (2 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{d}",1,"(a^2*(2*A*ArcSin[Sqrt[1 - Sec[c + d*x]]] - 3*B*ArcSin[Sqrt[Sec[c + d*x]]] + (B + 2*A*Cos[c + d*x])*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
235,1,109,125,0.6562809,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{2 a^2 \tan (c+d x) \left(\sqrt{1-\sec (c+d x)} (A \cos (c+d x)+5 A+3 B)+3 B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{3/2} B \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^2 (4 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\sec (c+d x)}}",1,"(2*a^2*((5*A + 3*B + A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] + 3*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Tan[c + d*x])/(3*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
236,1,73,131,0.5063765,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{a^2 \sin (c+d x) \sqrt{\sec (c+d x)} (2 (9 A+5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+39 A+50 B)}{15 d \sqrt{a (\sec (c+d x)+1)}}","\frac{8 a^2 (3 A+5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^2*(39*A + 50*B + 2*(9*A + 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*Sin[c + d*x])/(15*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
237,1,92,181,0.4943533,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{2 a^2 \sin (c+d x) \left(2 (52 A+63 B) \sec ^3(c+d x)+(52 A+63 B) \sec ^2(c+d x)+3 (13 A+7 B) \sec (c+d x)+15 A\right)}{105 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^2 (8 A+7 B) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (52 A+63 B) \sin (c+d x)}{105 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}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*a^2*(15*A + 3*(13*A + 7*B)*Sec[c + d*x] + (52*A + 63*B)*Sec[c + d*x]^2 + 2*(52*A + 63*B)*Sec[c + d*x]^3)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
238,1,110,228,0.6722177,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{2 a^2 \sin (c+d x) \left(8 (34 A+39 B) \sec ^4(c+d x)+4 (34 A+39 B) \sec ^3(c+d x)+3 (34 A+39 B) \sec ^2(c+d x)+5 (17 A+9 B) \sec (c+d x)+35 A\right)}{315 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^2 (34 A+39 B) \sin (c+d x)}{105 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (10 A+9 B) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \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}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(2*a^2*(35*A + 5*(17*A + 9*B)*Sec[c + d*x] + 3*(34*A + 39*B)*Sec[c + d*x]^2 + 4*(34*A + 39*B)*Sec[c + d*x]^3 + 8*(34*A + 39*B)*Sec[c + d*x]^4)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
239,1,178,274,2.2774806,"\int \sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(60 \sqrt{2} (326 A+283 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \sec ^5(c+d x) (36 (650 A+781 B) \cos (c+d x)+4 (6730 A+6509 B) \cos (2 (c+d x))+6520 A \cos (3 (c+d x))+4890 A \cos (4 (c+d x))+22030 A+5660 B \cos (3 (c+d x))+4245 B \cos (4 (c+d x))+24863 B)\right)}{15360 d \sqrt{\sec (c+d x)}}","\frac{a^{5/2} (326 A+283 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{128 d}+\frac{a^3 (170 A+157 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{240 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{192 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{128 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{40 d}+\frac{a B \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(326*A + 283*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (22030*A + 24863*B + 36*(650*A + 781*B)*Cos[c + d*x] + 4*(6730*A + 6509*B)*Cos[2*(c + d*x)] + 6520*A*Cos[3*(c + d*x)] + 5660*B*Cos[3*(c + d*x)] + 4890*A*Cos[4*(c + d*x)] + 4245*B*Cos[4*(c + d*x)])*Sec[c + d*x]^5*Sin[(c + d*x)/2]))/(15360*d*Sqrt[Sec[c + d*x]])","A",1
240,1,154,227,1.5949313,"\int \sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(6 \sqrt{2} (200 A+163 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \sec ^4(c+d x) ((2056 A+2203 B) \cos (c+d x)+(544 A+652 B) \cos (2 (c+d x))+600 A \cos (3 (c+d x))+544 A+489 B \cos (3 (c+d x))+844 B)\right)}{768 d \sqrt{\sec (c+d x)}}","\frac{a^{5/2} (200 A+163 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{64 d}+\frac{a^3 (104 A+95 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{96 d \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (200 A+163 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{64 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{24 d}+\frac{a B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(200*A + 163*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (544*A + 844*B + (2056*A + 2203*B)*Cos[c + d*x] + (544*A + 652*B)*Cos[2*(c + d*x)] + 600*A*Cos[3*(c + d*x)] + 489*B*Cos[3*(c + d*x)])*Sec[c + d*x]^4*Sin[(c + d*x)/2]))/(768*d*Sqrt[Sec[c + d*x]])","A",1
241,1,133,180,1.4487411,"\int \sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(3 \sqrt{2} (38 A+25 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\sin \left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) (4 (6 A+17 B) \cos (c+d x)+(66 A+75 B) \cos (2 (c+d x))+66 A+91 B)\right)}{48 d \sqrt{\sec (c+d x)}}","\frac{a^{5/2} (38 A+25 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{24 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(38*A + 25*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + (66*A + 91*B + 4*(6*A + 17*B)*Cos[c + d*x] + (66*A + 75*B)*Cos[2*(c + d*x)])*Sec[c + d*x]^3*Sin[(c + d*x)/2]))/(48*d*Sqrt[Sec[c + d*x]])","A",1
242,1,137,180,2.2664005,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{a^3 \left(\sqrt{-((\sec (c+d x)-1) \sec (c+d x))} (\tan (c+d x) (4 A+2 B \sec (c+d x)+11 B)+8 A \sin (c+d x))+20 A \tan (c+d x) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-19 B \tan (c+d x) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{a^{5/2} (20 A+19 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 d}+\frac{a^3 (4 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (4 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{4 d}+\frac{a B \sin (c+d x) \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}{2 d}",1,"(a^3*(20*A*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 19*B*ArcSin[Sqrt[Sec[c + d*x]]]*Tan[c + d*x] + Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*(8*A*Sin[c + d*x] + (4*A + 11*B + 2*B*Sec[c + d*x])*Tan[c + d*x])))/(4*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
243,1,133,177,1.0519748,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{a^3 \left(3 (2 A+5 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+\sqrt{1-\sec (c+d x)} (\tan (c+d x) (16 A+3 B \sec (c+d x)+6 B)+2 A \sin (c+d x))\right)}{3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\sec (c+d x)+1)}}","\frac{a^{5/2} (2 A+5 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{a^3 (14 A+3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (2 A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"(a^3*(3*(2*A + 5*B)*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + Sqrt[1 - Sec[c + d*x]]*(2*A*Sin[c + d*x] + (16*A + 6*B + 3*B*Sec[c + d*x])*Tan[c + d*x])))/(3*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
244,1,127,172,1.9567881,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{a^3 \tan (c+d x) \left(\sqrt{1-\sec (c+d x)} (2 (14 A+5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+89 A+80 B)+30 B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{15 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{5/2} B \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{d}+\frac{2 a^3 (32 A+35 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+5 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(a^3*((89*A + 80*B + 2*(14*A + 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[1 - Sec[c + d*x]] + 30*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Tan[c + d*x])/(15*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
245,1,91,178,0.5693458,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{2 a^3 \sin (c+d x) \left((230 A+301 B) \sec ^3(c+d x)+(115 A+98 B) \sec ^2(c+d x)+3 (20 A+7 B) \sec (c+d x)+15 A\right)}{105 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{64 a^3 (5 A+7 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (5 A+7 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{105 d \sqrt{\sec (c+d x)}}+\frac{2 a (5 A+7 B) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 A \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(2*a^3*(15*A + 3*(20*A + 7*B)*Sec[c + d*x] + (115*A + 98*B)*Sec[c + d*x]^2 + (230*A + 301*B)*Sec[c + d*x]^3)*Sin[c + d*x])/(105*d*Sec[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
246,1,108,228,0.7080668,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{2 a^3 \sin (c+d x) \left((584 A+690 B) \sec ^4(c+d x)+(292 A+345 B) \sec ^3(c+d x)+3 (73 A+60 B) \sec ^2(c+d x)+5 (26 A+9 B) \sec (c+d x)+35 A\right)}{315 d \sec ^{\frac{7}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^3 (124 A+135 B) \sin (c+d x)}{315 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (292 A+345 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (292 A+345 B) \sin (c+d x)}{315 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (4 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 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^3*(35*A + 5*(26*A + 9*B)*Sec[c + d*x] + 3*(73*A + 60*B)*Sec[c + d*x]^2 + (292*A + 345*B)*Sec[c + d*x]^3 + (584*A + 690*B)*Sec[c + d*x]^4)*Sin[c + d*x])/(315*d*Sec[c + d*x]^(7/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
247,1,127,275,4.2504546,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]","\frac{2 a^3 \sin (c+d x) \left(8 (710 A+803 B) \sec ^5(c+d x)+4 (710 A+803 B) \sec ^4(c+d x)+3 (710 A+803 B) \sec ^3(c+d x)+5 (355 A+286 B) \sec ^2(c+d x)+35 (32 A+11 B) \sec (c+d x)+315 A\right)}{3465 d \sec ^{\frac{9}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^3 (710 A+803 B) \sin (c+d x)}{1155 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (194 A+209 B) \sin (c+d x)}{693 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{16 a^3 (710 A+803 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^3 (710 A+803 B) \sin (c+d x)}{3465 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (14 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 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^3*(315*A + 35*(32*A + 11*B)*Sec[c + d*x] + 5*(355*A + 286*B)*Sec[c + d*x]^2 + 3*(710*A + 803*B)*Sec[c + d*x]^3 + 4*(710*A + 803*B)*Sec[c + d*x]^4 + 8*(710*A + 803*B)*Sec[c + d*x]^5)*Sin[c + d*x])/(3465*d*Sec[c + d*x]^(9/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
248,1,125,190,0.8887731,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(8 (A-B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} (4 A-7 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 A+2 B \sec (c+d x)-B)\right)}{4 d \sqrt{a (\sec (c+d x)+1)}}","\frac{(4 A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (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{(4 A-7 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{4 \sqrt{a} d}+\frac{B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(8*(A - B)*ArcTanh[Sin[(c + d*x)/2]] - Sqrt[2]*(4*A - 7*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*A - B + 2*B*Sec[c + d*x])*Sin[(c + d*x)/2]))/(4*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
249,1,106,141,0.4217231,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left(-2 (A-B) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+\sqrt{2} (2 A-B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{d \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (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) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}+\frac{B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*(-2*(A - B)*ArcTanh[Sin[(c + d*x)/2]] + Sqrt[2]*(2*A - B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*B*Sec[c + d*x]*Sin[(c + d*x)/2]))/(d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
250,1,95,100,0.1980314,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\tan (c+d x) \left(\sqrt{2} (B-A) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-2 B \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (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 B \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{\sqrt{a} d}",1,"((-2*B*ArcSin[Sqrt[Sec[c + d*x]]] + Sqrt[2]*(-A + B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]])*Tan[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
251,1,114,99,0.2720199,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\tan (c+d x) \left(\sqrt{2} (A-B) \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 A \sqrt{1-\sec (c+d x)}\right)}{d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 A \sin (c+d x) \sqrt{\sec (c+d x)}}{d \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (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}",1,"((2*A*Sqrt[1 - Sec[c + d*x]] + Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Tan[c + d*x])/(d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
252,1,132,142,0.3890582,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\tan (c+d x) \left(2 \sqrt{1-\sec (c+d x)} (A \cos (c+d x)-A+3 B)-3 \sqrt{2} (A-B) \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{3 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 (A-3 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x)}{3 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"((2*(-A + 3*B + A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] - 3*Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Tan[c + d*x])/(3*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
253,1,133,187,1.1918863,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\frac{15 \sqrt{2} (A-B) \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{\sqrt{1-\sec (c+d x)}}+\sin (c+d x) \sqrt{\sec (c+d x)} (-2 (A-5 B) \cos (c+d x)+3 A \cos (2 (c+d x))+29 A-10 B)}{15 d \sqrt{a (\sec (c+d x)+1)}}","\frac{2 (13 A-5 B) \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)}{15 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (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 \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"((29*A - 10*B - 2*(A - 5*B)*Cos[c + d*x] + 3*A*Cos[2*(c + d*x)])*Sqrt[Sec[c + d*x]]*Sin[c + d*x] + (15*Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x])/Sqrt[1 - Sec[c + d*x]])/(15*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
254,1,152,230,1.6404856,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{-\frac{2 \sin (c+d x) \left((43 A-91 B) \sec ^3(c+d x)+(7 B-31 A) \sec ^2(c+d x)+3 (A-7 B) \sec (c+d x)-15 A\right)}{\sec ^{\frac{5}{2}}(c+d x)}-\frac{105 \sqrt{2} (A-B) \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{\sqrt{1-\sec (c+d x)}}}{105 d \sqrt{a (\sec (c+d x)+1)}}","-\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 (43 A-91 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (31 A-7 B) \sin (c+d x)}{105 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"((-2*(-15*A + 3*(A - 7*B)*Sec[c + d*x] + (-31*A + 7*B)*Sec[c + d*x]^2 + (43*A - 91*B)*Sec[c + d*x]^3)*Sin[c + d*x])/Sec[c + d*x]^(5/2) - (105*Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x])/Sqrt[1 - Sec[c + d*x]])/(105*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
255,1,497,247,4.6356893,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{4 (6 A-7 B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)+8 (9 A-13 B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)+4 A \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)+6 A \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-9 \sqrt{2} A \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-9 \sqrt{2} A \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 B \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{7}{2}}(c+d x)-3 B \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)-7 B \sin (c+d x) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+13 \sqrt{2} B \tan (c+d x) \sec (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+13 \sqrt{2} B \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","\frac{(9 A-13 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(12 A-19 B) \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) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-2 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}+\frac{(6 A-7 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 a d \sqrt{a \sec (c+d x)+a}}",1,"(4*(6*A - 7*B)*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^3*Sec[c + d*x]^2*Sin[(c + d*x)/2] + 8*(9*A - 13*B)*ArcSin[Sqrt[Sec[c + d*x]]]*Cos[(c + d*x)/2]^3*Sec[c + d*x]^2*Sin[(c + d*x)/2] + 6*A*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] - 7*B*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 4*A*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] - 3*B*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2)*Sin[c + d*x] + 2*B*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(7/2)*Sin[c + d*x] - 9*Sqrt[2]*A*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] + 13*Sqrt[2]*B*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Tan[c + d*x] - 9*Sqrt[2]*A*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x] + 13*Sqrt[2]*B*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]*Tan[c + d*x])/(4*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","B",1
256,1,132,197,1.9408942,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec (c+d x)} \left((9 B-5 A) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sqrt{2} (2 A-3 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+\tan \left(\frac{1}{2} (c+d x)\right) \sec \left(\frac{1}{2} (c+d x)\right) (-A+2 B \sec (c+d x)+3 B)\right)}{2 a d \sqrt{a (\sec (c+d x)+1)}}","-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 A-3 B) \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) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-3 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 a d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*Sqrt[Sec[c + d*x]]*((-5*A + 9*B)*ArcTanh[Sin[(c + d*x)/2]] + 2*Sqrt[2]*(2*A - 3*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + Sec[(c + d*x)/2]*(-A + 3*B + 2*B*Sec[c + d*x])*Tan[(c + d*x)/2]))/(2*a*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
257,1,113,145,0.8569562,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left((A-B) \tan \left(\frac{1}{2} (c+d x)\right)+(A-5 B) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \sqrt{2} B \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 a d \sqrt{a (\sec (c+d x)+1)}}","\frac{(A-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 B \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) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Sqrt[Sec[c + d*x]]*((A - 5*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + 4*Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (A - B)*Tan[(c + d*x)/2]))/(2*a*d*Sqrt[a*(1 + Sec[c + d*x])])","A",1
258,1,84,107,0.2865891,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \left((B-A) \sin \left(\frac{1}{2} (c+d x)\right)+(3 A+B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{d (a (\sec (c+d x)+1))^{3/2}}","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(3/2)*((3*A + B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^2 + (-A + B)*Sin[(c + d*x)/2]))/(d*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
259,1,174,156,1.5231451,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) \left((5 A-B) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+4 A \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}\right)+2 \sqrt{2} (7 A-3 B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sec ^2(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{2 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"(2*Sqrt[2]*(7*A - 3*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^3*Sec[c + d*x]^2*Sin[(c + d*x)/2] + ((5*A - B)*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + 4*A*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x])/(2*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
260,1,173,203,1.7954561,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\tan (c+d x) \sqrt{1-\sec (c+d x)} (\sec (c+d x) (2 A \cos (2 (c+d x))-17 A+15 B)+12 (B-A))-6 \sqrt{2} (11 A-7 B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{6 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} (a (\sec (c+d x)+1))^{3/2}}","\frac{(11 A-7 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(19 A-15 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}+\frac{(7 A-3 B) \sin (c+d x)}{6 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(-6*Sqrt[2]*(11*A - 7*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^3*Sec[c + d*x]^(5/2)*Sin[(c + d*x)/2] + Sqrt[1 - Sec[c + d*x]]*(12*(-A + B) + (-17*A + 15*B + 2*A*Cos[2*(c + d*x)])*Sec[c + d*x])*Tan[c + d*x])/(6*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
261,1,171,250,1.4963547,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sec (c+d x) \left(\frac{15 \sqrt{2} (15 A-11 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{\sqrt{1-\sec (c+d x)}}+\sin (c+d x) \sqrt{\sec (c+d x)} (3 (39 A-20 B) \cos (c+d x)+(10 B-6 A) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+141 A-85 B)\right)}{30 d (a (\sec (c+d x)+1))^{3/2}}","-\frac{(15 A-11 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B) \sin (c+d x)}{10 a d \sec ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \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) \sin (c+d x) \sqrt{\sec (c+d x)}}{30 a d \sqrt{a \sec (c+d x)+a}}-\frac{(39 A-35 B) \sin (c+d x)}{30 a d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(Sec[c + d*x]*((141*A - 85*B + 3*(39*A - 20*B)*Cos[c + d*x] + (-6*A + 10*B)*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sqrt[Sec[c + d*x]]*Sin[c + d*x] + (15*Sqrt[2]*(15*A - 11*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Tan[c + d*x])/Sqrt[1 - Sec[c + d*x]]))/(30*d*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
262,1,941,246,6.1711242,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{7 B (\sec (c+d x)+1) \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{B \sin (c+d x) \sec ^{\frac{11}{2}}(c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}-\frac{7 B (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}+\frac{3 A (\sec (c+d x)+1) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{A \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}-\frac{3 A (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}+\frac{11 B (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}+\frac{7 A (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{15 B (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{11 A (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}+\frac{35 B (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{11 A \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{35 B \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{43 A \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{115 B \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{43 A \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{115 B \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{(43 A-115 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 A-5 B) \sinh ^{-1}\left(\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right)}{a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{(A-B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(7 A-15 B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"-1/4*(A*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(d*(a*(1 + Sec[c + d*x]))^(5/2)) - (B*Sec[c + d*x]^(11/2)*Sin[c + d*x])/(4*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*Sec[c + d*x]^(9/2)*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (7*B*Sec[c + d*x]^(11/2)*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*A*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (35*B*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (7*A*Sec[c + d*x]^(5/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (15*B*Sec[c + d*x]^(5/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (3*A*Sec[c + d*x]^(7/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (11*B*Sec[c + d*x]^(7/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (7*B*Sec[c + d*x]^(9/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*A*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (35*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (43*A*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (115*B*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (43*A*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (115*B*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","B",1
263,1,845,194,6.1622015,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{3 B (\sec (c+d x)+1) \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{B \sin (c+d x) \sec ^{\frac{9}{2}}(c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}-\frac{3 B (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{A (\sec (c+d x)+1) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{A \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{4 d (a (\sec (c+d x)+1))^{5/2}}+\frac{A (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}+\frac{7 B (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}+\frac{3 A (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}-\frac{11 B (\sec (c+d x)+1)^2 \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 d (a (\sec (c+d x)+1))^{5/2}}+\frac{3 A \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{11 B \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{3 A \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{43 B \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{3 A \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{43 B \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) (\sec (c+d x)+1)^2 \tan (c+d x)}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","\frac{(3 A-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 B \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) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}+\frac{(3 A-11 B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"-1/4*(A*Sec[c + d*x]^(7/2)*Sin[c + d*x])/(d*(a*(1 + Sec[c + d*x]))^(5/2)) - (B*Sec[c + d*x]^(9/2)*Sin[c + d*x])/(4*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (A*Sec[c + d*x]^(7/2)*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*B*Sec[c + d*x]^(9/2)*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*B*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (A*Sec[c + d*x]^(5/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (7*B*Sec[c + d*x]^(5/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) - (3*B*Sec[c + d*x]^(7/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (43*B*ArcSin[Sqrt[Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (3*A*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (43*B*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*(1 + Sec[c + d*x])^2*Tan[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","B",1
264,1,106,156,0.8122615,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \left(\frac{1}{2} \sin \left(\frac{1}{2} (c+d x)\right) ((5 A+3 B) \cos (c+d x)+A+7 B)+(5 A+3 B) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d (a (\sec (c+d x)+1))^{5/2}}","\frac{(5 A+3 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(A-B) \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) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{16 a d (a \sec (c+d x)+a)^{3/2}}",1,"(Cos[(c + d*x)/2]*Sec[c + d*x]^(5/2)*((5*A + 3*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + ((A + 7*B + (5*A + 3*B)*Cos[c + d*x])*Sin[(c + d*x)/2])/2))/(4*d*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
265,1,103,156,1.1555004,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\tan \left(\frac{1}{2} (c+d x)\right) ((B-9 A) \sec (c+d x)-13 A+5 B)+2 (19 A+5 B) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sec (c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{16 a d (a (\sec (c+d x)+1))^{3/2}}","\frac{(19 A+5 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B) \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) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(Sqrt[Sec[c + d*x]]*(2*(19*A + 5*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3*Sec[c + d*x] + (-13*A + 5*B + (-9*A + B)*Sec[c + d*x])*Tan[(c + d*x)/2]))/(16*a*d*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
266,1,206,203,2.6073209,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sin (c+d x) \left((49 A-9 B) \sqrt{1-\sec (c+d x)} \sec ^{\frac{5}{2}}(c+d x)+(85 A-13 B) \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)+32 A \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}\right)+4 \sqrt{2} (75 A-19 B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec ^3(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}","-\frac{(75 A-19 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A-9 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a^2 d \sqrt{a \sec (c+d x)+a}}-\frac{(13 A-5 B) \sin (c+d x) \sqrt{\sec (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(4*Sqrt[2]*(75*A - 19*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^5*Sec[c + d*x]^3*Sin[(c + d*x)/2] + ((85*A - 13*B)*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + (49*A - 9*B)*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(5/2) + 32*A*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])])*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
267,1,193,250,1.8888455,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{2 \tan (c+d x) \sqrt{1-\sec (c+d x)} \sec ^2(c+d x) ((255 B-479 A) \cos (c+d x)+(48 B-80 A) \cos (2 (c+d x))+8 A \cos (3 (c+d x))-379 A+195 B)-12 \sqrt{2} (163 A-75 B) \sin (c+d x) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{96 d \sqrt{-((\sec (c+d x)-1) \sec (c+d x))} (a (\sec (c+d x)+1))^{5/2}}","\frac{(163 A-75 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(299 A-147 B) \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) \sin (c+d x)}{48 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(17 A-9 B) \sin (c+d x)}{16 a d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(-12*Sqrt[2]*(163*A - 75*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^(7/2)*Sin[c + d*x] + 2*(-379*A + 195*B + (-479*A + 255*B)*Cos[c + d*x] + (-80*A + 48*B)*Cos[2*(c + d*x)] + 8*A*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^2*Tan[c + d*x])/(96*d*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
268,1,196,297,2.3954098,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sec ^2(c+d x) \left(\frac{30 \sqrt{2} (283 A-163 B) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tan (c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{\sqrt{1-\sec (c+d x)}}+\sin (c+d x) \sqrt{\sec (c+d x)} (5 (887 A-479 B) \cos (c+d x)+16 (52 A-25 B) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+12 A \cos (4 (c+d x))+3491 A+40 B \cos (3 (c+d x))-1895 B)\right)}{240 d (a (\sec (c+d x)+1))^{5/2}}","-\frac{(283 A-163 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A-85 B) \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) \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) \sin (c+d x)}{240 a^2 d \sqrt{\sec (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(21 A-13 B) \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) \sin (c+d x)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[c + d*x]^2*((3491*A - 1895*B + 5*(887*A - 479*B)*Cos[c + d*x] + 16*(52*A - 25*B)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 40*B*Cos[3*(c + d*x)] + 12*A*Cos[4*(c + d*x)])*Sqrt[Sec[c + d*x]]*Sin[c + d*x] + (30*Sqrt[2]*(283*A - 163*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Tan[c + d*x])/Sqrt[1 - Sec[c + d*x]]))/(240*d*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
269,1,4445,406,20.7168947,"\int (a+a \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{3 \sqrt{2} A \tan (c+d x) (a \sec (c+d x)+a)^{2/3} F_1\left(\frac{7}{6};\frac{1}{2},1;\frac{13}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{7 d \sqrt{1-\sec (c+d x)}}+\frac{3 B \tan (c+d x) (a \sec (c+d x)+a)^{2/3}}{2 d (\sec (c+d x)+1)}-\frac{3^{3/4} B \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)}{2 \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*B*Cos[c + d*x]*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*(A + B*Sec[c + d*x])*Tan[(c + d*x)/2])/(2*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^(2/3)) + (Cos[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*(A + B*Sec[c + d*x])*((A*Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/2 + Sec[(c + d*x)/2]^2*((A*(1 + Sec[c + d*x])^(2/3))/2 + (B*(1 + Sec[c + d*x])^(2/3))/4))*Tan[(c + d*x)/2]*(2*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^4 + 9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(3*(4*A + B)*Cos[(c + d*x)/2]^2 + B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2)))/(3*2^(1/3)*d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^(2/3)*(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)*((Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(2*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^4 + 9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(3*(4*A + B)*Cos[(c + d*x)/2]^2 + B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2)))/(6*2^(1/3)*(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) - ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*(2*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^4 + 9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(3*(4*A + B)*Cos[(c + d*x)/2]^2 + B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2))*(2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(3*2^(1/3)*(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2) + ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*(4*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^3 + 2*B*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^4*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (4*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^4*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9)*(3*(4*A + B)*Cos[(c + d*x)/2]^2 + B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2) + 2*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^4*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])) + 9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-3*(4*A + B)*Cos[(c + d*x)/2]*Sin[(c + d*x)/2] + B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + B*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)))))/(3*2^(1/3)*(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) + (2^(2/3)*Tan[(c + d*x)/2]*(2*B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^4 + 9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(3*(4*A + B)*Cos[(c + d*x)/2]^2 + B*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3)*(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))","B",0
270,1,2709,354,19.1553497,"\int \frac{A+B \sec (c+d x)}{\sqrt[3]{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(1/3),x]","\text{Result too large to show}","\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(\frac{1}{6};\frac{1}{2},1;\frac{7}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} \sqrt[3]{a \sec (c+d x)+a}}-\frac{3^{3/4} B \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)}{\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,"(2^(2/3)*Cos[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x])*((B*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/2 + (A*Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/2)*Tan[(c + d*x)/2]*((-A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(3*d*(B + A*Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^(1/3)*((Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*((-A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(3*2^(1/3)) + (2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*((-A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + (-A + B)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*(-A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (27*(A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (27*(A + B)*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (27*(A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/3 + (2*2^(2/3)*Tan[(c + d*x)/2]*((-A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
271,1,2901,415,19.5611555,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{4/3}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(4/3),x]","\text{Result too large to show}","-\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(-\frac{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 d \sqrt{1-\sec (c+d x)} (\sec (c+d x)+1) \sqrt[3]{a \sec (c+d x)+a}}+\frac{3 B \tan (c+d x)}{5 a d (\sec (c+d x)+1) \sqrt[3]{a \sec (c+d x)+a}}-\frac{3^{3/4} B \tan (c+d x) \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right) \sqrt{\frac{(\sec (c+d x)+1)^{2/3}+\sqrt[3]{2} \sqrt[3]{\sec (c+d x)+1}+2^{2/3}}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} F\left(\cos ^{-1}\left(\frac{\sqrt[3]{2}-\left(1-\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}{\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}}\right)|\frac{1}{4} \left(2+\sqrt{3}\right)\right)}{5 \sqrt[3]{2} a d (1-\sec (c+d x)) \sqrt{-\frac{\sqrt[3]{\sec (c+d x)+1} \left(\sqrt[3]{2}-\sqrt[3]{\sec (c+d x)+1}\right)}{\left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right)^2}} \sqrt[3]{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]*((1 + Cos[c + d*x])*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x])*((3*Sec[(c + d*x)/2]*(-(A*Sin[(c + d*x)/2]) + B*Sin[(c + d*x)/2]))/5 - (3*Sec[(c + d*x)/2]^3*(-(A*Sin[(c + d*x)/2]) + B*Sin[(c + d*x)/2]))/10))/(d*(B + A*Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^(4/3)) + (2^(2/3)*Cos[c + d*x]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x])*((A*Cos[c + d*x]*Sec[(c + d*x)/2]^2*(1 + Sec[c + d*x])^(2/3))/2 + Sec[(c + d*x)/2]^2*(-1/10*(A*(1 + Sec[c + d*x])^(2/3)) + (B*(1 + Sec[c + d*x])^(2/3))/10))*Tan[(c + d*x)/2]*((-6*A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(4*A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(15*d*(B + A*Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^(4/3)*((Sec[(c + d*x)/2]^2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*((-6*A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(4*A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))/(15*2^(1/3)) + (2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*Tan[(c + d*x)/2]*((-6*A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2] + (-6*A + B)*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 2/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + (2*(-6*A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(1/3)) - (27*(4*A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]*Sin[(c + d*x)/2])/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) + (27*(4*A + B)*Cos[(c + d*x)/2]^2*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2) - (27*(4*A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2*(2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 9*(-1/3*(AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*(-3*((-6*AppellF1[5/2, 2/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (2*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5) + 2*((-3*AppellF1[5/2, 5/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + AppellF1[5/2, 8/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))))/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2))/15 + (2*2^(2/3)*Tan[(c + d*x)/2]*((-6*A + B)*AppellF1[3/2, 2/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)*Tan[(c + d*x)/2]^2 + (27*(4*A + B)*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Cos[(c + d*x)/2]^2)/(9*AppellF1[1/2, 2/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(-3*AppellF1[3/2, 2/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*AppellF1[3/2, 5/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(45*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(1/3))))","B",0
272,1,4110,787,19.6091243,"\int (a+a \sec (c+d x))^{4/3} (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^(4/3)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{3 \sqrt{2} a A \tan (c+d x) (\sec (c+d x)+1) \sqrt[3]{a \sec (c+d x)+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)}{11 d \sqrt{1-\sec (c+d x)}}+\frac{3 a B \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a}}{4 d}-\frac{15 \left(1+\sqrt{3}\right) a B \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{5\ 3^{3/4} \left(1-\sqrt{3}\right) a B \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{15 \sqrt[4]{3} a B \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,"(Cos[c + d*x]*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(a*(1 + Sec[c + d*x]))^(4/3)*(A + B*Sec[c + d*x])*((3*(4*A + 5*B)*Sin[c + d*x])/4 + (3*B*Tan[c + d*x])/4))/(d*(B + A*Cos[c + d*x])*(1 + Sec[c + d*x])^(4/3)) + (Cos[c + d*x]*(a*(1 + Sec[c + d*x]))^(4/3)*(A + B*Sec[c + d*x])*(2*A*(1 + Sec[c + d*x])^(1/3) + (5*B*(1 + Sec[c + d*x])^(1/3))/4 + Cos[c + d*x]*(-3*A*(1 + Sec[c + d*x])^(1/3) - (15*B*(1 + Sec[c + d*x])^(1/3))/4))*Tan[(c + d*x)/2]*(-(((4*A + 5*B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*A + 5*B + 5*(4*A + 7*B)*Cos[c + d*x]) - 4*(4*A + 5*B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(6*2^(2/3)*d*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(1 + Sec[c + d*x])^(4/3)*((Sec[(c + d*x)/2]^2*(-(((4*A + 5*B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*A + 5*B + 5*(4*A + 7*B)*Cos[c + d*x]) - 4*(4*A + 5*B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(12*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (Tan[(c + d*x)/2]*(-(((4*A + 5*B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - ((4*A + 5*B)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + (2*(4*A + 5*B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3)) + (9*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*A + 5*B + 5*(4*A + 7*B)*Cos[c + d*x]) - 4*(4*A + 5*B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)^2*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) + (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*A + 5*B + 5*(4*A + 7*B)*Cos[c + d*x]) - 4*(4*A + 5*B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2)*(2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2) - (9*(-15*(4*A + 7*B)*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[c + d*x] - 4*(4*A + 5*B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] + 4*(4*A + 5*B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sin[c + d*x]*Tan[(c + d*x)/2]^2 + 3*(-4*A + 5*B + 5*(4*A + 7*B)*Cos[c + d*x])*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) - 4*(4*A + 5*B)*Cos[c + d*x]*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(6*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) - (Tan[(c + d*x)/2]*(-(((4*A + 5*B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3)) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(-4*A + 5*B + 5*(4*A + 7*B)*Cos[c + d*x]) - 4*(4*A + 5*B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
273,1,5094,739,21.3380646,"\int \sqrt[3]{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[(a + a*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{3 \sqrt{2} A \tan (c+d x) \sqrt[3]{a \sec (c+d x)+a} F_1\left(\frac{5}{6};\frac{1}{2},1;\frac{11}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{5 d \sqrt{1-\sec (c+d x)}}-\frac{3 \left(1+\sqrt{3}\right) B \tan (c+d x) \sqrt[3]{a \sec (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^{3/4} \left(1-\sqrt{3}\right) B \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} 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} B \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)}{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,"Result too large to show","B",0
274,1,4066,764,19.3628265,"\int \frac{A+B \sec (c+d x)}{(a+a \sec (c+d x))^{2/3}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + a*Sec[c + d*x])^(2/3),x]","\text{Result too large to show}","-\frac{3 \sqrt{2} A \tan (c+d x) F_1\left(-\frac{1}{6};\frac{1}{2},1;\frac{5}{6};\frac{1}{2} (\sec (c+d x)+1),\sec (c+d x)+1\right)}{d \sqrt{1-\sec (c+d x)} (a \sec (c+d x)+a)^{2/3}}+\frac{3 B \tan (c+d x)}{d (a \sec (c+d x)+a)^{2/3}}+\frac{3 \left(1+\sqrt{3}\right) B \tan (c+d x) \sqrt[3]{\sec (c+d x)+1}}{d \left(\sqrt[3]{2}-\left(1+\sqrt{3}\right) \sqrt[3]{\sec (c+d x)+1}\right) (a \sec (c+d x)+a)^{2/3}}-\frac{3^{3/4} \left(1-\sqrt{3}\right) B \tan (c+d x) \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}} 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} 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}} (a \sec (c+d x)+a)^{2/3}}-\frac{3 \sqrt[3]{2} \sqrt[4]{3} B \tan (c+d x) \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}} 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)}{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}} (a \sec (c+d x)+a)^{2/3}}",1,"(Cos[c + d*x]*((1 + Cos[c + d*x])*Sec[c + d*x])^(1/3)*(1 + Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x])*(3*Sec[(c + d*x)/2]*(-(A*Sin[(c + d*x)/2]) + B*Sin[(c + d*x)/2]) - 3*(-A + B)*Sin[c + d*x]))/(d*(B + A*Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^(2/3)) - (2^(1/3)*Cos[c + d*x]*(1 + Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x])*(2*A*(1 + Sec[c + d*x])^(1/3) - B*(1 + Sec[c + d*x])^(1/3) + Cos[c + d*x]*(-3*A*(1 + Sec[c + d*x])^(1/3) + 3*B*(1 + Sec[c + d*x])^(1/3)))*Tan[(c + d*x)/2]*(((A - B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B + (-5*A + 7*B)*Cos[c + d*x]) + 4*(A - B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(3*d*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)*(a*(1 + Sec[c + d*x]))^(2/3)*(-1/3*(Sec[(c + d*x)/2]^2*(((A - B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B + (-5*A + 7*B)*Cos[c + d*x]) + 4*(A - B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(2^(2/3)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) - (2^(1/3)*Tan[(c + d*x)/2]*(((A - B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) + ((A - B)*Tan[(c + d*x)/2]^2*((-3*AppellF1[5/2, 1/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) - (2*(A - B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2*(-(Sec[(c + d*x)/2]^2*Sin[c + d*x]) + Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(5/3)) + (9*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B + (-5*A + 7*B)*Cos[c + d*x]) + 4*(A - B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)^2*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)) + (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B + (-5*A + 7*B)*Cos[c + d*x]) + 4*(A - B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2)*(2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 9*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 2*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)^2) - (9*(-3*(-5*A + 7*B)*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sin[c + d*x] + 4*(A - B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2] - 4*(A - B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Sin[c + d*x]*Tan[(c + d*x)/2]^2 + 3*(A + B + (-5*A + 7*B)*Cos[c + d*x])*(-1/3*(AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]) + (AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/9) + 4*(A - B)*Cos[c + d*x]*Tan[(c + d*x)/2]^2*((3*AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 - (4*AppellF1[5/2, 7/3, 1, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + 3*((-6*AppellF1[5/2, 1/3, 3, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5 + (AppellF1[5/2, 4/3, 2, 7/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/5))))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2))))/(3*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(2/3)) + (2*2^(1/3)*Tan[(c + d*x)/2]*(((A - B)*AppellF1[3/2, 1/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*Tan[(c + d*x)/2]^2)/(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(2/3) - (9*(3*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2]*(A + B + (-5*A + 7*B)*Cos[c + d*x]) + 4*(A - B)*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Cos[c + d*x]*Tan[(c + d*x)/2]^2))/(2*(-1 + Tan[(c + d*x)/2]^2)*(-9*AppellF1[1/2, 1/3, 1, 3/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] + 2*(3*AppellF1[3/2, 1/3, 2, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2] - AppellF1[3/2, 4/3, 1, 5/2, Tan[(c + d*x)/2]^2, -Tan[(c + d*x)/2]^2])*Tan[(c + d*x)/2]^2)))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(9*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(5/3))))","B",0
275,1,4897,197,23.0748943,"\int (c \sec (e+f x))^n (a+a \sec (e+f x))^m (A+B \sec (e+f x)) \, dx","Integrate[(c*Sec[e + f*x])^n*(a + a*Sec[e + f*x])^m*(A + B*Sec[e + f*x]),x]","\text{Result too large to show}","-\frac{(A-B) \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m (c \sec (e+f x))^n F_1\left(n;\frac{1}{2},\frac{1}{2}-m;n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)}}-\frac{B \tan (e+f x) (\sec (e+f x)+1)^{-m-\frac{1}{2}} (a \sec (e+f x)+a)^m (c \sec (e+f x))^n F_1\left(n;\frac{1}{2},-m-\frac{1}{2};n+1;\sec (e+f x),-\sec (e+f x)\right)}{f n \sqrt{1-\sec (e+f x)}}",1,"(2^(1 + m)*(Sec[(e + f*x)/2]^2)^n*Sec[e + f*x]^(-1 - n)*(c*Sec[e + f*x])^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*(a*(1 + Sec[e + f*x]))^m*(A + B*Sec[e + f*x])*(A*Sec[e + f*x]^n*(1 + Sec[e + f*x])^m + B*Sec[e + f*x]^(1 + n)*(1 + Sec[e + f*x])^m)*Tan[(e + f*x)/2]*((-3*A*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (B*AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)))/(f*(B + A*Cos[e + f*x])*(1 + Sec[e + f*x])^m*(-1 + Tan[(e + f*x)/2]^2)*(-((2^(1 + m)*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2*((-3*A*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (B*AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)))/(-1 + Tan[(e + f*x)/2]^2)^2) + (2^m*(Sec[(e + f*x)/2]^2)^(1 + n)*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*((-3*A*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (B*AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)))/(-1 + Tan[(e + f*x)/2]^2) + (2^(1 + m)*n*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]^2*((-3*A*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (B*AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)))/(-1 + Tan[(e + f*x)/2]^2) + (2^(1 + m)*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(m + n)*Tan[(e + f*x)/2]*((3*A*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sin[e + f*x])/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (3*A*Cos[e + f*x]*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (B*((n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + ((1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3))/(AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3) + (3*A*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x]*(2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2] + 3*(-1/3*((1 - n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2]) + ((m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3) + 2*Tan[(e + f*x)/2]^2*((-1 + n)*((-3*(2 - n)*AppellF1[5/2, m + n, 3 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(m + n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (m + n)*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5))))/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)^2 + (B*AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*((n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + ((1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/3 + (2*Tan[(e + f*x)/2]^2*(n*((-3*(1 - n)*AppellF1[5/2, 1 + m + n, 2 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(1 + m + n)*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5) + (1 + m + n)*((3*n*AppellF1[5/2, 2 + m + n, 1 - n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5 + (3*(2 + m + n)*AppellF1[5/2, 3 + m + n, -n, 7/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Sec[(e + f*x)/2]^2*Tan[(e + f*x)/2])/5)))/3))/(AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3)^2))/(-1 + Tan[(e + f*x)/2]^2) + (2^(1 + m)*(m + n)*(Sec[(e + f*x)/2]^2)^n*(Cos[(e + f*x)/2]^2*Sec[e + f*x])^(-1 + m + n)*Tan[(e + f*x)/2]*((-3*A*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2]*Cos[e + f*x])/(3*AppellF1[1/2, m + n, 1 - n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + 2*((-1 + n)*AppellF1[3/2, m + n, 2 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (m + n)*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2) - (B*AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])/(AppellF1[1/2, 1 + m + n, -n, 3/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (2*(n*AppellF1[3/2, 1 + m + n, 1 - n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2] + (1 + m + n)*AppellF1[3/2, 2 + m + n, -n, 5/2, Tan[(e + f*x)/2]^2, -Tan[(e + f*x)/2]^2])*Tan[(e + f*x)/2]^2)/3))*(-(Cos[(e + f*x)/2]*Sec[e + f*x]*Sin[(e + f*x)/2]) + Cos[(e + f*x)/2]^2*Sec[e + f*x]*Tan[e + f*x]))/(-1 + Tan[(e + f*x)/2]^2)))","B",0
276,1,111,164,1.1267381,"\int \sec ^{-1-n}(c+d x) (a+a \sec (c+d x))^n (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(-1 - n)*(a + a*Sec[c + d*x])^n*(A + B*Sec[c + d*x]),x]","\frac{\sin (c+d x) \sec ^{-n}(c+d x) (a (\sec (c+d x)+1))^n \left(\frac{(A n+B n+B) \left(-\cot ^2\left(\frac{1}{2} (c+d x)\right)\right)^{\frac{1}{2}-n} \, _2F_1\left(\frac{1}{2}-n,-n;1-n;\csc ^2\left(\frac{1}{2} (c+d x)\right)\right)}{n (\cos (c+d x)+1)}+A\right)}{d (n+1)}","\frac{(A n+B n+B) \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)}",1,"((A + ((B + A*n + B*n)*(-Cot[(c + d*x)/2]^2)^(1/2 - n)*Hypergeometric2F1[1/2 - n, -n, 1 - n, Csc[(c + d*x)/2]^2])/(n*(1 + Cos[c + d*x])))*(a*(1 + Sec[c + d*x]))^n*Sin[c + d*x])/(d*(1 + n)*Sec[c + d*x]^n)","A",1
277,1,85,114,0.6339583,"\int \sec ^3(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{3 (4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x) \left(8 (a B+A b) (\cos (2 (c+d x))+2) \sec (c+d x)+12 a A+6 b B \sec ^2(c+d x)+9 b B\right)}{24 d}","\frac{(a B+A b) \tan ^3(c+d x)}{3 d}+\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(4 a A+3 b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a A+3 b B) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{b B \tan (c+d x) \sec ^3(c+d x)}{4 d}",1,"(3*(4*a*A + 3*b*B)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(12*a*A + 9*b*B + 8*(A*b + a*B)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*b*B*Sec[c + d*x]^2)*Tan[c + d*x])/(24*d)","A",1
278,1,67,93,0.2913136,"\int \sec ^2(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{3 (a B+A b) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 (a B+A b) \sec (c+d x)+6 a A+2 b B \tan ^2(c+d x)+6 b B\right)}{6 d}","\frac{(3 a A+2 b B) \tan (c+d x)}{3 d}+\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{(a B+A b) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{b B \tan (c+d x) \sec ^2(c+d x)}{3 d}",1,"(3*(A*b + a*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(6*a*A + 6*b*B + 3*(A*b + a*B)*Sec[c + d*x] + 2*b*B*Tan[c + d*x]^2))/(6*d)","A",1
279,1,75,61,0.0258767,"\int \sec (c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a A \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a B \tan (c+d x)}{d}+\frac{A b \tan (c+d x)}{d}+\frac{b B \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b B \tan (c+d x) \sec (c+d x)}{2 d}","\frac{(a B+A b) \tan (c+d x)}{d}+\frac{(2 a A+b B) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b B \tan (c+d x) \sec (c+d x)}{2 d}",1,"(a*A*ArcTanh[Sin[c + d*x]])/d + (b*B*ArcTanh[Sin[c + d*x]])/(2*d) + (A*b*Tan[c + d*x])/d + (a*B*Tan[c + d*x])/d + (b*B*Sec[c + d*x]*Tan[c + d*x])/(2*d)","A",1
280,1,43,35,0.0109273,"\int (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","a A x+\frac{a B \tanh ^{-1}(\sin (c+d x))}{d}+\frac{A b \tanh ^{-1}(\sin (c+d x))}{d}+\frac{b B \tan (c+d x)}{d}","\frac{(a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a A x+\frac{b B \tan (c+d x)}{d}",1,"a*A*x + (A*b*ArcTanh[Sin[c + d*x]])/d + (a*B*ArcTanh[Sin[c + d*x]])/d + (b*B*Tan[c + d*x])/d","A",1
281,1,46,35,0.0267456,"\int \cos (c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a A \sin (c) \cos (d x)}{d}+\frac{a A \cos (c) \sin (d x)}{d}+a B x+A b x+\frac{b B \tanh ^{-1}(\sin (c+d x))}{d}","x (a B+A b)+\frac{a A \sin (c+d x)}{d}+\frac{b B \tanh ^{-1}(\sin (c+d x))}{d}",1,"A*b*x + a*B*x + (b*B*ArcTanh[Sin[c + d*x]])/d + (a*A*Cos[d*x]*Sin[c])/d + (a*A*Cos[c]*Sin[d*x])/d","A",1
282,1,51,52,0.0861486,"\int \cos ^2(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{4 (a B+A b) \sin (c+d x)+a A \sin (2 (c+d x))+2 a A c+2 a A d x+4 b B d x}{4 d}","\frac{(a B+A b) \sin (c+d x)}{d}+\frac{1}{2} x (a A+2 b B)+\frac{a A \sin (c+d x) \cos (c+d x)}{2 d}",1,"(2*a*A*c + 2*a*A*d*x + 4*b*B*d*x + 4*(A*b + a*B)*Sin[c + d*x] + a*A*Sin[2*(c + d*x)])/(4*d)","A",1
283,1,75,84,0.1624193,"\int \cos ^3(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{3 (3 a A+4 b B) \sin (c+d x)+3 (a B+A b) \sin (2 (c+d x))+a A \sin (3 (c+d x))+6 a B c+6 a B d x+6 A b c+6 A b d x}{12 d}","\frac{(2 a A+3 b B) \sin (c+d x)}{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)+\frac{a A \sin (c+d x) \cos ^2(c+d x)}{3 d}",1,"(6*A*b*c + 6*a*B*c + 6*A*b*d*x + 6*a*B*d*x + 3*(3*a*A + 4*b*B)*Sin[c + d*x] + 3*(A*b + a*B)*Sin[2*(c + d*x)] + a*A*Sin[3*(c + d*x)])/(12*d)","A",1
284,1,91,105,0.2363529,"\int \cos ^4(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{-32 (a B+A b) \sin ^3(c+d x)+96 (a B+A b) \sin (c+d x)+24 (a A+b B) \sin (2 (c+d x))+3 a A \sin (4 (c+d x))+36 a A c+36 a A d x+48 b B c+48 b B d x}{96 d}","-\frac{(a B+A b) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x)}{d}+\frac{(3 a A+4 b B) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x (3 a A+4 b B)+\frac{a A \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(36*a*A*c + 48*b*B*c + 36*a*A*d*x + 48*b*B*d*x + 96*(A*b + a*B)*Sin[c + d*x] - 32*(A*b + a*B)*Sin[c + d*x]^3 + 24*(a*A + b*B)*Sin[2*(c + d*x)] + 3*a*A*Sin[4*(c + d*x)])/(96*d)","A",1
285,1,150,198,1.558687,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{15 \left(4 a^2 A+6 a b B+3 A b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 \left(a^2 B+2 a A b+2 b^2 B\right) \tan ^2(c+d x)+15 \left(a^2 B+2 a A b+b^2 B\right)+3 b^2 B \tan ^4(c+d x)\right)+15 \left(4 a^2 A+6 a b B+3 A b^2\right) \sec (c+d x)+30 b (2 a B+A b) \sec ^3(c+d x)\right)}{120 d}","\frac{\left(4 a^2 A+6 a b B+3 A b^2\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(4 a^2 A+6 a b B+3 A b^2\right) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \tan ^3(c+d x)}{15 d}+\frac{\left(5 a (a B+2 A b)+4 b^2 B\right) \tan (c+d x)}{5 d}+\frac{b (6 a B+5 A b) \tan (c+d x) \sec ^3(c+d x)}{20 d}+\frac{b B \tan (c+d x) \sec ^3(c+d x) (a+b \sec (c+d x))}{5 d}",1,"(15*(4*a^2*A + 3*A*b^2 + 6*a*b*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(4*a^2*A + 3*A*b^2 + 6*a*b*B)*Sec[c + d*x] + 30*b*(A*b + 2*a*B)*Sec[c + d*x]^3 + 8*(15*(2*a*A*b + a^2*B + b^2*B) + 5*(2*a*A*b + a^2*B + 2*b^2*B)*Tan[c + d*x]^2 + 3*b^2*B*Tan[c + d*x]^4)))/(120*d)","A",1
286,1,120,179,0.7543829,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{3 \left(4 a^2 B+8 a A b+3 b^2 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(3 \left(4 a^2 B+8 a A b+3 b^2 B\right) \sec (c+d x)+24 \left(a^2 A+2 a b B+A b^2\right)+8 b (2 a B+A b) \tan ^2(c+d x)+6 b^2 B \sec ^3(c+d x)\right)}{24 d}","\frac{\left(4 a^2 B+8 a A b+3 b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(-2 a^2 B+8 a A b+9 b^2 B\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{\left(a^3 (-B)+4 a^2 A b+8 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 b d}+\frac{(4 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^2}{12 b d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^3}{4 b d}",1,"(3*(8*a*A*b + 4*a^2*B + 3*b^2*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(a^2*A + A*b^2 + 2*a*b*B) + 3*(8*a*A*b + 4*a^2*B + 3*b^2*B)*Sec[c + d*x] + 6*b^2*B*Sec[c + d*x]^3 + 8*b*(A*b + 2*a*B)*Tan[c + d*x]^2))/(24*d)","A",1
287,1,92,116,0.4873206,"\int \sec (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{3 \left(2 a^2 A+2 a b B+A b^2\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(2 \left(3 a^2 B+6 a A b+b^2 B \tan ^2(c+d x)+3 b^2 B\right)+3 b (2 a B+A b) \sec (c+d x)\right)}{6 d}","\frac{2 \left(a^2 B+3 a A b+b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(2 a^2 A+2 a b B+A b^2\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b (2 a B+3 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(3*(2*a^2*A + A*b^2 + 2*a*b*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(3*b*(A*b + 2*a*B)*Sec[c + d*x] + 2*(6*a*A*b + 3*a^2*B + 3*b^2*B + b^2*B*Tan[c + d*x]^2)))/(6*d)","A",1
288,1,67,86,0.2811106,"\int (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\left(2 a^2 B+4 a A b+b^2 B\right) \tanh ^{-1}(\sin (c+d x))+2 a^2 A d x+b \tan (c+d x) (4 a B+2 A b+b B \sec (c+d x))}{2 d}","\frac{\left(2 a^2 B+4 a A b+b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+a^2 A x+\frac{b (3 a B+2 A b) \tan (c+d x)}{2 d}+\frac{b B \tan (c+d x) (a+b \sec (c+d x))}{2 d}",1,"(2*a^2*A*d*x + (4*a*A*b + 2*a^2*B + b^2*B)*ArcTanh[Sin[c + d*x]] + b*(2*A*b + 4*a*B + b*B*Sec[c + d*x])*Tan[c + d*x])/(2*d)","A",1
289,1,109,60,0.4948864,"\int \cos (c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{a^2 A \sin (c+d x)+a (c+d x) (a B+2 A b)-b (2 a B+A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b (2 a B+A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b^2 B \tan (c+d x)}{d}","\frac{a^2 A \sin (c+d x)}{d}+\frac{b (2 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+a x (a B+2 A b)+\frac{b^2 B \tan (c+d x)}{d}",1,"(a*(2*A*b + a*B)*(c + d*x) - b*(A*b + 2*a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*(A*b + 2*a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + a^2*A*Sin[c + d*x] + b^2*B*Tan[c + d*x])/d","A",1
290,1,120,80,0.2290725,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{2 (c+d x) \left(a^2 A+4 a b B+2 A b^2\right)+a^2 A \sin (2 (c+d x))+4 a (a B+2 A b) \sin (c+d x)-4 b^2 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{4 d}","\frac{1}{2} x \left(a^2 A+4 a b B+2 A b^2\right)+\frac{a^2 A \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a (a B+2 A b) \sin (c+d x)}{d}+\frac{b^2 B \tanh ^{-1}(\sin (c+d x))}{d}",1,"(2*(a^2*A + 2*A*b^2 + 4*a*b*B)*(c + d*x) - 4*b^2*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*b^2*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 4*a*(2*A*b + a*B)*Sin[c + d*x] + a^2*A*Sin[2*(c + d*x)])/(4*d)","A",1
291,1,90,107,0.2384593,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{6 (c+d x) \left(a^2 B+2 a A b+2 b^2 B\right)+3 \left(3 a^2 A+8 a b B+4 A b^2\right) \sin (c+d x)+a^2 A \sin (3 (c+d x))+3 a (a B+2 A b) \sin (2 (c+d x))}{12 d}","\frac{\left(2 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x)}{3 d}+\frac{1}{2} x \left(a^2 B+2 a A b+2 b^2 B\right)+\frac{a^2 A \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac{a (a B+2 A b) \sin (c+d x) \cos (c+d x)}{2 d}",1,"(6*(2*a*A*b + a^2*B + 2*b^2*B)*(c + d*x) + 3*(3*a^2*A + 4*A*b^2 + 8*a*b*B)*Sin[c + d*x] + 3*a*(2*A*b + a*B)*Sin[2*(c + d*x)] + a^2*A*Sin[3*(c + d*x)])/(12*d)","A",1
292,1,118,136,0.4605076,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{12 (c+d x) \left(3 a^2 A+8 a b B+4 A b^2\right)+24 \left(3 a^2 B+6 a A b+4 b^2 B\right) \sin (c+d x)+24 \left(a^2 A+2 a b B+A b^2\right) \sin (2 (c+d x))+3 a^2 A \sin (4 (c+d x))+8 a (a B+2 A b) \sin (3 (c+d x))}{96 d}","\frac{\left(a^2 B+2 a A b+b^2 B\right) \sin (c+d x)}{d}+\frac{\left(3 a^2 A+8 a b B+4 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2 A+8 a b B+4 A b^2\right)+\frac{a^2 A \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac{a (a B+2 A b) \sin ^3(c+d x)}{3 d}",1,"(12*(3*a^2*A + 4*A*b^2 + 8*a*b*B)*(c + d*x) + 24*(6*a*A*b + 3*a^2*B + 4*b^2*B)*Sin[c + d*x] + 24*(a^2*A + A*b^2 + 2*a*b*B)*Sin[2*(c + d*x)] + 8*a*(2*A*b + a*B)*Sin[3*(c + d*x)] + 3*a^2*A*Sin[4*(c + d*x)])/(96*d)","A",1
293,1,146,180,0.5456383,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{60 (c+d x) \left(3 a^2 B+6 a A b+4 b^2 B\right)+60 \left(5 a^2 A+12 a b B+6 A b^2\right) \sin (c+d x)+120 \left(a^2 B+2 a A b+b^2 B\right) \sin (2 (c+d x))+10 \left(5 a^2 A+8 a b B+4 A b^2\right) \sin (3 (c+d x))+6 a^2 A \sin (5 (c+d x))+15 a (a B+2 A b) \sin (4 (c+d x))}{480 d}","-\frac{\left(4 a^2 A+10 a b B+5 A b^2\right) \sin ^3(c+d x)}{15 d}+\frac{\left(4 a^2 A+10 a b B+5 A b^2\right) \sin (c+d x)}{5 d}+\frac{\left(3 a^2 B+6 a A b+4 b^2 B\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^2 B+6 a A b+4 b^2 B\right)+\frac{a^2 A \sin (c+d x) \cos ^4(c+d x)}{5 d}+\frac{a (a B+2 A b) \sin (c+d x) \cos ^3(c+d x)}{4 d}",1,"(60*(6*a*A*b + 3*a^2*B + 4*b^2*B)*(c + d*x) + 60*(5*a^2*A + 6*A*b^2 + 12*a*b*B)*Sin[c + d*x] + 120*(2*a*A*b + a^2*B + b^2*B)*Sin[2*(c + d*x)] + 10*(5*a^2*A + 4*A*b^2 + 8*a*b*B)*Sin[3*(c + d*x)] + 15*a*(2*A*b + a*B)*Sin[4*(c + d*x)] + 6*a^2*A*Sin[5*(c + d*x)])/(480*d)","A",1
294,1,181,252,3.4957902,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{15 \left(4 a^3 B+12 a^2 A b+9 a b^2 B+3 A b^3\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(8 \left(5 b \left(3 a^2 B+3 a A b+2 b^2 B\right) \tan ^2(c+d x)+15 \left(a^3 A+3 a^2 b B+3 a A b^2+b^3 B\right)+3 b^3 B \tan ^4(c+d x)\right)+15 \left(4 a^3 B+12 a^2 A b+9 a b^2 B+3 A b^3\right) \sec (c+d x)+30 b^2 (3 a B+A b) \sec ^3(c+d x)\right)}{120 d}","\frac{\left(-3 a^2 B+15 a A b+16 b^2 B\right) \tan (c+d x) (a+b \sec (c+d x))^2}{60 b d}+\frac{\left(4 a^3 B+12 a^2 A b+9 a b^2 B+3 A b^3\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left(-6 a^3 B+30 a^2 A b+71 a b^2 B+45 A b^3\right) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{\left(-3 a^4 B+15 a^3 A b+52 a^2 b^2 B+60 a A b^3+16 b^4 B\right) \tan (c+d x)}{30 b d}+\frac{(5 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^3}{20 b d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^4}{5 b d}",1,"(15*(12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(15*(12*a^2*A*b + 3*A*b^3 + 4*a^3*B + 9*a*b^2*B)*Sec[c + d*x] + 30*b^2*(A*b + 3*a*B)*Sec[c + d*x]^3 + 8*(15*(a^3*A + 3*a*A*b^2 + 3*a^2*b*B + b^3*B) + 5*b*(3*a*A*b + 3*a^2*B + 2*b^2*B)*Tan[c + d*x]^2 + 3*b^3*B*Tan[c + d*x]^4)))/(120*d)","A",1
295,1,140,180,0.9491789,"\int \sec (c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{3 \left(8 a^3 A+12 a^2 b B+12 a A b^2+3 b^3 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(9 b \left(4 a^2 B+4 a A b+b^2 B\right) \sec (c+d x)+24 \left(a^3 B+3 a^2 A b+3 a b^2 B+A b^3\right)+8 b^2 (3 a B+A b) \tan ^2(c+d x)+6 b^3 B \sec ^3(c+d x)\right)}{24 d}","\frac{b \left(6 a^2 B+20 a A b+9 b^2 B\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{\left(3 a^3 B+16 a^2 A b+12 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 d}+\frac{\left(8 a^3 A+12 a^2 b B+12 a A b^2+3 b^3 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(3 a B+4 A b) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(3*(8*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 3*b^3*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(24*(3*a^2*A*b + A*b^3 + a^3*B + 3*a*b^2*B) + 9*b*(4*a*A*b + 4*a^2*B + b^2*B)*Sec[c + d*x] + 6*b^3*B*Sec[c + d*x]^3 + 8*b^2*(A*b + 3*a*B)*Tan[c + d*x]^2))/(24*d)","A",1
296,1,108,137,0.5909971,"\int (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{6 a^3 A d x+3 b \tan (c+d x) \left(6 a^2 B+b (3 a B+A b) \sec (c+d x)+6 a A b+2 b^2 B\right)+3 \left(2 a^3 B+6 a^2 A b+3 a b^2 B+A b^3\right) \tanh ^{-1}(\sin (c+d x))+2 b^3 B \tan ^3(c+d x)}{6 d}","a^3 A x+\frac{b \left(8 a^2 B+9 a A b+2 b^2 B\right) \tan (c+d x)}{3 d}+\frac{\left(2 a^3 B+6 a^2 A b+3 a b^2 B+A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{b^2 (5 a B+3 A b) \tan (c+d x) \sec (c+d x)}{6 d}+\frac{b B \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}",1,"(6*a^3*A*d*x + 3*(6*a^2*A*b + A*b^3 + 2*a^3*B + 3*a*b^2*B)*ArcTanh[Sin[c + d*x]] + 3*b*(6*a*A*b + 6*a^2*B + 2*b^2*B + b*(A*b + 3*a*B)*Sec[c + d*x])*Tan[c + d*x] + 2*b^3*B*Tan[c + d*x]^3)/(6*d)","A",1
297,1,399,119,0.9787481,"\int \cos (c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{\sec ^2(c+d x) \left(\left(a^3 A+2 b^3 B\right) \sin (c+d x)+a^3 A \sin (3 (c+d x))+2 a^3 B c+2 a^3 B d x+\cos (2 (c+d x)) \left(-b \left(6 a^2 B+6 a A b+b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b \left(6 a^2 B+6 a A b+b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+2 a^2 (c+d x) (a B+3 A b)\right)+6 a^2 A b c+6 a^2 A b d x-6 a^2 b B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 a^2 b B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-6 a A b^2 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+6 a A b^2 \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+6 a b^2 B \sin (2 (c+d x))+2 A b^3 \sin (2 (c+d x))-b^3 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+b^3 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d}","\frac{b \left(6 a^2 B+6 a A b+b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a^2 (2 a A-b B) \sin (c+d x)}{2 d}+a^2 x (a B+3 A b)+\frac{b^2 (2 a B+A b) \tan (c+d x)}{d}+\frac{b B \sin (c+d x) (a+b \sec (c+d x))^2}{2 d}",1,"(Sec[c + d*x]^2*(6*a^2*A*b*c + 2*a^3*B*c + 6*a^2*A*b*d*x + 2*a^3*B*d*x - 6*a*A*b^2*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*a^2*b*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - b^3*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 6*a*A*b^2*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 6*a^2*b*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + b^3*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + Cos[2*(c + d*x)]*(2*a^2*(3*A*b + a*B)*(c + d*x) - b*(6*a*A*b + 6*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + b*(6*a*A*b + 6*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + (a^3*A + 2*b^3*B)*Sin[c + d*x] + 2*A*b^3*Sin[2*(c + d*x)] + 6*a*b^2*B*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)]))/(4*d)","B",1
298,1,217,124,0.7082932,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 A \sin (2 (c+d x))+2 a (c+d x) \left(a^2 A+6 a b B+6 A b^2\right)+4 a^2 (a B+3 A b) \sin (c+d x)-4 b^2 (3 a B+A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 b^2 (3 a B+A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 b^3 B \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b^3 B \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{4 d}","\frac{1}{2} a x \left(a^2 A+6 a b B+6 A b^2\right)+\frac{a^2 (a B+2 A b) \sin (c+d x)}{d}-\frac{b^2 (a A-2 b B) \tan (c+d x)}{2 d}+\frac{b^2 (3 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}",1,"(2*a*(a^2*A + 6*A*b^2 + 6*a*b*B)*(c + d*x) - 4*b^2*(A*b + 3*a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 4*b^2*(A*b + 3*a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (4*b^3*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (4*b^3*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*a^2*(3*A*b + a*B)*Sin[c + d*x] + a^3*A*Sin[2*(c + d*x)])/(4*d)","A",1
299,1,159,145,0.375293,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{a^3 A \sin (3 (c+d x))+9 a \left(a^2 A+4 a b B+4 A b^2\right) \sin (c+d x)+3 a^2 (a B+3 A b) \sin (2 (c+d x))+6 (c+d x) \left(a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right)-12 b^3 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^3 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{12 d}","\frac{a \left(2 a^2 A+9 a b B+8 A b^2\right) \sin (c+d x)}{3 d}+\frac{a^2 (3 a B+5 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} x \left(a^3 B+3 a^2 A b+6 a b^2 B+2 A b^3\right)+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{b^3 B \tanh ^{-1}(\sin (c+d x))}{d}",1,"(6*(3*a^2*A*b + 2*A*b^3 + a^3*B + 6*a*b^2*B)*(c + d*x) - 12*b^3*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^3*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 9*a*(a^2*A + 4*A*b^2 + 4*a*b*B)*Sin[c + d*x] + 3*a^2*(3*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*d)","A",1
300,1,140,179,0.4267405,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{3 a^3 A \sin (4 (c+d x))+24 a \left(a^2 A+3 a b B+3 A b^2\right) \sin (2 (c+d x))+8 a^2 (a B+3 A b) \sin (3 (c+d x))+12 (c+d x) \left(3 a^3 A+12 a^2 b B+12 a A b^2+8 b^3 B\right)+24 \left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right) \sin (c+d x)}{96 d}","\frac{a \left(3 a^2 A+12 a b B+10 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{a^2 (2 a B+3 A b) \sin (c+d x) \cos ^2(c+d x)}{6 d}+\frac{\left(2 a^3 B+6 a^2 A b+9 a b^2 B+3 A b^3\right) \sin (c+d x)}{3 d}+\frac{1}{8} x \left(3 a^3 A+12 a^2 b B+12 a A b^2+8 b^3 B\right)+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}",1,"(12*(3*a^3*A + 12*a*A*b^2 + 12*a^2*b*B + 8*b^3*B)*(c + d*x) + 24*(9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*Sin[c + d*x] + 24*a*(a^2*A + 3*A*b^2 + 3*a*b*B)*Sin[2*(c + d*x)] + 8*a^2*(3*A*b + a*B)*Sin[3*(c + d*x)] + 3*a^3*A*Sin[4*(c + d*x)])/(96*d)","A",1
301,1,176,221,0.727198,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{6 a^3 A \sin (5 (c+d x))+10 a \left(5 a^2 A+12 a b B+12 A b^2\right) \sin (3 (c+d x))+15 a^2 (a B+3 A b) \sin (4 (c+d x))+60 (c+d x) \left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right)+60 \left(5 a^3 A+18 a^2 b B+18 a A b^2+8 b^3 B\right) \sin (c+d x)+120 \left(a^3 B+3 a^2 A b+3 a b^2 B+A b^3\right) \sin (2 (c+d x))}{480 d}","-\frac{a \left(4 a^2 A+15 a b B+12 A b^2\right) \sin ^3(c+d x)}{15 d}+\frac{a^2 (5 a B+7 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac{\left(4 a^3 A+15 a^2 b B+14 a A b^2+5 b^3 B\right) \sin (c+d x)}{5 d}+\frac{\left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right) \sin (c+d x) \cos (c+d x)}{8 d}+\frac{1}{8} x \left(3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right)+\frac{a A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}",1,"(60*(9*a^2*A*b + 4*A*b^3 + 3*a^3*B + 12*a*b^2*B)*(c + d*x) + 60*(5*a^3*A + 18*a*A*b^2 + 18*a^2*b*B + 8*b^3*B)*Sin[c + d*x] + 120*(3*a^2*A*b + A*b^3 + a^3*B + 3*a*b^2*B)*Sin[2*(c + d*x)] + 10*a*(5*a^2*A + 12*A*b^2 + 12*a*b*B)*Sin[3*(c + d*x)] + 15*a^2*(3*A*b + a*B)*Sin[4*(c + d*x)] + 6*a^3*A*Sin[5*(c + d*x)])/(480*d)","A",1
302,1,244,334,2.8542604,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{15 \left(8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(10 b^2 \left(36 a^2 B+24 a A b+5 b^2 B\right) \sec ^3(c+d x)+160 b \left(2 a^3 B+3 a^2 A b+4 a b^2 B+A b^3\right) \tan ^2(c+d x)+15 \left(8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right) \sec (c+d x)+240 \left(a^4 A+4 a^3 b B+6 a^2 A b^2+4 a b^3 B+A b^4\right)+48 b^3 (4 a B+A b) \tan ^4(c+d x)+40 b^4 B \sec ^5(c+d x)\right)}{240 d}","\frac{\left(-4 a^2 B+24 a A b+25 b^2 B\right) \tan (c+d x) (a+b \sec (c+d x))^3}{120 b d}+\frac{\left(-4 a^3 B+24 a^2 A b+53 a b^2 B+32 A b^3\right) \tan (c+d x) (a+b \sec (c+d x))^2}{120 b d}+\frac{\left(8 a^4 B+32 a^3 A b+36 a^2 b^2 B+24 a A b^3+5 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\left(-8 a^4 B+48 a^3 A b+178 a^2 b^2 B+232 a A b^3+75 b^4 B\right) \tan (c+d x) \sec (c+d x)}{240 d}+\frac{\left(-4 a^5 B+24 a^4 A b+121 a^3 b^2 B+224 a^2 A b^3+128 a b^4 B+32 A b^5\right) \tan (c+d x)}{60 b d}+\frac{(6 A b-a B) \tan (c+d x) (a+b \sec (c+d x))^4}{30 b d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^5}{6 b d}",1,"(15*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(240*(a^4*A + 6*a^2*A*b^2 + A*b^4 + 4*a^3*b*B + 4*a*b^3*B) + 15*(32*a^3*A*b + 24*a*A*b^3 + 8*a^4*B + 36*a^2*b^2*B + 5*b^4*B)*Sec[c + d*x] + 10*b^2*(24*a*A*b + 36*a^2*B + 5*b^2*B)*Sec[c + d*x]^3 + 40*b^4*B*Sec[c + d*x]^5 + 160*b*(3*a^2*A*b + A*b^3 + 2*a^3*B + 4*a*b^2*B)*Tan[c + d*x]^2 + 48*b^3*(A*b + 4*a*B)*Tan[c + d*x]^4))/(240*d)","A",1
303,1,198,250,3.9400835,"\int \sec (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{15 \left(8 a^4 A+16 a^3 b B+24 a^2 A b^2+12 a b^3 B+3 A b^4\right) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left(80 b^2 \left(3 a^2 B+2 a A b+b^2 B\right) \tan ^2(c+d x)+15 b \left(16 a^3 B+24 a^2 A b+12 a b^2 B+3 A b^3\right) \sec (c+d x)+120 \left(a^4 B+4 a^3 A b+6 a^2 b^2 B+4 a A b^3+b^4 B\right)+30 b^3 (4 a B+A b) \sec ^3(c+d x)+24 b^4 B \tan ^4(c+d x)\right)}{120 d}","\frac{\left(12 a^2 B+35 a A b+16 b^2 B\right) \tan (c+d x) (a+b \sec (c+d x))^2}{60 d}+\frac{b \left(24 a^3 B+130 a^2 A b+116 a b^2 B+45 A b^3\right) \tan (c+d x) \sec (c+d x)}{120 d}+\frac{\left(12 a^4 B+95 a^3 A b+112 a^2 b^2 B+80 a A b^3+16 b^4 B\right) \tan (c+d x)}{30 d}+\frac{\left(8 a^4 A+16 a^3 b B+24 a^2 A b^2+12 a b^3 B+3 A b^4\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{(4 a B+5 A b) \tan (c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac{B \tan (c+d x) (a+b \sec (c+d x))^4}{5 d}",1,"(15*(8*a^4*A + 24*a^2*A*b^2 + 3*A*b^4 + 16*a^3*b*B + 12*a*b^3*B)*ArcTanh[Sin[c + d*x]] + Tan[c + d*x]*(120*(4*a^3*A*b + 4*a*A*b^3 + a^4*B + 6*a^2*b^2*B + b^4*B) + 15*b*(24*a^2*A*b + 3*A*b^3 + 16*a^3*B + 12*a*b^2*B)*Sec[c + d*x] + 30*b^3*(A*b + 4*a*B)*Sec[c + d*x]^3 + 80*b^2*(2*a*A*b + 3*a^2*B + b^2*B)*Tan[c + d*x]^2 + 24*b^4*B*Tan[c + d*x]^4))/(120*d)","A",1
304,1,160,200,1.0315069,"\int (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{24 a^4 A d x+3 b \tan (c+d x) \left(b \left(24 a^2 B+16 a A b+3 b^2 B\right) \sec (c+d x)+8 \left(4 a^3 B+6 a^2 A b+4 a b^2 B+A b^3\right)+2 b^3 B \sec ^3(c+d x)\right)+3 \left(8 a^4 B+32 a^3 A b+24 a^2 b^2 B+16 a A b^3+3 b^4 B\right) \tanh ^{-1}(\sin (c+d x))+8 b^3 (4 a B+A b) \tan ^3(c+d x)}{24 d}","a^4 A x+\frac{b^2 \left(26 a^2 B+32 a A b+9 b^2 B\right) \tan (c+d x) \sec (c+d x)}{24 d}+\frac{b \left(19 a^3 B+34 a^2 A b+16 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 d}+\frac{\left(8 a^4 B+32 a^3 A b+24 a^2 b^2 B+16 a A b^3+3 b^4 B\right) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{b (7 a B+4 A b) \tan (c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{b B \tan (c+d x) (a+b \sec (c+d x))^3}{4 d}",1,"(24*a^4*A*d*x + 3*(32*a^3*A*b + 16*a*A*b^3 + 8*a^4*B + 24*a^2*b^2*B + 3*b^4*B)*ArcTanh[Sin[c + d*x]] + 3*b*(8*(6*a^2*A*b + A*b^3 + 4*a^3*B + 4*a*b^2*B) + b*(16*a*A*b + 24*a^2*B + 3*b^2*B)*Sec[c + d*x] + 2*b^3*B*Sec[c + d*x]^3)*Tan[c + d*x] + 8*b^3*(A*b + 4*a*B)*Tan[c + d*x]^3)/(24*d)","A",1
305,1,1051,195,6.2944435,"\int \cos (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{\left(-A b^4-4 a B b^3-12 a^2 A b^2-8 a^3 B b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{2 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac{\left(A b^4+4 a B b^3+12 a^2 A b^2+8 a^3 B b\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{2 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac{a^3 (4 A b+a B) (c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac{b^4 B (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x)}{6 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \left(B \sin \left(\frac{1}{2} (c+d x)\right) b^4+6 a A \sin \left(\frac{1}{2} (c+d x)\right) b^3+9 a^2 B \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) \cos ^5(c+d x)}{3 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{2 (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \left(B \sin \left(\frac{1}{2} (c+d x)\right) b^4+6 a A \sin \left(\frac{1}{2} (c+d x)\right) b^3+9 a^2 B \sin \left(\frac{1}{2} (c+d x)\right) b^2\right) \cos ^5(c+d x)}{3 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{a^4 A (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \sin (c+d x) \cos ^5(c+d x)}{d (b+a \cos (c+d x))^4 (B+A \cos (c+d x))}+\frac{\left(3 A b^4+B b^4+12 a B b^3\right) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{12 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{\left(-3 A b^4-B b^4-12 a B b^3\right) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \cos ^5(c+d x)}{12 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{b^4 B (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5(c+d x)}{6 d (b+a \cos (c+d x))^4 (B+A \cos (c+d x)) \left(\cos \left(\frac{1}{2} (c+d x)\right)+\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}","a^3 x (a B+4 A b)-\frac{b^2 \left(6 a^2 A-8 a b B-3 A b^2\right) \tan (c+d x) \sec (c+d x)}{6 d}-\frac{b \left(6 a^3 A-17 a^2 b B-12 a A b^2-2 b^3 B\right) \tan (c+d x)}{3 d}+\frac{b \left(8 a^3 B+12 a^2 A b+4 a b^2 B+A b^3\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b (3 a A-b B) \tan (c+d x) (a+b \sec (c+d x))^2}{3 d}+\frac{a A \sin (c+d x) (a+b \sec (c+d x))^3}{d}",1,"(a^3*(4*A*b + a*B)*(c + d*x)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])) + ((-12*a^2*A*b^2 - A*b^4 - 8*a^3*b*B - 4*a*b^3*B)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(2*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])) + ((12*a^2*A*b^2 + A*b^4 + 8*a^3*b*B + 4*a*b^3*B)*Cos[c + d*x]^5*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(2*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])) + ((3*A*b^4 + 12*a*b^3*B + b^4*B)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(12*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (b^4*B*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sin[(c + d*x)/2])/(6*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (b^4*B*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sin[(c + d*x)/2])/(6*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((-3*A*b^4 - 12*a*b^3*B - b^4*B)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]))/(12*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(6*a*A*b^3*Sin[(c + d*x)/2] + 9*a^2*b^2*B*Sin[(c + d*x)/2] + b^4*B*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*(6*a*A*b^3*Sin[(c + d*x)/2] + 9*a^2*b^2*B*Sin[(c + d*x)/2] + b^4*B*Sin[(c + d*x)/2]))/(3*d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (a^4*A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x]))","B",1
306,1,310,209,2.009953,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 A \sin (2 (c+d x))+4 a^3 (a B+4 A b) \sin (c+d x)+2 a^2 (c+d x) \left(a^2 A+8 a b B+12 A b^2\right)-2 b^2 \left(12 a^2 B+8 a A b+b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 b^2 \left(12 a^2 B+8 a A b+b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 b^3 (4 a B+A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b^3 (4 a B+A b) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{b^4 B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^4 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 d}","\frac{b^2 \left(12 a^2 B+8 a A b+b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{b^2 \left(2 a^2 B+6 a A b-b^2 B\right) \tan (c+d x) \sec (c+d x)}{2 d}+\frac{1}{2} a^2 x \left(a^2 A+8 a b B+12 A b^2\right)-\frac{b \left(4 a^3 B+13 a^2 A b-8 a b^2 B-2 A b^3\right) \tan (c+d x)}{2 d}+\frac{a (2 a B+5 A b) \sin (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{a A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^3}{2 d}",1,"(2*a^2*(a^2*A + 12*A*b^2 + 8*a*b*B)*(c + d*x) - 2*b^2*(8*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*b^2*(8*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^4*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*b^3*(A*b + 4*a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^4*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b^3*(A*b + 4*a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 4*a^3*(4*A*b + a*B)*Sin[c + d*x] + a^4*A*Sin[2*(c + d*x)])/(4*d)","A",1
307,1,257,198,1.118628,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{a^4 A \sin (3 (c+d x))+3 a^3 (a B+4 A b) \sin (2 (c+d x))+3 a^2 \left(3 a^2 A+16 a b B+24 A b^2\right) \sin (c+d x)+6 a (c+d x) \left(a^3 B+4 a^2 A b+12 a b^2 B+8 A b^3\right)-12 b^3 (4 a B+A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+12 b^3 (4 a B+A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{12 b^4 B \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{12 b^4 B \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}}{12 d}","\frac{a^2 \left(2 a^2 A+9 a b B+9 A b^2\right) \sin (c+d x)}{3 d}-\frac{b^2 \left(3 a^2 B+8 a A b-6 b^2 B\right) \tan (c+d x)}{6 d}+\frac{1}{2} a x \left(a^3 B+4 a^2 A b+12 a b^2 B+8 A b^3\right)+\frac{b^3 (4 a B+A b) \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a (a B+2 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^2}{2 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^3}{3 d}",1,"(6*a*(4*a^2*A*b + 8*A*b^3 + a^3*B + 12*a*b^2*B)*(c + d*x) - 12*b^3*(A*b + 4*a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 12*b^3*(A*b + 4*a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (12*b^4*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (12*b^4*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]) + 3*a^2*(3*a^2*A + 24*A*b^2 + 16*a*b*B)*Sin[c + d*x] + 3*a^3*(4*A*b + a*B)*Sin[2*(c + d*x)] + a^4*A*Sin[3*(c + d*x)])/(12*d)","A",1
308,1,210,216,0.6156673,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{3 a^4 A \sin (4 (c+d x))+8 a^3 (a B+4 A b) \sin (3 (c+d x))+24 a^2 \left(a^2 A+4 a b B+6 A b^2\right) \sin (2 (c+d x))+24 a \left(3 a^3 B+12 a^2 A b+24 a b^2 B+16 A b^3\right) \sin (c+d x)+12 (c+d x) \left(3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right)-96 b^4 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+96 b^4 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{96 d}","\frac{a^2 \left(9 a^2 A+32 a b B+26 A b^2\right) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{a \left(4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right) \sin (c+d x)}{6 d}+\frac{1}{8} x \left(3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right)+\frac{a (4 a B+7 A b) \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^2}{12 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{4 d}+\frac{b^4 B \tanh ^{-1}(\sin (c+d x))}{d}",1,"(12*(3*a^4*A + 24*a^2*A*b^2 + 8*A*b^4 + 16*a^3*b*B + 32*a*b^3*B)*(c + d*x) - 96*b^4*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*b^4*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 24*a*(12*a^2*A*b + 16*A*b^3 + 3*a^3*B + 24*a*b^2*B)*Sin[c + d*x] + 24*a^2*(a^2*A + 6*A*b^2 + 4*a*b*B)*Sin[2*(c + d*x)] + 8*a^3*(4*A*b + a*B)*Sin[3*(c + d*x)] + 3*a^4*A*Sin[4*(c + d*x)])/(96*d)","A",1
309,1,263,258,0.6493626,"\int \cos ^5(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{50 a^4 A \sin (3 (c+d x))+6 a^4 A \sin (5 (c+d x))+15 a^4 B \sin (4 (c+d x))+180 a^4 B c+180 a^4 B d x+60 a^3 A b \sin (4 (c+d x))+720 a^3 A b c+720 a^3 A b d x+160 a^3 b B \sin (3 (c+d x))+240 a^2 A b^2 \sin (3 (c+d x))+1440 a^2 b^2 B c+1440 a^2 b^2 B d x+120 a \left(a^3 B+4 a^2 A b+6 a b^2 B+4 A b^3\right) \sin (2 (c+d x))+60 \left(5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right) \sin (c+d x)+960 a A b^3 c+960 a A b^3 d x+480 b^4 B c+480 b^4 B d x}{480 d}","\frac{a^2 \left(8 a^2 A+25 a b B+18 A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{30 d}+\frac{a \left(15 a^3 B+60 a^2 A b+110 a b^2 B+56 A b^3\right) \sin (c+d x) \cos (c+d x)}{40 d}+\frac{\left(8 a^4 A+40 a^3 b B+60 a^2 A b^2+60 a b^3 B+15 A b^4\right) \sin (c+d x)}{15 d}+\frac{1}{8} x \left(3 a^4 B+12 a^3 A b+24 a^2 b^2 B+16 a A b^3+8 b^4 B\right)+\frac{a (5 a B+8 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{20 d}+\frac{a A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}",1,"(720*a^3*A*b*c + 960*a*A*b^3*c + 180*a^4*B*c + 1440*a^2*b^2*B*c + 480*b^4*B*c + 720*a^3*A*b*d*x + 960*a*A*b^3*d*x + 180*a^4*B*d*x + 1440*a^2*b^2*B*d*x + 480*b^4*B*d*x + 60*(5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*Sin[c + d*x] + 120*a*(4*a^2*A*b + 4*A*b^3 + a^3*B + 6*a*b^2*B)*Sin[2*(c + d*x)] + 50*a^4*A*Sin[3*(c + d*x)] + 240*a^2*A*b^2*Sin[3*(c + d*x)] + 160*a^3*b*B*Sin[3*(c + d*x)] + 60*a^3*A*b*Sin[4*(c + d*x)] + 15*a^4*B*Sin[4*(c + d*x)] + 6*a^4*A*Sin[5*(c + d*x)])/(480*d)","A",1
310,1,333,309,1.241982,"\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{45 a^4 A \sin (4 (c+d x))+5 a^4 A \sin (6 (c+d x))+300 a^4 A c+300 a^4 A d x+100 a^4 B \sin (3 (c+d x))+12 a^4 B \sin (5 (c+d x))+400 a^3 A b \sin (3 (c+d x))+48 a^3 A b \sin (5 (c+d x))+120 a^3 b B \sin (4 (c+d x))+1440 a^3 b B c+1440 a^3 b B d x+180 a^2 A b^2 \sin (4 (c+d x))+2160 a^2 A b^2 c+2160 a^2 A b^2 d x+480 a^2 b^2 B \sin (3 (c+d x))+120 \left(5 a^4 B+20 a^3 A b+36 a^2 b^2 B+24 a A b^3+8 b^4 B\right) \sin (c+d x)+15 \left(15 a^4 A+64 a^3 b B+96 a^2 A b^2+64 a b^3 B+16 A b^4\right) \sin (2 (c+d x))+320 a A b^3 \sin (3 (c+d x))+1920 a b^3 B c+1920 a b^3 B d x+480 A b^4 c+480 A b^4 d x}{960 d}","\frac{a^2 \left(25 a^2 A+72 a b B+48 A b^2\right) \sin (c+d x) \cos ^3(c+d x)}{120 d}-\frac{a \left(4 a^3 B+16 a^2 A b+27 a b^2 B+13 A b^3\right) \sin ^3(c+d x)}{15 d}+\frac{\left(12 a^4 B+48 a^3 A b+87 a^2 b^2 B+53 a A b^3+15 b^4 B\right) \sin (c+d x)}{15 d}+\frac{\left(5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right) \sin (c+d x) \cos (c+d x)}{16 d}+\frac{1}{16} x \left(5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right)+\frac{a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac{a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}",1,"(300*a^4*A*c + 2160*a^2*A*b^2*c + 480*A*b^4*c + 1440*a^3*b*B*c + 1920*a*b^3*B*c + 300*a^4*A*d*x + 2160*a^2*A*b^2*d*x + 480*A*b^4*d*x + 1440*a^3*b*B*d*x + 1920*a*b^3*B*d*x + 120*(20*a^3*A*b + 24*a*A*b^3 + 5*a^4*B + 36*a^2*b^2*B + 8*b^4*B)*Sin[c + d*x] + 15*(15*a^4*A + 96*a^2*A*b^2 + 16*A*b^4 + 64*a^3*b*B + 64*a*b^3*B)*Sin[2*(c + d*x)] + 400*a^3*A*b*Sin[3*(c + d*x)] + 320*a*A*b^3*Sin[3*(c + d*x)] + 100*a^4*B*Sin[3*(c + d*x)] + 480*a^2*b^2*B*Sin[3*(c + d*x)] + 45*a^4*A*Sin[4*(c + d*x)] + 180*a^2*A*b^2*Sin[4*(c + d*x)] + 120*a^3*b*B*Sin[4*(c + d*x)] + 48*a^3*A*b*Sin[5*(c + d*x)] + 12*a^4*B*Sin[5*(c + d*x)] + 5*a^4*A*Sin[6*(c + d*x)])/(960*d)","A",1
311,1,422,187,3.0075301,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{\frac{4 b \left(3 a^2 B-3 a A b+2 b^2 B\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b \left(3 a^2 B-3 a A b+2 b^2 B\right) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+6 \left(2 a^2+b^2\right) (a B-A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-6 \left(2 a^2+b^2\right) (a B-A b) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{24 a^3 (A b-a B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{b^2 (B (b-3 a)+3 A b)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 (B (b-3 a)+3 A b)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}+\frac{2 b^3 B \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^3}+\frac{2 b^3 B \sin \left(\frac{1}{2} (c+d x)\right)}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^3}}{12 b^4 d}","-\frac{2 a^3 (A b-a B) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d \sqrt{a-b} \sqrt{a+b}}+\frac{\left(2 a^2+b^2\right) (A b-a B) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}-\frac{\left(-3 a^2 B+3 a A b-2 b^2 B\right) \tan (c+d x)}{3 b^3 d}+\frac{(A b-a B) \tan (c+d x) \sec (c+d x)}{2 b^2 d}+\frac{B \tan (c+d x) \sec ^2(c+d x)}{3 b d}",1,"((24*a^3*(A*b - a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 6*(2*a^2 + b^2)*(-(A*b) + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 6*(2*a^2 + b^2)*(-(A*b) + a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*(3*A*b + (-3*a + b)*B))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (2*b^3*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (4*b*(-3*a*A*b + 3*a^2*B + 2*b^2*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) + (2*b^3*B*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3 - (b^2*(3*A*b + (-3*a + b)*B))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b*(-3*a*A*b + 3*a^2*B + 2*b^2*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(12*b^4*d)","B",1
312,1,300,143,1.9167675,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{\frac{8 a^2 (a B-A b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-2 \left(2 a^2 B-2 a A b+b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \left(2 a^2 B-2 a A b+b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{4 b (A b-a B) \sin \left(\frac{1}{2} (c+d x)\right)}{\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)}+\frac{4 b (A b-a B) \sin \left(\frac{1}{2} (c+d x)\right)}{\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)}+\frac{b^2 B}{\left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{b^2 B}{\left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}}{4 b^3 d}","\frac{2 a^2 (A b-a B) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{\left(-2 a^2 B+2 a A b-b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^3 d}+\frac{(A b-a B) \tan (c+d x)}{b^2 d}+\frac{B \tan (c+d x) \sec (c+d x)}{2 b d}",1,"((8*a^2*(-(A*b) + a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 2*(-2*a*A*b + 2*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*(-2*a*A*b + 2*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (b^2*B)/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (4*b*(A*b - a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]) - (b^2*B)/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 + (4*b*(A*b - a*B)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))/(4*b^3*d)","B",1
313,1,130,98,0.7113897,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 a (a B-A b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}-(A b-a B) \left(\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)+b B \tan (c+d x)}{b^2 d}","\frac{(A b-a B) \tanh ^{-1}(\sin (c+d x))}{b^2 d}-\frac{2 a (A b-a B) \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 \tan (c+d x)}{b d}",1,"((-2*a*(-(A*b) + a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - (A*b - a*B)*(Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]) + b*B*Tan[c + d*x])/(b^2*d)","A",1
314,1,112,76,0.1879784,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 (a B-A b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+B \left(\log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-\log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{b d}","\frac{2 (A b-a B) \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{B \tanh ^{-1}(\sin (c+d x))}{b d}",1,"((2*(-(A*b) + a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + B*(-Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]))/(b*d)","A",1
315,1,68,67,0.1333289,"\int \frac{A+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 (A b-a B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+A (c+d x)}{a d}","\frac{A x}{a}-\frac{2 (A b-a B) \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,"(A*(c + d*x) + (2*(A*b - a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2])/(a*d)","A",1
316,1,85,90,0.2259232,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{-\frac{2 b (A b-a B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+(c+d x) (a B-A b)+a A \sin (c+d x)}{a^2 d}","\frac{2 b (A b-a B) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{x (A b-a B)}{a^2}+\frac{A \sin (c+d x)}{a d}",1,"((-(A*b) + a*B)*(c + d*x) - (2*b*(A*b - a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + a*A*Sin[c + d*x])/(a^2*d)","A",1
317,1,121,134,0.3598912,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{2 (c+d x) \left(a^2 A-2 a b B+2 A b^2\right)+\frac{8 b^2 (A b-a B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+a^2 A \sin (2 (c+d x))+4 a (a B-A b) \sin (c+d x)}{4 a^3 d}","-\frac{2 b^2 (A b-a B) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \sin (c+d x)}{a^2 d}+\frac{x \left(a^2 A-2 a b B+2 A b^2\right)}{2 a^3}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d}",1,"(2*(a^2*A + 2*A*b^2 - 2*a*b*B)*(c + d*x) + (8*b^2*(A*b - a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 4*a*(-(A*b) + a*B)*Sin[c + d*x] + a^2*A*Sin[2*(c + d*x)])/(4*a^3*d)","A",1
318,1,152,178,0.5369583,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{a^3 A \sin (3 (c+d x))+6 \left(a^2+2 b^2\right) (c+d x) (a B-A b)+3 a \left(3 a^2 A-4 a b B+4 A b^2\right) \sin (c+d x)-\frac{24 b^3 (A b-a B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+3 a^2 (a B-A b) \sin (2 (c+d x))}{12 a^4 d}","\frac{2 b^3 (A b-a B) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \sin (c+d x) \cos (c+d x)}{2 a^2 d}-\frac{x \left(a^2+2 b^2\right) (A b-a B)}{2 a^4}+\frac{\left(2 a^2 A-3 a b B+3 A b^2\right) \sin (c+d x)}{3 a^3 d}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d}",1,"(6*(a^2 + 2*b^2)*(-(A*b) + a*B)*(c + d*x) - (24*b^3*(A*b - a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 3*a*(3*a^2*A + 4*A*b^2 - 4*a*b*B)*Sin[c + d*x] + 3*a^2*(-(A*b) + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^4*d)","A",1
319,1,202,240,0.6713636,"\int \frac{\cos ^4(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{3 a^4 A \sin (4 (c+d x))+8 a^3 (a B-A b) \sin (3 (c+d x))+24 a^2 \left(a^2 A-a b B+A b^2\right) \sin (2 (c+d x))+24 a \left(3 a^2+4 b^2\right) (a B-A b) \sin (c+d x)+\frac{192 b^4 (A b-a B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+12 (c+d x) \left(3 a^4 A-4 a^3 b B+4 a^2 A b^2-8 a b^3 B+8 A b^4\right)}{96 a^5 d}","-\frac{2 b^4 (A b-a B) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^5 d \sqrt{a-b} \sqrt{a+b}}-\frac{(A b-a B) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d}-\frac{\left(2 a^2+3 b^2\right) (A b-a B) \sin (c+d x)}{3 a^4 d}+\frac{\left(3 a^2 A-4 a b B+4 A b^2\right) \sin (c+d x) \cos (c+d x)}{8 a^3 d}+\frac{x \left(3 a^4 A-4 a^3 b B+4 a^2 A b^2-8 a b^3 B+8 A b^4\right)}{8 a^5}+\frac{A \sin (c+d x) \cos ^3(c+d x)}{4 a d}",1,"(12*(3*a^4*A + 4*a^2*A*b^2 + 8*A*b^4 - 4*a^3*b*B - 8*a*b^3*B)*(c + d*x) + (192*b^4*(A*b - a*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + 24*a*(3*a^2 + 4*b^2)*(-(A*b) + a*B)*Sin[c + d*x] + 24*a^2*(a^2*A + A*b^2 - a*b*B)*Sin[2*(c + d*x)] + 8*a^3*(-(A*b) + a*B)*Sin[3*(c + d*x)] + 3*a^4*A*Sin[4*(c + d*x)])/(96*a^5*d)","A",1
320,1,438,272,6.297327,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\left(-6 a^2 B+4 a A b-b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{\left(6 a^2 B-4 a A b+b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 b^4 d}+\frac{a^4 B \sin (c+d x)-a^3 A b \sin (c+d x)}{b^3 d (b-a) (a+b) (a \cos (c+d x)+b)}-\frac{2 a^2 \left(3 a^3 B-2 a^2 A b-4 a b^2 B+3 A b^3\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2} \left(b^2-a^2\right)}+\frac{A b \sin \left(\frac{1}{2} (c+d x)\right)-2 a B \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{A b \sin \left(\frac{1}{2} (c+d x)\right)-2 a B \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{B}{4 b^2 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)^2}-\frac{B}{4 b^2 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)^2}","\frac{a (A b-a B) \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 B+2 a A b+b^2 B\right) \tan (c+d x) \sec (c+d x)}{2 b^2 d \left(a^2-b^2\right)}-\frac{\left(-6 a^2 B+4 a A b-b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^4 d}+\frac{\left(-3 a^3 B+2 a^2 A b+2 a b^2 B-A b^3\right) \tan (c+d x)}{b^3 d \left(a^2-b^2\right)}+\frac{2 a^2 \left(-3 a^3 B+2 a^2 A b+4 a b^2 B-3 A b^3\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(-2*a^2*(-2*a^2*A*b + 3*A*b^3 + 3*a^3*B - 4*a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*(-a^2 + b^2)*d) + ((4*a*A*b - 6*a^2*B - b^2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(2*b^4*d) + ((-4*a*A*b + 6*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(2*b^4*d) + B/(4*b^2*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) - B/(4*b^2*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (A*b*Sin[(c + d*x)/2] - 2*a*B*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (A*b*Sin[(c + d*x)/2] - 2*a*B*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (-(a^3*A*b*Sin[c + d*x]) + a^4*B*Sin[c + d*x])/(b^3*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x]))","A",0
321,1,240,164,2.2311151,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{a^2 b (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-\frac{2 a \left(2 a^3 B-a^2 A b-3 a b^2 B+2 A b^3\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+2 a B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)-2 a B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)-A b \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+A b \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+b B \tan (c+d x)}{b^3 d}","-\frac{a^2 (A b-a B) \tan (c+d x)}{b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{2 a \left(-2 a^3 B+a^2 A b+3 a b^2 B-2 A b^3\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 b-2 a B) \tanh ^{-1}(\sin (c+d x))}{b^3 d}+\frac{B \tan (c+d x)}{b^2 d}",1,"((-2*a*(-(a^2*A*b) + 2*A*b^3 + 2*a^3*B - 3*a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - A*b*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*a*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + A*b*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 2*a*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a^2*b*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + b*B*Tan[c + d*x])/(b^3*d)","A",1
322,1,191,131,0.7152281,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\cos (c+d x) (A+B \sec (c+d x)) \left(\frac{2 \left(a B \left(a^2-2 b^2\right)+A b^3\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a b (a B-A b) \sin (c+d x)}{(b-a) (a+b) (a \cos (c+d x)+b)}-B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{b^2 d (A \cos (c+d x)+B)}","-\frac{2 \left(a^3 B-2 a b^2 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)}{b^2 d (a-b)^{3/2} (a+b)^{3/2}}+\frac{a (A b-a B) \tan (c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{B \tanh ^{-1}(\sin (c+d x))}{b^2 d}",1,"(Cos[c + d*x]*(A + B*Sec[c + d*x])*((2*(A*b^3 + a*(a^2 - 2*b^2)*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b*(-(A*b) + a*B)*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c + d*x]))))/(b^2*d*(B + A*Cos[c + d*x]))","A",1
323,1,97,100,0.370038,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{(a B-A b) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}-\frac{2 (a A-b B) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}}{d}","\frac{2 (a A-b B) \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{(A b-a B) \tan (c+d x)}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}",1,"((-2*(a*A - b*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + ((-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])))/d","A",1
324,1,155,124,0.6688491,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{A b \left(a^2-b^2\right) (c+d x)+a A \left(a^2-b^2\right) (c+d x) \cos (c+d x)-a b (a B-A b) \sin (c+d x)}{a \cos (c+d x)+b}-\frac{2 \left(a^3 B-2 a^2 A b+A b^3\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}}{a^2 d (a-b) (a+b)}","\frac{b (A b-a B) \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 \left(a^3 (-B)+2 a^2 A 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}}",1,"((-2*(-2*a^2*A*b + A*b^3 + a^3*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (A*b*(a^2 - b^2)*(c + d*x) + a*A*(a^2 - b^2)*(c + d*x)*Cos[c + d*x] - a*b*(-(A*b) + a*B)*Sin[c + d*x])/(b + a*Cos[c + d*x]))/(a^2*(a - b)*(a + b)*d)","A",1
325,1,221,180,1.1306977,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{(a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(\frac{2 b \left(2 a^3 B-3 a^2 A b-a b^2 B+2 A b^3\right) \sec (c+d x) (a \cos (c+d x)+b) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{a b^2 (a B-A b) \tan (c+d x)}{(a-b) (a+b)}+(c+d x) (a B-2 A b) \sec (c+d x) (a \cos (c+d x)+b)+a A \tan (c+d x) (a \cos (c+d x)+b)\right)}{a^3 d (a+b \sec (c+d x))^2 (A \cos (c+d x)+B)}","-\frac{x (2 A b-a B)}{a^3}+\frac{\left(a^2 A+a b B-2 A b^2\right) \sin (c+d x)}{a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{2 b \left(-2 a^3 B+3 a^2 A b+a b^2 B-2 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^3 d (a-b)^{3/2} (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])*(A + B*Sec[c + d*x])*((-2*A*b + a*B)*(c + d*x)*(b + a*Cos[c + d*x])*Sec[c + d*x] + (2*b*(-3*a^2*A*b + 2*A*b^3 + 2*a^3*B - a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])*Sec[c + d*x])/(a^2 - b^2)^(3/2) + (a*b^2*(-(A*b) + a*B)*Tan[c + d*x])/((a - b)*(a + b)) + a*A*(b + a*Cos[c + d*x])*Tan[c + d*x]))/(a^3*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^2)","A",1
326,1,184,261,1.1382208,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{2 (c+d x) \left(a^2 A-4 a b B+6 A b^2\right)+a^2 A \sin (2 (c+d x))-\frac{8 b^2 \left(3 a^3 B-4 a^2 A b-2 a b^2 B+3 A b^3\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}-\frac{4 a b^3 (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}+4 a (a B-2 A b) \sin (c+d x)}{4 a^4 d}","\frac{\left(a^2 A+2 a b B-3 A b^2\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \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-4 a b B+6 A b^2\right)}{2 a^4}-\frac{\left(a^3 (-B)+2 a^2 A b+2 a b^2 B-3 A b^3\right) \sin (c+d x)}{a^3 d \left(a^2-b^2\right)}-\frac{2 b^2 \left(-3 a^3 B+4 a^2 A b+2 a b^2 B-3 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^4 d (a-b)^{3/2} (a+b)^{3/2}}",1,"(2*(a^2*A + 6*A*b^2 - 4*a*b*B)*(c + d*x) - (8*b^2*(-4*a^2*A*b + 3*A*b^3 + 3*a^3*B - 2*a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 4*a*(-2*A*b + a*B)*Sin[c + d*x] - (4*a*b^3*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + a^2*A*Sin[2*(c + d*x)])/(4*a^4*d)","A",1
327,1,224,346,1.3626364,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{a^3 A \sin (3 (c+d x))+3 a \left(3 a^2 A-8 a b B+12 A b^2\right) \sin (c+d x)+3 a^2 (a B-2 A b) \sin (2 (c+d x))+6 (c+d x) \left(a^3 B-2 a^2 A b+6 a b^2 B-8 A b^3\right)+\frac{24 b^3 \left(4 a^3 B-5 a^2 A b-3 a b^2 B+4 A b^3\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{3/2}}+\frac{12 a b^4 (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)}}{12 a^5 d}","\frac{\left(a^2 A+3 a b B-4 A b^2\right) \sin (c+d x) \cos ^2(c+d x)}{3 a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{\left(a^3 (-B)+2 a^2 A b+3 a b^2 B-4 A b^3\right) \sin (c+d x) \cos (c+d x)}{2 a^3 d \left(a^2-b^2\right)}+\frac{2 b^3 \left(-4 a^3 B+5 a^2 A b+3 a b^2 B-4 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^5 d (a-b)^{3/2} (a+b)^{3/2}}-\frac{x \left(a^3 (-B)+2 a^2 A b-6 a b^2 B+8 A b^3\right)}{2 a^5}+\frac{\left(2 a^4 A-6 a^3 b B+7 a^2 A b^2+9 a b^3 B-12 A b^4\right) \sin (c+d x)}{3 a^4 d \left(a^2-b^2\right)}",1,"(6*(-2*a^2*A*b - 8*A*b^3 + a^3*B + 6*a*b^2*B)*(c + d*x) + (24*b^3*(-5*a^2*A*b + 4*A*b^3 + 4*a^3*B - 3*a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 3*a*(3*a^2*A + 12*A*b^2 - 8*a*b*B)*Sin[c + d*x] + (12*a*b^4*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])) + 3*a^2*(-2*A*b + a*B)*Sin[2*(c + d*x)] + a^3*A*Sin[3*(c + d*x)])/(12*a^5*d)","A",1
328,1,507,407,3.080825,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{-8 \left(12 a^2 B-6 a A b+b^2 B\right) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+8 \left(12 a^2 B-6 a A b+b^2 B\right) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)+\frac{16 a^2 \left(12 a^5 B-6 a^4 A b-29 a^3 b^2 B+15 a^2 A b^3+20 a b^4 B-12 A b^5\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{2 b \tan (c+d x) \sec (c+d x) \left(-12 a^7 B \cos (3 (c+d x))+6 a^6 A b \cos (3 (c+d x))-36 a^6 b B+18 a^5 A b^2+21 a^5 b^2 B \cos (3 (c+d x))-11 a^4 A b^3 \cos (3 (c+d x))+68 a^4 b^3 B-32 a^3 A b^4-6 a^3 b^4 B \cos (3 (c+d x))+2 a^2 A b^5 \cos (3 (c+d x))-30 a^2 b^5 B-2 a b \left(18 a^5 B-9 a^4 A b-32 a^3 b^2 B+16 a^2 A b^3+11 a b^4 B-4 A b^5\right) \cos (2 (c+d x))+\left(-36 a^7 B+18 a^6 A b+47 a^5 b^2 B-25 a^4 A b^3+14 a^3 b^4 B-10 a^2 A b^5-16 a b^6 B+8 A b^7\right) \cos (c+d x)+8 a A b^6+4 b^7 B\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{16 b^5 d}","\frac{a (A b-a B) \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(-12 a^2 B+6 a A b-b^2 B\right) \tanh ^{-1}(\sin (c+d x))}{2 b^5 d}+\frac{a \left(-4 a^3 B+2 a^2 A b+7 a b^2 B-5 A b^3\right) \tan (c+d x) \sec ^2(c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(-6 a^4 B+3 a^3 A b+10 a^2 b^2 B-6 a A b^3-b^4 B\right) \tan (c+d x) \sec (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-12 a^5 B+6 a^4 A b+21 a^3 b^2 B-11 a^2 A b^3-6 a b^4 B+2 A b^5\right) \tan (c+d x)}{2 b^4 d \left(a^2-b^2\right)^2}+\frac{a^2 \left(-12 a^5 B+6 a^4 A b+29 a^3 b^2 B-15 a^2 A b^3-20 a b^4 B+12 A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((16*a^2*(-6*a^4*A*b + 15*a^2*A*b^3 - 12*A*b^5 + 12*a^5*B - 29*a^3*b^2*B + 20*a*b^4*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 8*(-6*a*A*b + 12*a^2*B + b^2*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*(18*a^5*A*b^2 - 32*a^3*A*b^4 + 8*a*A*b^6 - 36*a^6*b*B + 68*a^4*b^3*B - 30*a^2*b^5*B + 4*b^7*B + (18*a^6*A*b - 25*a^4*A*b^3 - 10*a^2*A*b^5 + 8*A*b^7 - 36*a^7*B + 47*a^5*b^2*B + 14*a^3*b^4*B - 16*a*b^6*B)*Cos[c + d*x] - 2*a*b*(-9*a^4*A*b + 16*a^2*A*b^3 - 4*A*b^5 + 18*a^5*B - 32*a^3*b^2*B + 11*a*b^4*B)*Cos[2*(c + d*x)] + 6*a^6*A*b*Cos[3*(c + d*x)] - 11*a^4*A*b^3*Cos[3*(c + d*x)] + 2*a^2*A*b^5*Cos[3*(c + d*x)] - 12*a^7*B*Cos[3*(c + d*x)] + 21*a^5*b^2*B*Cos[3*(c + d*x)] - 6*a^3*b^4*B*Cos[3*(c + d*x)])*Sec[c + d*x]*Tan[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(16*b^5*d)","A",1
329,1,418,289,6.5151113,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{a^2 A b \sin (c+d x)-a^3 B \sin (c+d x)}{2 b^2 d (b-a) (a+b) (a \cos (c+d x)+b)^2}+\frac{4 a^5 B \sin (c+d x)-2 a^4 A b \sin (c+d x)-7 a^3 b^2 B \sin (c+d x)+5 a^2 A b^3 \sin (c+d x)}{2 b^3 d (b-a)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{a \left(-6 a^5 B+2 a^4 A b+15 a^3 b^2 B-5 a^2 A b^3-12 a b^4 B+6 A b^5\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{b^4 d \sqrt{a^2-b^2} \left(b^2-a^2\right)^2}+\frac{(3 a B-A b) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}{b^4 d}+\frac{(A b-3 a B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{b^4 d}+\frac{B \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)}+\frac{B \sin \left(\frac{1}{2} (c+d x)\right)}{b^3 d \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}","\frac{a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)}-\frac{a^2 \left(-3 a^3 B+a^2 A b+6 a b^2 B-4 A b^3\right) \tan (c+d x)}{2 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{a \left(-6 a^5 B+2 a^4 A b+15 a^3 b^2 B-5 a^2 A b^3-12 a b^4 B+6 A b^5\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-3 a B) \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(a*(2*a^4*A*b - 5*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 15*a^3*b^2*B - 12*a*b^4*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]*(-a^2 + b^2)^2*d) + ((-(A*b) + 3*a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]])/(b^4*d) + ((A*b - 3*a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])/(b^4*d) + (B*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (B*Sin[(c + d*x)/2])/(b^3*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])) + (a^2*A*b*Sin[c + d*x] - a^3*B*Sin[c + d*x])/(2*b^2*(-a + b)*(a + b)*d*(b + a*Cos[c + d*x])^2) + (-2*a^4*A*b*Sin[c + d*x] + 5*a^2*A*b^3*Sin[c + d*x] + 4*a^5*B*Sin[c + d*x] - 7*a^3*b^2*B*Sin[c + d*x])/(2*b^3*(-a + b)^2*(a + b)^2*d*(b + a*Cos[c + d*x]))","A",0
330,1,270,220,2.0539829,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\cos (c+d x) (A+B \sec (c+d x)) \left(\frac{a b \left(-2 a^3 B+5 a b^2 B-3 A b^3\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{2 \left(2 a^5 B-5 a^3 b^2 B-a^2 A b^3+6 a b^4 B-2 A b^5\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^2 (a B-A b) \sin (c+d x)}{(b-a) (a+b) (a \cos (c+d x)+b)^2}-2 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{2 b^3 d (A \cos (c+d x)+B)}","-\frac{a^2 (A b-a B) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(-3 a^3 B+a^2 A b+6 a b^2 B-4 A b^3\right) \tan (c+d x)}{2 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(-2 a^5 B+5 a^3 b^2 B+a^2 A b^3-6 a b^4 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)}{b^3 d (a-b)^{5/2} (a+b)^{5/2}}+\frac{B \tanh ^{-1}(\sin (c+d x))}{b^3 d}",1,"(Cos[c + d*x]*(A + B*Sec[c + d*x])*((2*(-(a^2*A*b^3) - 2*A*b^5 + 2*a^5*B - 5*a^3*b^2*B + 6*a*b^4*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - 2*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 2*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (a*b^2*(-(A*b) + a*B)*Sin[c + d*x])/((-a + b)*(a + b)*(b + a*Cos[c + d*x])^2) + (a*b*(-3*A*b^3 - 2*a^3*B + 5*a*b^2*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x]))))/(2*b^3*d*(B + A*Cos[c + d*x]))","A",1
331,1,157,180,0.7308991,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{\left(2 a^2 A-3 a b B+A b^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}-\frac{2 \left(a^2 B-3 a A b+2 b^2 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{(a B-A b) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^2}}{2 d}","-\frac{\left(a^2 (-B)+3 a A b-2 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{a (A b-a B) \tan (c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(a^3 B+a^2 A b-4 a b^2 B+2 A b^3\right) \tan (c+d x)}{2 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}",1,"((-2*(-3*a*A*b + a^2*B + 2*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + ((-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + ((2*a^2*A + A*b^2 - 3*a*b*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])))/(2*d)","A",1
332,1,172,164,0.9187807,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{-\frac{2 \left(2 a^2 A-3 a b B+A b^2\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{\left(2 a^3 B-4 a^2 A b+a b^2 B+A b^3\right) \sin (c+d x)}{a (a-b)^2 (a+b)^2 (a \cos (c+d x)+b)}+\frac{b (A b-a B) \sin (c+d x)}{a (a-b) (a+b) (a \cos (c+d x)+b)^2}}{2 d}","\frac{\left(2 a^2 A-3 a b B+A b^2\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)+3 a A b-2 b^2 B\right) \tan (c+d x)}{2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{(A b-a B) \tan (c+d x)}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}",1,"((-2*(2*a^2*A + A*b^2 - 3*a*b*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b*(A*b - a*B)*Sin[c + d*x])/(a*(a - b)*(a + b)*(b + a*Cos[c + d*x])^2) + ((-4*a^2*A*b + A*b^3 + 2*a^3*B + a*b^2*B)*Sin[c + d*x])/(a*(a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])))/(2*d)","A",1
333,1,267,205,1.4657979,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(-\frac{a b \left(4 a^3 B-6 a^2 A b-a b^2 B+3 A b^3\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{2 \left(2 a^5 B-6 a^4 A b+a^3 b^2 B+5 a^2 A b^3-2 A b^5\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^2 (a B-A b) \sin (c+d x)}{(a-b) (a+b)}+2 A (c+d x) (a \cos (c+d x)+b)^2\right)}{2 a^3 d (a+b \sec (c+d x))^3 (A \cos (c+d x)+B)}","\frac{A x}{a^3}+\frac{b (A b-a B) \tan (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{b \left(-3 a^3 B+5 a^2 A b-2 A b^3\right) \tan (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(-2 a^5 B+6 a^4 A b-a^3 b^2 B-5 a^2 A b^3+2 A b^5\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^3 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(2*A*(c + d*x)*(b + a*Cos[c + d*x])^2 - (2*(-6*a^4*A*b + 5*a^2*A*b^3 - 2*A*b^5 + 2*a^5*B + a^3*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) + (a*b^2*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)) - (a*b*(-6*a^2*A*b + 3*A*b^3 + 4*a^3*B - a*b^2*B)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2)))/(2*a^3*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^3)","A",1
334,1,306,290,2.0723532,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(\frac{a b^2 \left(6 a^3 B-8 a^2 A b-3 a b^2 B+5 A b^3\right) \sin (c+d x) (a \cos (c+d x)+b)}{(a-b)^2 (a+b)^2}-\frac{2 b \left(-6 a^5 B+12 a^4 A b+5 a^3 b^2 B-15 a^2 A b^3-2 a b^4 B+6 A b^5\right) (a \cos (c+d x)+b)^2 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{a b^3 (A b-a B) \sin (c+d x)}{(a-b) (a+b)}+2 (c+d x) (a B-3 A b) (a \cos (c+d x)+b)^2+2 a A \sin (c+d x) (a \cos (c+d x)+b)^2\right)}{2 a^4 d (a+b \sec (c+d x))^3 (A \cos (c+d x)+B)}","-\frac{x (3 A b-a B)}{a^4}+\frac{b (A b-a B) \sin (c+d x)}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{b \left(-4 a^3 B+6 a^2 A b+a b^2 B-3 A b^3\right) \sin (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(2 a^4 A+5 a^3 b B-11 a^2 A b^2-2 a b^3 B+6 A b^4\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^2}+\frac{b \left(-6 a^5 B+12 a^4 A b+5 a^3 b^2 B-15 a^2 A b^3-2 a b^4 B+6 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^4 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*(2*(-3*A*b + a*B)*(c + d*x)*(b + a*Cos[c + d*x])^2 - (2*b*(12*a^4*A*b - 15*a^2*A*b^3 + 6*A*b^5 - 6*a^5*B + 5*a^3*b^2*B - 2*a*b^4*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^2)/(a^2 - b^2)^(5/2) + (a*b^3*(A*b - a*B)*Sin[c + d*x])/((a - b)*(a + b)) + (a*b^2*(-8*a^2*A*b + 5*A*b^3 + 6*a^3*B - 3*a*b^2*B)*(b + a*Cos[c + d*x])*Sin[c + d*x])/((a - b)^2*(a + b)^2) + 2*a*A*(b + a*Cos[c + d*x])^2*Sin[c + d*x]))/(2*a^4*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^3)","A",1
335,1,734,393,4.6918959,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{16 b^2 \left(-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 A b^5\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{5/2}}+\frac{2 a^8 A \sin (2 (c+d x))+a^8 A \sin (4 (c+d x))+4 a^8 A c+4 a^8 A d x+4 a^8 B \sin (c+d x)+4 a^8 B \sin (3 (c+d x))-8 a^7 A b \sin (c+d x)-8 a^7 A b \sin (3 (c+d x))+16 a^7 b B \sin (2 (c+d x))-24 a^7 b B c-24 a^7 b B d x-48 a^6 A b^2 \sin (2 (c+d x))-2 a^6 A b^2 \sin (4 (c+d x))+48 a^6 A b^2 c+48 a^6 A b^2 d x+8 a^6 b^2 B \sin (c+d x)-8 a^6 b^2 B \sin (3 (c+d x))-32 a^5 A b^3 \sin (c+d x)+16 a^5 A b^3 \sin (3 (c+d x))-64 a^5 b^3 B \sin (2 (c+d x))+130 a^4 A b^4 \sin (2 (c+d x))+a^4 A b^4 \sin (4 (c+d x))-12 a^4 A b^4 c-12 a^4 A b^4 d x-84 a^4 b^4 B \sin (c+d x)+4 a^4 b^4 B \sin (3 (c+d x))+160 a^3 A b^5 \sin (c+d x)-8 a^3 A b^5 \sin (3 (c+d x))+36 a^3 b^5 B \sin (2 (c+d x))+72 a^3 b^5 B c+72 a^3 b^5 B d x-72 a^2 A b^6 \sin (2 (c+d x))-136 a^2 A b^6 c-136 a^2 A b^6 d x+16 a b \left(a^2-b^2\right)^2 (c+d x) \left(a^2 A-6 a b B+12 A b^2\right) \cos (c+d x)+48 a^2 b^6 B \sin (c+d x)+4 \left(a^3-a b^2\right)^2 (c+d x) \left(a^2 A-6 a b B+12 A b^2\right) \cos (2 (c+d x))-96 a A b^7 \sin (c+d x)-48 a b^7 B c-48 a b^7 B d x+96 A b^8 c+96 A b^8 d x}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{16 a^5 d}","\frac{b (A b-a B) \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-6 a b B+12 A b^2\right)}{2 a^5}+\frac{b \left(-5 a^3 B+7 a^2 A b+2 a b^2 B-4 A b^3\right) \sin (c+d x) \cos (c+d x)}{2 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(a^4 A+6 a^3 b B-10 a^2 A b^2-3 a b^3 B+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{\left(-2 a^5 B+6 a^4 A b+11 a^3 b^2 B-21 a^2 A b^3-6 a b^4 B+12 A b^5\right) \sin (c+d x)}{2 a^4 d \left(a^2-b^2\right)^2}-\frac{b^2 \left(-12 a^5 B+20 a^4 A b+15 a^3 b^2 B-29 a^2 A b^3-6 a b^4 B+12 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^5 d (a-b)^{5/2} (a+b)^{5/2}}",1,"((16*b^2*(20*a^4*A*b - 29*a^2*A*b^3 + 12*A*b^5 - 12*a^5*B + 15*a^3*b^2*B - 6*a*b^4*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (4*a^8*A*c + 48*a^6*A*b^2*c - 12*a^4*A*b^4*c - 136*a^2*A*b^6*c + 96*A*b^8*c - 24*a^7*b*B*c + 72*a^3*b^5*B*c - 48*a*b^7*B*c + 4*a^8*A*d*x + 48*a^6*A*b^2*d*x - 12*a^4*A*b^4*d*x - 136*a^2*A*b^6*d*x + 96*A*b^8*d*x - 24*a^7*b*B*d*x + 72*a^3*b^5*B*d*x - 48*a*b^7*B*d*x + 16*a*b*(a^2 - b^2)^2*(a^2*A + 12*A*b^2 - 6*a*b*B)*(c + d*x)*Cos[c + d*x] + 4*(a^3 - a*b^2)^2*(a^2*A + 12*A*b^2 - 6*a*b*B)*(c + d*x)*Cos[2*(c + d*x)] - 8*a^7*A*b*Sin[c + d*x] - 32*a^5*A*b^3*Sin[c + d*x] + 160*a^3*A*b^5*Sin[c + d*x] - 96*a*A*b^7*Sin[c + d*x] + 4*a^8*B*Sin[c + d*x] + 8*a^6*b^2*B*Sin[c + d*x] - 84*a^4*b^4*B*Sin[c + d*x] + 48*a^2*b^6*B*Sin[c + d*x] + 2*a^8*A*Sin[2*(c + d*x)] - 48*a^6*A*b^2*Sin[2*(c + d*x)] + 130*a^4*A*b^4*Sin[2*(c + d*x)] - 72*a^2*A*b^6*Sin[2*(c + d*x)] + 16*a^7*b*B*Sin[2*(c + d*x)] - 64*a^5*b^3*B*Sin[2*(c + d*x)] + 36*a^3*b^5*B*Sin[2*(c + d*x)] - 8*a^7*A*b*Sin[3*(c + d*x)] + 16*a^5*A*b^3*Sin[3*(c + d*x)] - 8*a^3*A*b^5*Sin[3*(c + d*x)] + 4*a^8*B*Sin[3*(c + d*x)] - 8*a^6*b^2*B*Sin[3*(c + d*x)] + 4*a^4*b^4*B*Sin[3*(c + d*x)] + a^8*A*Sin[4*(c + d*x)] - 2*a^6*A*b^2*Sin[4*(c + d*x)] + a^4*A*b^4*Sin[4*(c + d*x)])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(16*a^5*d)","A",1
336,1,548,418,3.2473526,"\int \frac{\sec ^5(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^5*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{-\frac{48 a \left(8 a^7 B-2 a^6 A b-28 a^5 b^2 B+7 a^4 A b^3+35 a^3 b^4 B-8 a^2 A b^5-20 a b^6 B+8 A b^7\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{2 b \tan (c+d x) \left(-24 a^9 B \cos (3 (c+d x))+6 a^8 A b \cos (3 (c+d x))-120 a^8 b B+30 a^7 A b^2+68 a^7 b^2 B \cos (3 (c+d x))-17 a^6 A b^3 \cos (3 (c+d x))+318 a^6 b^3 B-90 a^5 A b^4-65 a^5 b^4 B \cos (3 (c+d x))+26 a^4 A b^5 \cos (3 (c+d x))-246 a^4 b^5 B+120 a^3 A b^6+6 a^3 b^6 B \cos (3 (c+d x))-36 a^2 b^7 B-6 a^2 b \left(20 a^6 B-5 a^5 A b-57 a^4 b^2 B+15 a^3 A b^3+53 a^2 b^4 B-20 a A b^5-6 b^6 B\right) \cos (2 (c+d x))+a \left(-72 a^8 B+18 a^7 A b+28 a^6 b^2 B-7 a^5 A b^3+305 a^4 b^4 B-50 a^3 A b^5-438 a^2 b^6 B+144 a A b^7+72 b^8 B\right) \cos (c+d x)+24 b^9 B\right)}{\left(b^2-a^2\right)^3 (a \cos (c+d x)+b)^3}-48 (A b-4 a B) \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+48 (A b-4 a B) \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{48 b^5 d}","\frac{a (A b-a B) \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{a \left(-4 a^3 B+a^2 A b+9 a b^2 B-6 A b^3\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{\left(-12 a^4 B+3 a^3 A b+23 a^2 b^2 B-8 a A b^3-6 b^4 B\right) \tan (c+d x)}{6 b^4 d \left(a^2-b^2\right)^2}-\frac{a^2 \left(-4 a^5 B+a^4 A b+11 a^3 b^2 B-2 a^2 A b^3-12 a b^4 B+6 A b^5\right) \tan (c+d x)}{2 b^4 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{a \left(-8 a^7 B+2 a^6 A b+28 a^5 b^2 B-7 a^4 A b^3-35 a^3 b^4 B+8 a^2 A b^5+20 a b^6 B-8 A b^7\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{(A b-4 a B) \tanh ^{-1}(\sin (c+d x))}{b^5 d}",1,"((-48*a*(-2*a^6*A*b + 7*a^4*A*b^3 - 8*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 28*a^5*b^2*B + 35*a^3*b^4*B - 20*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - 48*(A*b - 4*a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 48*(A*b - 4*a*B)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (2*b*(30*a^7*A*b^2 - 90*a^5*A*b^4 + 120*a^3*A*b^6 - 120*a^8*b*B + 318*a^6*b^3*B - 246*a^4*b^5*B - 36*a^2*b^7*B + 24*b^9*B + a*(18*a^7*A*b - 7*a^5*A*b^3 - 50*a^3*A*b^5 + 144*a*A*b^7 - 72*a^8*B + 28*a^6*b^2*B + 305*a^4*b^4*B - 438*a^2*b^6*B + 72*b^8*B)*Cos[c + d*x] - 6*a^2*b*(-5*a^5*A*b + 15*a^3*A*b^3 - 20*a*A*b^5 + 20*a^6*B - 57*a^4*b^2*B + 53*a^2*b^4*B - 6*b^6*B)*Cos[2*(c + d*x)] + 6*a^8*A*b*Cos[3*(c + d*x)] - 17*a^6*A*b^3*Cos[3*(c + d*x)] + 26*a^4*A*b^5*Cos[3*(c + d*x)] - 24*a^9*B*Cos[3*(c + d*x)] + 68*a^7*b^2*B*Cos[3*(c + d*x)] - 65*a^5*b^4*B*Cos[3*(c + d*x)] + 6*a^3*b^6*B*Cos[3*(c + d*x)])*Tan[c + d*x])/((-a^2 + b^2)^3*(b + a*Cos[c + d*x])^3))/(48*b^5*d)","A",1
337,1,369,310,1.9924963,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{\cos (c+d x) (A+B \sec (c+d x)) \left(\frac{24 \left(2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 B+2 A b^7\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}-\frac{2 a b \sin (c+d x) \left(-6 a^7 B-5 a^5 b^2 B+8 a^4 A b^3+38 a^3 b^4 B+a^2 A b^5+a^2 \left(-6 a^5 B+17 a^3 b^2 B+4 a^2 A b^3-26 a b^4 B+11 A b^5\right) \cos (2 (c+d x))-6 a b \left(5 a^5 B-15 a^3 b^2 B-a^2 A b^3+20 a b^4 B-9 A b^5\right) \cos (c+d x)-72 a b^6 B+36 A b^7\right)}{\left(b^2-a^2\right)^3 (a \cos (c+d x)+b)^3}-24 B \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sin \left(\frac{1}{2} (c+d x)\right)\right)+24 B \log \left(\sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)\right)}{24 b^4 d (A \cos (c+d x)+B)}","\frac{a (A b-a B) \tan (c+d x) \sec ^2(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{a^2 \left(3 a^3 B-8 a b^2 B+5 A b^3\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{a \left(9 a^5 B-28 a^3 b^2 B+a^2 A b^3+34 a b^4 B-16 A b^5\right) \tan (c+d x)}{6 b^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\left(2 a^7 B-7 a^5 b^2 B+8 a^3 b^4 B+3 a^2 A b^5-8 a b^6 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)}{b^4 d (a-b)^{7/2} (a+b)^{7/2}}+\frac{B \tanh ^{-1}(\sin (c+d x))}{b^4 d}",1,"(Cos[c + d*x]*(A + B*Sec[c + d*x])*((24*(3*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B - 7*a^5*b^2*B + 8*a^3*b^4*B - 8*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) - 24*B*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 24*B*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - (2*a*b*(8*a^4*A*b^3 + a^2*A*b^5 + 36*A*b^7 - 6*a^7*B - 5*a^5*b^2*B + 38*a^3*b^4*B - 72*a*b^6*B - 6*a*b*(-(a^2*A*b^3) - 9*A*b^5 + 5*a^5*B - 15*a^3*b^2*B + 20*a*b^4*B)*Cos[c + d*x] + a^2*(4*a^2*A*b^3 + 11*A*b^5 - 6*a^5*B + 17*a^3*b^2*B - 26*a*b^4*B)*Cos[2*(c + d*x)])*Sin[c + d*x])/((-a^2 + b^2)^3*(b + a*Cos[c + d*x])^3)))/(24*b^4*d*(B + A*Cos[c + d*x]))","A",1
338,1,226,274,2.8287907,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{\left(3 a^2 A-5 a b B+2 A b^2\right) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a \cos (c+d x)+b)^2}+\frac{\left(4 a^3 B-13 a^2 A b+11 a b^2 B-2 A b^3\right) \sin (c+d x)}{(a-b)^3 (a+b)^3 (a \cos (c+d x)+b)}-\frac{6 \left(a^3 A-3 a^2 b B+4 a A b^2-2 b^3 B\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}+\frac{2 (a B-A b) \sin (c+d x)}{(a-b) (a+b) (a \cos (c+d x)+b)^3}}{6 d}","-\frac{a^2 (A b-a B) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{\left(a^3 A-3 a^2 b B+4 a A b^2-2 b^3 B\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{a \left(-4 a^3 B+a^2 A b+9 a b^2 B-6 A b^3\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{\left(2 a^5 B+a^4 A b-5 a^3 b^2 B-10 a^2 A b^3+18 a b^4 B-6 A b^5\right) \tan (c+d x)}{6 b^2 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((-6*(a^3*A + 4*a*A*b^2 - 3*a^2*b*B - 2*b^3*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (2*(-(A*b) + a*B)*Sin[c + d*x])/((a - b)*(a + b)*(b + a*Cos[c + d*x])^3) + ((3*a^2*A + 2*A*b^2 - 5*a*b*B)*Sin[c + d*x])/((a - b)^2*(a + b)^2*(b + a*Cos[c + d*x])^2) + ((-13*a^2*A*b - 2*A*b^3 + 4*a^3*B + 11*a*b^2*B)*Sin[c + d*x])/((a - b)^3*(a + b)^3*(b + a*Cos[c + d*x])))/(6*d)","A",1
339,1,252,263,1.2672024,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{24 \left(a^3 B-4 a^2 A b+4 a b^2 B-A b^3\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+\frac{2 \sin (c+d x) \left(-6 a^5 A+11 a^4 b B-14 a^3 A b^2+22 a^2 b^3 B+a \left(-6 a^4 A+13 a^3 b B-10 a^2 A b^2+2 a b^3 B+A b^4\right) \cos (2 (c+d x))-6 \left(a^5 B+2 a^4 A b-9 a^3 b^2 B+9 a^2 A b^3-2 a b^4 B-A b^5\right) \cos (c+d x)-25 a A b^4+12 b^5 B\right)}{(a \cos (c+d x)+b)^3}}{24 d \left(b^2-a^2\right)^3}","\frac{a (A b-a B) \tan (c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}-\frac{\left(a^3 (-B)+4 a^2 A b-4 a b^2 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)}{d (a-b)^{7/2} (a+b)^{7/2}}+\frac{\left(a^3 B+2 a^2 A b-6 a b^2 B+3 A b^3\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^4 B+2 a^3 A b-10 a^2 b^2 B+13 a A b^3-6 b^4 B\right) \tan (c+d x)}{6 b d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((24*(-4*a^2*A*b - A*b^3 + a^3*B + 4*a*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (2*(-6*a^5*A - 14*a^3*A*b^2 - 25*a*A*b^4 + 11*a^4*b*B + 22*a^2*b^3*B + 12*b^5*B - 6*(2*a^4*A*b + 9*a^2*A*b^3 - A*b^5 + a^5*B - 9*a^3*b^2*B - 2*a*b^4*B)*Cos[c + d*x] + a*(-6*a^4*A - 10*a^2*A*b^2 + A*b^4 + 13*a^3*b*B + 2*a*b^3*B)*Cos[2*(c + d*x)])*Sin[c + d*x])/(b + a*Cos[c + d*x])^3)/(24*(-a^2 + b^2)^3*d)","A",1
340,1,404,237,1.1063869,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(-6 a^5 B \sin (c+d x)-6 a^5 B \sin (3 (c+d x))+18 a^4 A b \sin (c+d x)+18 a^4 A b \sin (3 (c+d x))-12 a^4 b B \sin (2 (c+d x))+54 a^3 A b^2 \sin (2 (c+d x))-18 a^3 b^2 B \sin (c+d x)-10 a^3 b^2 B \sin (3 (c+d x))+39 a^2 A b^3 \sin (c+d x)-5 a^2 A b^3 \sin (3 (c+d x))-54 a^2 b^3 B \sin (2 (c+d x))+\frac{24 \left(2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right) (a \cos (c+d x)+b)^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\sqrt{a^2-b^2}}+6 a A b^4 \sin (2 (c+d x))-51 a b^4 B \sin (c+d x)+a b^4 B \sin (3 (c+d x))+18 A b^5 \sin (c+d x)+2 A b^5 \sin (3 (c+d x))+6 b^5 B \sin (2 (c+d x))\right)}{24 d \left(b^2-a^2\right)^3 (a+b \sec (c+d x))^4 (A \cos (c+d x)+B)}","-\frac{\left(-2 a^2 B+5 a A b-3 b^2 B\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}-\frac{(A b-a B) \tan (c+d x)}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{\left(2 a^3 A-4 a^2 b B+3 a A b^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)}{d (a-b)^{7/2} (a+b)^{7/2}}-\frac{\left(-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right) \tan (c+d x)}{6 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((24*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/Sqrt[a^2 - b^2] + 18*a^4*A*b*Sin[c + d*x] + 39*a^2*A*b^3*Sin[c + d*x] + 18*A*b^5*Sin[c + d*x] - 6*a^5*B*Sin[c + d*x] - 18*a^3*b^2*B*Sin[c + d*x] - 51*a*b^4*B*Sin[c + d*x] + 54*a^3*A*b^2*Sin[2*(c + d*x)] + 6*a*A*b^4*Sin[2*(c + d*x)] - 12*a^4*b*B*Sin[2*(c + d*x)] - 54*a^2*b^3*B*Sin[2*(c + d*x)] + 6*b^5*B*Sin[2*(c + d*x)] + 18*a^4*A*b*Sin[3*(c + d*x)] - 5*a^2*A*b^3*Sin[3*(c + d*x)] + 2*A*b^5*Sin[3*(c + d*x)] - 6*a^5*B*Sin[3*(c + d*x)] - 10*a^3*b^2*B*Sin[3*(c + d*x)] + a*b^4*B*Sin[3*(c + d*x)]))/(24*(-a^2 + b^2)^3*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^4)","A",1
341,1,769,292,3.5063657,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^4} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^4,x]","\frac{\sec ^3(c+d x) (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(\frac{6 a^9 A c \cos (3 (c+d x))+6 a^9 A d x \cos (3 (c+d x))+36 a^8 A b c+36 a^8 A b d x-18 a^8 b B \sin (c+d x)-18 a^8 b B \sin (3 (c+d x))+36 a^7 A b^2 \sin (c+d x)+36 a^7 A b^2 \sin (3 (c+d x))-18 a^7 A b^2 c \cos (3 (c+d x))-18 a^7 A b^2 d x \cos (3 (c+d x))-54 a^7 b^2 B \sin (2 (c+d x))+120 a^6 A b^3 \sin (2 (c+d x))-84 a^6 A b^3 c-84 a^6 A b^3 d x-39 a^6 b^3 B \sin (c+d x)+5 a^6 b^3 B \sin (3 (c+d x))+72 a^5 A b^4 \sin (c+d x)-32 a^5 A b^4 \sin (3 (c+d x))+18 a^5 A b^4 c \cos (3 (c+d x))+18 a^5 A b^4 d x \cos (3 (c+d x))-6 a^5 b^4 B \sin (2 (c+d x))-90 a^4 A b^5 \sin (2 (c+d x))+36 a^4 A b^5 c+36 a^4 A b^5 d x-18 a^4 b^5 B \sin (c+d x)-2 a^4 b^5 B \sin (3 (c+d x))-57 a^3 A b^6 \sin (c+d x)+11 a^3 A b^6 \sin (3 (c+d x))-6 a^3 A b^6 c \cos (3 (c+d x))-6 a^3 A b^6 d x \cos (3 (c+d x))+30 a^2 A b^7 \sin (2 (c+d x))+36 a^2 A b^7 c+36 a^2 A b^7 d x+36 a^2 A b \left(a^2-b^2\right)^3 (c+d x) \cos (2 (c+d x))+18 a A \left(a^2-b^2\right)^3 \left(a^2+4 b^2\right) (c+d x) \cos (c+d x)+24 a A b^8 \sin (c+d x)-24 A b^9 c-24 A b^9 d x}{\left(a^2-b^2\right)^3}-\frac{24 \left(2 a^7 B-8 a^6 A b+3 a^5 b^2 B+8 a^4 A b^3-7 a^2 A b^5+2 A b^7\right) (a \cos (c+d x)+b)^3 \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}\right)}{24 a^4 d (a+b \sec (c+d x))^4 (A \cos (c+d x)+B)}","\frac{A x}{a^4}+\frac{b (A b-a B) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{b \left(-5 a^3 B+8 a^2 A b-3 A b^3\right) \tan (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b \left(-11 a^5 B+26 a^4 A b-4 a^3 b^2 B-17 a^2 A b^3+6 A b^5\right) \tan (c+d x)}{6 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}-\frac{\left(-2 a^7 B+8 a^6 A b-3 a^5 b^2 B-8 a^4 A b^3+7 a^2 A b^5-2 A b^7\right) \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^4 d (a-b)^{7/2} (a+b)^{7/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((-24*(-8*a^6*A*b + 8*a^4*A*b^3 - 7*a^2*A*b^5 + 2*A*b^7 + 2*a^7*B + 3*a^5*b^2*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (36*a^8*A*b*c - 84*a^6*A*b^3*c + 36*a^4*A*b^5*c + 36*a^2*A*b^7*c - 24*A*b^9*c + 36*a^8*A*b*d*x - 84*a^6*A*b^3*d*x + 36*a^4*A*b^5*d*x + 36*a^2*A*b^7*d*x - 24*A*b^9*d*x + 18*a*A*(a^2 - b^2)^3*(a^2 + 4*b^2)*(c + d*x)*Cos[c + d*x] + 36*a^2*A*b*(a^2 - b^2)^3*(c + d*x)*Cos[2*(c + d*x)] + 6*a^9*A*c*Cos[3*(c + d*x)] - 18*a^7*A*b^2*c*Cos[3*(c + d*x)] + 18*a^5*A*b^4*c*Cos[3*(c + d*x)] - 6*a^3*A*b^6*c*Cos[3*(c + d*x)] + 6*a^9*A*d*x*Cos[3*(c + d*x)] - 18*a^7*A*b^2*d*x*Cos[3*(c + d*x)] + 18*a^5*A*b^4*d*x*Cos[3*(c + d*x)] - 6*a^3*A*b^6*d*x*Cos[3*(c + d*x)] + 36*a^7*A*b^2*Sin[c + d*x] + 72*a^5*A*b^4*Sin[c + d*x] - 57*a^3*A*b^6*Sin[c + d*x] + 24*a*A*b^8*Sin[c + d*x] - 18*a^8*b*B*Sin[c + d*x] - 39*a^6*b^3*B*Sin[c + d*x] - 18*a^4*b^5*B*Sin[c + d*x] + 120*a^6*A*b^3*Sin[2*(c + d*x)] - 90*a^4*A*b^5*Sin[2*(c + d*x)] + 30*a^2*A*b^7*Sin[2*(c + d*x)] - 54*a^7*b^2*B*Sin[2*(c + d*x)] - 6*a^5*b^4*B*Sin[2*(c + d*x)] + 36*a^7*A*b^2*Sin[3*(c + d*x)] - 32*a^5*A*b^4*Sin[3*(c + d*x)] + 11*a^3*A*b^6*Sin[3*(c + d*x)] - 18*a^8*b*B*Sin[3*(c + d*x)] + 5*a^6*b^3*B*Sin[3*(c + d*x)] - 2*a^4*b^5*B*Sin[3*(c + d*x)])/(a^2 - b^2)^3))/(24*a^4*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^4)","B",1
342,1,1205,411,6.3572292,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{(b+a \cos (c+d x)) \sec ^3(c+d x) (A+B \sec (c+d x)) \left(\frac{24 b \left(8 B a^7-20 A b a^6-8 b^2 B a^5+35 A b^3 a^4+7 b^4 B a^3-28 A b^5 a^2-2 b^6 B a+8 A b^7\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right) (b+a \cos (c+d x))^3}{\left(a^2-b^2\right)^{7/2}}+\frac{6 B c \cos (3 (c+d x)) a^{10}+6 B d x \cos (3 (c+d x)) a^{10}+6 A \sin (2 (c+d x)) a^{10}+3 A \sin (4 (c+d x)) a^{10}+36 b B c a^9+36 b B d x a^9-24 A b c \cos (3 (c+d x)) a^9-24 A b d x \cos (3 (c+d x)) a^9+18 A b \sin (c+d x) a^9+18 A b \sin (3 (c+d x)) a^9-144 A b^2 c a^8-144 A b^2 d x a^8-18 b^2 B c \cos (3 (c+d x)) a^8-18 b^2 B d x \cos (3 (c+d x)) a^8+36 b^2 B \sin (c+d x) a^8+18 A b^2 \sin (2 (c+d x)) a^8+36 b^2 B \sin (3 (c+d x)) a^8-9 A b^2 \sin (4 (c+d x)) a^8-84 b^3 B c a^7-84 b^3 B d x a^7+72 A b^3 c \cos (3 (c+d x)) a^7+72 A b^3 d x \cos (3 (c+d x)) a^7-90 A b^3 \sin (c+d x) a^7+120 b^3 B \sin (2 (c+d x)) a^7-114 A b^3 \sin (3 (c+d x)) a^7+336 A b^4 c a^6+336 A b^4 d x a^6+18 b^4 B c \cos (3 (c+d x)) a^6+18 b^4 B d x \cos (3 (c+d x)) a^6+72 b^4 B \sin (c+d x) a^6-300 A b^4 \sin (2 (c+d x)) a^6-32 b^4 B \sin (3 (c+d x)) a^6+9 A b^4 \sin (4 (c+d x)) a^6+36 b^5 B c a^5+36 b^5 B d x a^5-72 A b^5 c \cos (3 (c+d x)) a^5-72 A b^5 d x \cos (3 (c+d x)) a^5-135 A b^5 \sin (c+d x) a^5-90 b^5 B \sin (2 (c+d x)) a^5+125 A b^5 \sin (3 (c+d x)) a^5-144 A b^6 c a^4-144 A b^6 d x a^4-6 b^6 B c \cos (3 (c+d x)) a^4-6 b^6 B d x \cos (3 (c+d x)) a^4-57 b^6 B \sin (c+d x) a^4+336 A b^6 \sin (2 (c+d x)) a^4+11 b^6 B \sin (3 (c+d x)) a^4-3 A b^6 \sin (4 (c+d x)) a^4+36 b^7 B c a^3+36 b^7 B d x a^3+24 A b^7 c \cos (3 (c+d x)) a^3+24 A b^7 d x \cos (3 (c+d x)) a^3+228 A b^7 \sin (c+d x) a^3+30 b^7 B \sin (2 (c+d x)) a^3-44 A b^7 \sin (3 (c+d x)) a^3-144 A b^8 c a^2-144 A b^8 d x a^2+36 b \left(a^2-b^2\right)^3 (a B-4 A b) (c+d x) \cos (2 (c+d x)) a^2+24 b^8 B \sin (c+d x) a^2-120 A b^8 \sin (2 (c+d x)) a^2-24 b^9 B c a-24 b^9 B d x a+18 \left(a^2-b^2\right)^3 \left(a^2+4 b^2\right) (a B-4 A b) (c+d x) \cos (c+d x) a-96 A b^9 \sin (c+d x) a+96 A b^{10} c+96 A b^{10} d x}{\left(a^2-b^2\right)^3}\right)}{24 a^5 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^4}","-\frac{x (4 A b-a B)}{a^5}+\frac{b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^3}+\frac{b \left(-6 a^3 B+9 a^2 A b+a b^2 B-4 A b^3\right) \sin (c+d x)}{6 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^2}+\frac{b \left(-6 a^5 B+12 a^4 A b+2 a^3 b^2 B-11 a^2 A b^3-a b^4 B+4 A b^5\right) \sin (c+d x)}{2 a^3 d \left(a^2-b^2\right)^3 (a+b \sec (c+d x))}+\frac{\left(6 a^6 A+26 a^5 b B-65 a^4 A b^2-17 a^3 b^3 B+68 a^2 A b^4+6 a b^5 B-24 A b^6\right) \sin (c+d x)}{6 a^4 d \left(a^2-b^2\right)^3}+\frac{b \left(-8 a^7 B+20 a^6 A b+8 a^5 b^2 B-35 a^4 A b^3-7 a^3 b^4 B+28 a^2 A b^5+2 a b^6 B-8 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^5 d (a-b)^{7/2} (a+b)^{7/2}}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]^3*(A + B*Sec[c + d*x])*((24*b*(-20*a^6*A*b + 35*a^4*A*b^3 - 28*a^2*A*b^5 + 8*A*b^7 + 8*a^7*B - 8*a^5*b^2*B + 7*a^3*b^4*B - 2*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]*(b + a*Cos[c + d*x])^3)/(a^2 - b^2)^(7/2) + (-144*a^8*A*b^2*c + 336*a^6*A*b^4*c - 144*a^4*A*b^6*c - 144*a^2*A*b^8*c + 96*A*b^10*c + 36*a^9*b*B*c - 84*a^7*b^3*B*c + 36*a^5*b^5*B*c + 36*a^3*b^7*B*c - 24*a*b^9*B*c - 144*a^8*A*b^2*d*x + 336*a^6*A*b^4*d*x - 144*a^4*A*b^6*d*x - 144*a^2*A*b^8*d*x + 96*A*b^10*d*x + 36*a^9*b*B*d*x - 84*a^7*b^3*B*d*x + 36*a^5*b^5*B*d*x + 36*a^3*b^7*B*d*x - 24*a*b^9*B*d*x + 18*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(-4*A*b + a*B)*(c + d*x)*Cos[c + d*x] + 36*a^2*b*(a^2 - b^2)^3*(-4*A*b + a*B)*(c + d*x)*Cos[2*(c + d*x)] - 24*a^9*A*b*c*Cos[3*(c + d*x)] + 72*a^7*A*b^3*c*Cos[3*(c + d*x)] - 72*a^5*A*b^5*c*Cos[3*(c + d*x)] + 24*a^3*A*b^7*c*Cos[3*(c + d*x)] + 6*a^10*B*c*Cos[3*(c + d*x)] - 18*a^8*b^2*B*c*Cos[3*(c + d*x)] + 18*a^6*b^4*B*c*Cos[3*(c + d*x)] - 6*a^4*b^6*B*c*Cos[3*(c + d*x)] - 24*a^9*A*b*d*x*Cos[3*(c + d*x)] + 72*a^7*A*b^3*d*x*Cos[3*(c + d*x)] - 72*a^5*A*b^5*d*x*Cos[3*(c + d*x)] + 24*a^3*A*b^7*d*x*Cos[3*(c + d*x)] + 6*a^10*B*d*x*Cos[3*(c + d*x)] - 18*a^8*b^2*B*d*x*Cos[3*(c + d*x)] + 18*a^6*b^4*B*d*x*Cos[3*(c + d*x)] - 6*a^4*b^6*B*d*x*Cos[3*(c + d*x)] + 18*a^9*A*b*Sin[c + d*x] - 90*a^7*A*b^3*Sin[c + d*x] - 135*a^5*A*b^5*Sin[c + d*x] + 228*a^3*A*b^7*Sin[c + d*x] - 96*a*A*b^9*Sin[c + d*x] + 36*a^8*b^2*B*Sin[c + d*x] + 72*a^6*b^4*B*Sin[c + d*x] - 57*a^4*b^6*B*Sin[c + d*x] + 24*a^2*b^8*B*Sin[c + d*x] + 6*a^10*A*Sin[2*(c + d*x)] + 18*a^8*A*b^2*Sin[2*(c + d*x)] - 300*a^6*A*b^4*Sin[2*(c + d*x)] + 336*a^4*A*b^6*Sin[2*(c + d*x)] - 120*a^2*A*b^8*Sin[2*(c + d*x)] + 120*a^7*b^3*B*Sin[2*(c + d*x)] - 90*a^5*b^5*B*Sin[2*(c + d*x)] + 30*a^3*b^7*B*Sin[2*(c + d*x)] + 18*a^9*A*b*Sin[3*(c + d*x)] - 114*a^7*A*b^3*Sin[3*(c + d*x)] + 125*a^5*A*b^5*Sin[3*(c + d*x)] - 44*a^3*A*b^7*Sin[3*(c + d*x)] + 36*a^8*b^2*B*Sin[3*(c + d*x)] - 32*a^6*b^4*B*Sin[3*(c + d*x)] + 11*a^4*b^6*B*Sin[3*(c + d*x)] + 3*a^10*A*Sin[4*(c + d*x)] - 9*a^8*A*b^2*Sin[4*(c + d*x)] + 9*a^6*A*b^4*Sin[4*(c + d*x)] - 3*a^4*A*b^6*Sin[4*(c + d*x)])/(a^2 - b^2)^3))/(24*a^5*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^4)","B",1
343,1,1452,538,6.120428,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^4} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^4,x]","\frac{\frac{12 A c \cos (3 (c+d x)) a^{11}+12 A d x \cos (3 (c+d x)) a^{11}+6 A \sin (c+d x) a^{11}+24 B \sin (2 (c+d x)) a^{11}+9 A \sin (3 (c+d x)) a^{11}+12 B \sin (4 (c+d x)) a^{11}+3 A \sin (5 (c+d x)) a^{11}+72 A b c a^{10}+72 A b d x a^{10}-96 b B c \cos (3 (c+d x)) a^{10}-96 b B d x \cos (3 (c+d x)) a^{10}+72 b B \sin (c+d x) a^{10}-60 A b \sin (2 (c+d x)) a^{10}+72 b B \sin (3 (c+d x)) a^{10}-30 A b \sin (4 (c+d x)) a^{10}-576 b^2 B c a^9-576 b^2 B d x a^9+204 A b^2 c \cos (3 (c+d x)) a^9+204 A b^2 d x \cos (3 (c+d x)) a^9-270 A b^2 \sin (c+d x) a^9+72 b^2 B \sin (2 (c+d x)) a^9-279 A b^2 \sin (3 (c+d x)) a^9-36 b^2 B \sin (4 (c+d x)) a^9-9 A b^2 \sin (5 (c+d x)) a^9+1272 A b^3 c a^8+1272 A b^3 d x a^8+288 b^3 B c \cos (3 (c+d x)) a^8+288 b^3 B d x \cos (3 (c+d x)) a^8-360 b^3 B \sin (c+d x) a^8-372 A b^3 \sin (2 (c+d x)) a^8-456 b^3 B \sin (3 (c+d x)) a^8+90 A b^3 \sin (4 (c+d x)) a^8+1344 b^4 B c a^7+1344 b^4 B d x a^7-684 A b^4 c \cos (3 (c+d x)) a^7-684 A b^4 d x \cos (3 (c+d x)) a^7+750 A b^4 \sin (c+d x) a^7-1200 b^4 B \sin (2 (c+d x)) a^7+1143 A b^4 \sin (3 (c+d x)) a^7+36 b^4 B \sin (4 (c+d x)) a^7+9 A b^4 \sin (5 (c+d x)) a^7-3288 A b^5 c a^6-3288 A b^5 d x a^6-288 b^5 B c \cos (3 (c+d x)) a^6-288 b^5 B d x \cos (3 (c+d x)) a^6-540 b^5 B \sin (c+d x) a^6+2772 A b^5 \sin (2 (c+d x)) a^6+500 b^5 B \sin (3 (c+d x)) a^6-90 A b^5 \sin (4 (c+d x)) a^6-576 b^6 B c a^5-576 b^6 B d x a^5+708 A b^6 c \cos (3 (c+d x)) a^5+708 A b^6 d x \cos (3 (c+d x)) a^5+1086 A b^6 \sin (c+d x) a^5+1344 b^6 B \sin (2 (c+d x)) a^5-1253 A b^6 \sin (3 (c+d x)) a^5-12 b^6 B \sin (4 (c+d x)) a^5-3 A b^6 \sin (5 (c+d x)) a^5+1512 A b^7 c a^4+1512 A b^7 d x a^4+96 b^7 B c \cos (3 (c+d x)) a^4+96 b^7 B d x \cos (3 (c+d x)) a^4+912 b^7 B \sin (c+d x) a^4-3300 A b^7 \sin (2 (c+d x)) a^4-176 b^7 B \sin (3 (c+d x)) a^4+30 A b^7 \sin (4 (c+d x)) a^4-576 b^8 B c a^3-576 b^8 B d x a^3-240 A b^8 c \cos (3 (c+d x)) a^3-240 A b^8 d x \cos (3 (c+d x)) a^3-2232 A b^8 \sin (c+d x) a^3-480 b^8 B \sin (2 (c+d x)) a^3+440 A b^8 \sin (3 (c+d x)) a^3+1392 A b^9 c a^2+1392 A b^9 d x a^2+72 b \left(a^2-b^2\right)^3 \left(A a^2-8 b B a+20 A b^2\right) (c+d x) \cos (2 (c+d x)) a^2-384 b^9 B \sin (c+d x) a^2+1200 A b^9 \sin (2 (c+d x)) a^2+384 b^{10} B c a+384 b^{10} B d x a+36 \left(a^2-b^2\right)^3 \left(a^2+4 b^2\right) \left(A a^2-8 b B a+20 A b^2\right) (c+d x) \cos (c+d x) a+960 A b^{10} \sin (c+d x) a-960 A b^{11} c-960 A b^{11} d x}{\left(a^2-b^2\right)^3 (b+a \cos (c+d x))^3}-\frac{96 b^2 \left(20 B a^7-40 A b a^6-35 b^2 B a^5+84 A b^3 a^4+28 b^4 B a^3-69 A b^5 a^2-8 b^6 B a+20 A b^7\right) \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)}{\left(a^2-b^2\right)^{7/2}}}{96 a^6 d}","\frac{b (A b-a B) \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-8 a b B+20 A b^2\right)}{2 a^6}+\frac{b \left(-7 a^3 B+10 a^2 A b+2 a b^2 B-5 A b^3\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{b \left(-27 a^5 B+48 a^4 A b+20 a^3 b^2 B-53 a^2 A b^3-8 a b^4 B+20 A b^5\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^6 A+12 a^5 b B-23 a^4 A b^2-11 a^3 b^3 B+27 a^2 A b^4+4 a b^5 B-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(-6 a^7 B+24 a^6 A b+65 a^5 b^2 B-146 a^4 A b^3-68 a^3 b^4 B+167 a^2 A b^5+24 a b^6 B-60 A b^7\right) \sin (c+d x)}{6 a^5 d \left(a^2-b^2\right)^3}-\frac{b^2 \left(-20 a^7 B+40 a^6 A b+35 a^5 b^2 B-84 a^4 A b^3-28 a^3 b^4 B+69 a^2 A b^5+8 a b^6 B-20 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^6 d (a-b)^{7/2} (a+b)^{7/2}}",1,"((-96*b^2*(-40*a^6*A*b + 84*a^4*A*b^3 - 69*a^2*A*b^5 + 20*A*b^7 + 20*a^7*B - 35*a^5*b^2*B + 28*a^3*b^4*B - 8*a*b^6*B)*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (72*a^10*A*b*c + 1272*a^8*A*b^3*c - 3288*a^6*A*b^5*c + 1512*a^4*A*b^7*c + 1392*a^2*A*b^9*c - 960*A*b^11*c - 576*a^9*b^2*B*c + 1344*a^7*b^4*B*c - 576*a^5*b^6*B*c - 576*a^3*b^8*B*c + 384*a*b^10*B*c + 72*a^10*A*b*d*x + 1272*a^8*A*b^3*d*x - 3288*a^6*A*b^5*d*x + 1512*a^4*A*b^7*d*x + 1392*a^2*A*b^9*d*x - 960*A*b^11*d*x - 576*a^9*b^2*B*d*x + 1344*a^7*b^4*B*d*x - 576*a^5*b^6*B*d*x - 576*a^3*b^8*B*d*x + 384*a*b^10*B*d*x + 36*a*(a^2 - b^2)^3*(a^2 + 4*b^2)*(a^2*A + 20*A*b^2 - 8*a*b*B)*(c + d*x)*Cos[c + d*x] + 72*a^2*b*(a^2 - b^2)^3*(a^2*A + 20*A*b^2 - 8*a*b*B)*(c + d*x)*Cos[2*(c + d*x)] + 12*a^11*A*c*Cos[3*(c + d*x)] + 204*a^9*A*b^2*c*Cos[3*(c + d*x)] - 684*a^7*A*b^4*c*Cos[3*(c + d*x)] + 708*a^5*A*b^6*c*Cos[3*(c + d*x)] - 240*a^3*A*b^8*c*Cos[3*(c + d*x)] - 96*a^10*b*B*c*Cos[3*(c + d*x)] + 288*a^8*b^3*B*c*Cos[3*(c + d*x)] - 288*a^6*b^5*B*c*Cos[3*(c + d*x)] + 96*a^4*b^7*B*c*Cos[3*(c + d*x)] + 12*a^11*A*d*x*Cos[3*(c + d*x)] + 204*a^9*A*b^2*d*x*Cos[3*(c + d*x)] - 684*a^7*A*b^4*d*x*Cos[3*(c + d*x)] + 708*a^5*A*b^6*d*x*Cos[3*(c + d*x)] - 240*a^3*A*b^8*d*x*Cos[3*(c + d*x)] - 96*a^10*b*B*d*x*Cos[3*(c + d*x)] + 288*a^8*b^3*B*d*x*Cos[3*(c + d*x)] - 288*a^6*b^5*B*d*x*Cos[3*(c + d*x)] + 96*a^4*b^7*B*d*x*Cos[3*(c + d*x)] + 6*a^11*A*Sin[c + d*x] - 270*a^9*A*b^2*Sin[c + d*x] + 750*a^7*A*b^4*Sin[c + d*x] + 1086*a^5*A*b^6*Sin[c + d*x] - 2232*a^3*A*b^8*Sin[c + d*x] + 960*a*A*b^10*Sin[c + d*x] + 72*a^10*b*B*Sin[c + d*x] - 360*a^8*b^3*B*Sin[c + d*x] - 540*a^6*b^5*B*Sin[c + d*x] + 912*a^4*b^7*B*Sin[c + d*x] - 384*a^2*b^9*B*Sin[c + d*x] - 60*a^10*A*b*Sin[2*(c + d*x)] - 372*a^8*A*b^3*Sin[2*(c + d*x)] + 2772*a^6*A*b^5*Sin[2*(c + d*x)] - 3300*a^4*A*b^7*Sin[2*(c + d*x)] + 1200*a^2*A*b^9*Sin[2*(c + d*x)] + 24*a^11*B*Sin[2*(c + d*x)] + 72*a^9*b^2*B*Sin[2*(c + d*x)] - 1200*a^7*b^4*B*Sin[2*(c + d*x)] + 1344*a^5*b^6*B*Sin[2*(c + d*x)] - 480*a^3*b^8*B*Sin[2*(c + d*x)] + 9*a^11*A*Sin[3*(c + d*x)] - 279*a^9*A*b^2*Sin[3*(c + d*x)] + 1143*a^7*A*b^4*Sin[3*(c + d*x)] - 1253*a^5*A*b^6*Sin[3*(c + d*x)] + 440*a^3*A*b^8*Sin[3*(c + d*x)] + 72*a^10*b*B*Sin[3*(c + d*x)] - 456*a^8*b^3*B*Sin[3*(c + d*x)] + 500*a^6*b^5*B*Sin[3*(c + d*x)] - 176*a^4*b^7*B*Sin[3*(c + d*x)] - 30*a^10*A*b*Sin[4*(c + d*x)] + 90*a^8*A*b^3*Sin[4*(c + d*x)] - 90*a^6*A*b^5*Sin[4*(c + d*x)] + 30*a^4*A*b^7*Sin[4*(c + d*x)] + 12*a^11*B*Sin[4*(c + d*x)] - 36*a^9*b^2*B*Sin[4*(c + d*x)] + 36*a^7*b^4*B*Sin[4*(c + d*x)] - 12*a^5*b^6*B*Sin[4*(c + d*x)] + 3*a^11*A*Sin[5*(c + d*x)] - 9*a^9*A*b^2*Sin[5*(c + d*x)] + 9*a^7*A*b^4*Sin[5*(c + d*x)] - 3*a^5*A*b^6*Sin[5*(c + d*x)])/((a^2 - b^2)^3*(b + a*Cos[c + d*x])^3))/(96*a^6*d)","B",1
344,1,61,61,0.1657529,"\int \frac{\frac{b B}{a}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[((b*B)/a + B*Sec[c + d*x])/(a + b*Sec[c + d*x]),x]","\frac{B \left(b (c+d x)-2 \sqrt{a^2-b^2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a^2-b^2}}\right)\right)}{a^2 d}","\frac{2 B \sqrt{a-b} \sqrt{a+b} \tanh ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{a^2 d}+\frac{b B x}{a^2}",1,"(B*(b*(c + d*x) - 2*Sqrt[a^2 - b^2]*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]]))/(a^2*d)","A",1
345,1,6,6,0.0007811,"\int \frac{\frac{a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx","Integrate[((a*B)/b + B*Sec[c + d*x])/(a + b*Sec[c + d*x]),x]","\frac{B x}{b}","\frac{B x}{b}",1,"(B*x)/b","A",1
346,1,97,86,0.3759489,"\int \frac{a+b \sec (c+d x)}{(b+a \sec (c+d x))^2} \, dx","Integrate[(a + b*Sec[c + d*x])/(b + a*Sec[c + d*x])^2,x]","\frac{2 \sqrt{b^2-a^2} \tanh ^{-1}\left(\frac{(b-a) \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{b^2-a^2}}\right)+\frac{a (a c+a d x-b \sin (c+d x)+b (c+d x) \cos (c+d x))}{a+b \cos (c+d x)}}{b^2 d}","-\frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left(\frac{\sqrt{a-b} \tan \left(\frac{1}{2} (c+d x)\right)}{\sqrt{a+b}}\right)}{b^2 d}+\frac{a x}{b^2}-\frac{a \tan (c+d x)}{b d (a \sec (c+d x)+b)}",1,"(2*Sqrt[-a^2 + b^2]*ArcTanh[((-a + b)*Tan[(c + d*x)/2])/Sqrt[-a^2 + b^2]] + (a*(a*c + a*d*x + b*(c + d*x)*Cos[c + d*x] - b*Sin[c + d*x]))/(a + b*Cos[c + d*x]))/(b^2*d)","A",1
347,1,39,87,0.0961437,"\int \frac{3+\sec (c+d x)}{2-\sec (c+d x)} \, dx","Integrate[(3 + Sec[c + d*x])/(2 - Sec[c + d*x]),x]","\frac{9 (c+d x)+10 \sqrt{3} \tanh ^{-1}\left(\sqrt{3} \tan \left(\frac{1}{2} (c+d x)\right)\right)}{6 d}","-\frac{5 \log \left(\cos \left(\frac{1}{2} (c+d x)\right)-\sqrt{3} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 \sqrt{3} d}+\frac{5 \log \left(\sqrt{3} \sin \left(\frac{1}{2} (c+d x)\right)+\cos \left(\frac{1}{2} (c+d x)\right)\right)}{2 \sqrt{3} d}+\frac{3 x}{2}",1,"(9*(c + d*x) + 10*Sqrt[3]*ArcTanh[Sqrt[3]*Tan[(c + d*x)/2]])/(6*d)","A",1
348,1,3734,485,26.1396304,"\int \sec ^4(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^4*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 \left(-6 a^2 B+9 a A b+49 b^2 B\right) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 \left(-8 a^3 B+12 a^2 A b-13 a b^2 B-75 A b^3\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^3 d}-\frac{2 (a-b) \sqrt{a+b} \left(-16 a^3 B+12 a^2 b (2 A-B)+18 a b^2 (A-2 B)+3 b^3 (25 A-49 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)}{315 b^4 d}-\frac{2 (a-b) \sqrt{a+b} \left(-16 a^4 B+24 a^3 A b-24 a^2 b^2 B+57 a A b^3+147 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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{315 b^5 d}+\frac{2 (a B+9 A b) \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{63 b d}+\frac{2 B \tan (c+d x) \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((2*(24*a^3*A*b + 57*a*A*b^3 - 16*a^4*B - 24*a^2*b^2*B + 147*b^4*B)*Sin[c + d*x])/(315*b^4) + (2*Sec[c + d*x]^3*(9*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/(63*b) + (2*Sec[c + d*x]^2*(9*a*A*b*Sin[c + d*x] - 6*a^2*B*Sin[c + d*x] + 49*b^2*B*Sin[c + d*x]))/(315*b^2) + (2*Sec[c + d*x]*(-12*a^2*A*b*Sin[c + d*x] + 75*A*b^3*Sin[c + d*x] + 8*a^3*B*Sin[c + d*x] + 13*a*b^2*B*Sin[c + d*x]))/(315*b^3) + (2*B*Sec[c + d*x]^3*Tan[c + d*x])/9))/d + (2*((-19*a*A)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*A)/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^4*B)/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*a^2*B)/(105*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*A*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (17*a^2*A*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) + (5*A*b*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (4*a*B*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (16*a^5*B*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x]]) + (4*a^3*B*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (19*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) - (7*a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (16*a^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^4*Sqrt[b + a*Cos[c + d*x]]) + (8*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(2*(a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^4*d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^4*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^3*B + 12*a^2*b*(2*A + B) - 18*a*b^2*(A + 2*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-24*a^3*A*b - 57*a*A*b^3 + 16*a^4*B + 24*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^4*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
349,1,3330,397,25.0161759,"\int \sec ^3(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 \left(-4 a^2 B+7 a A b+25 b^2 B\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 B+2 a b (7 A-3 B)+b^2 (63 A-25 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)}{105 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(-8 a^3 B+14 a^2 A b-19 a b^2 B-63 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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{105 b^4 d}+\frac{2 (a B+7 A b) \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{35 b d}+\frac{2 B \tan (c+d x) \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((2*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sin[c + d*x])/(105*b^3) + (2*Sec[c + d*x]^2*(7*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/(35*b) + (2*Sec[c + d*x]*(7*a*A*b*Sin[c + d*x] - 4*a^2*B*Sin[c + d*x] + 25*b^2*B*Sin[c + d*x]))/(105*b^2) + (2*B*Sec[c + d*x]^2*Tan[c + d*x])/7))/d - (2*((2*a^2*A)/(15*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*A*b)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (19*a*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*B)/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*A*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*A*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*B*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (17*a^2*B*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) + (5*b*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (3*a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b^3*Sqrt[b + a*Cos[c + d*x]]) - (19*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(2*(a + b)*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^2*B - 2*a*b*(7*A + 3*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^3*d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*(-1/105*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^2*B - 2*a*b*(7*A + 3*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^2*B - 2*a*b*(7*A + 3*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(8*a^2*B - 2*a*b*(7*A + 3*B) + b^2*(63*A + 25*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(8*a^2*B - 2*a*b*(7*A + 3*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(8*a^2*B - 2*a*b*(7*A + 3*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^2*B - 2*a*b*(7*A + 3*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-14*a^2*A*b + 63*A*b^3 + 8*a^3*B + 19*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
350,1,2905,314,22.0184916,"\int \sec ^2(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \left(-2 a^2 B+5 a A b+9 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)}{15 b^3 d}-\frac{2 (a-b) \sqrt{a+b} (-2 a B+5 A b-9 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)}{15 b^2 d}+\frac{2 (5 A b-2 a B) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b d}+\frac{2 B \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 b d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((2*(5*a*A*b - 2*a^2*B + 9*b^2*B)*Sin[c + d*x])/(15*b^2) + (2*Sec[c + d*x]*(5*A*b*Sin[c + d*x] + a*B*Sin[c + d*x]))/(15*b) + (2*B*Sec[c + d*x]*Tan[c + d*x])/5))/d + (2*(-1/3*(a*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^2*B)/(15*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*b*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^2*A*Sqrt[Sec[c + d*x]])/(3*b*Sqrt[b + a*Cos[c + d*x]]) + (A*b*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) - (2*a*B*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*B*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]) - (a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*Sqrt[b + a*Cos[c + d*x]]) - (3*a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(2*(a + b)*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*A*b - 2*a*B + 9*b*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*A*b + 2*a^2*B - 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^2*d*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*A*b - 2*a*B + 9*b*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*A*b + 2*a^2*B - 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*A*b - 2*a*B + 9*b*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*A*b + 2*a^2*B - 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-5*a*A*b + 2*a^2*B - 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(5*A*b - 2*a*B + 9*b*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(5*A*b - 2*a*B + 9*b*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-5*a*A*b + 2*a^2*B - 9*b^2*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-5*a*A*b + 2*a^2*B - 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(5*A*b - 2*a*B + 9*b*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-5*a*A*b + 2*a^2*B - 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(5*A*b - 2*a*B + 9*b*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-5*a*A*b + 2*a^2*B - 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
351,1,408,256,14.6519821,"\int \sec (c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\cos (c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \left(\frac{2 (a B+3 A b) \sin (c+d x)}{3 b}+\frac{2}{3} B \tan (c+d x)\right)}{d (A \cos (c+d x)+B)}+\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \left(-\left((a B+3 A b) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)+2 b (a+b) (3 A+B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 (a+b) (a B+3 A b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b) (A \cos (c+d x)+B)}","-\frac{2 (a-b) \sqrt{a+b} (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}} 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 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 B \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x])*(-2*(a + b)*(3*A*b + a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*A + B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (3*A*b + a*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b*d*(b + a*Cos[c + d*x])*(B + A*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)) + (Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x])*((2*(3*A*b + a*B)*Sin[c + d*x])/(3*b) + (2*B*Tan[c + d*x])/3))/(d*(B + A*Cos[c + d*x]))","A",0
352,1,913,320,17.9605882,"\int \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 B \cos (c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \sin (c+d x)}{d (B+A \cos (c+d x))}+\frac{2 \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \left(a \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 i a A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) B E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-i (a-b) (A-B) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i a A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{\sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} (B+A \cos (c+d x)) \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{2 \sqrt{a+b} (B (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{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 (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*B*Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x])*Sin[c + d*x])/(d*(B + A*Cos[c + d*x])) + (2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x])*(a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 2*a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + a*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + (2*I)*a*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*a*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*B*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(A - B)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*(B + A*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",1
353,1,1107,344,18.3074305,"\int \cos (c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(a A \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 i A b \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 i a B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a A \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i A (a-b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i (a-b) B F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i A b \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 i a B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{\sqrt{\frac{b-a}{a+b}} d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-b \tan ^4\left(\frac{1}{2} (c+d x)\right)+a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)^2+b\right)}","\frac{\sqrt{a+b} (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)}{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 \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{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)}{b d}",1,"(Sqrt[a + b*Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*A*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 2*a*A*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + a*A*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - (2*I)*A*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*A*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*A*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*B*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(Sqrt[(-a + b)/(a + b)]*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(b - b*Tan[(c + d*x)/2]^4 + a*(-1 + Tan[(c + d*x)/2]^2)^2))","C",1
354,1,1149,429,18.9372616,"\int \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{A \sqrt{a+b \sec (c+d x)} \sin (2 (c+d x))}{4 d}+\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 a^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a^2 A \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a A b \tan \left(\frac{1}{2} (c+d x)\right)+4 a^2 B \tan \left(\frac{1}{2} (c+d x)\right)+4 a b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (A b+4 a B) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a (2 a A-b A+4 b B) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 a^2 A \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 a b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \left(4 a^2 A+4 a b B-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^2 d}+\frac{(4 a B+A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a d}+\frac{\sqrt{a+b} (2 a (A+2 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)}{4 a d}+\frac{(a-b) \sqrt{a+b} (4 a B+A b) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a b d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"(A*Sqrt[a + b*Sec[c + d*x]]*Sin[2*(c + d*x)])/(4*d) + (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a*A*b*Tan[(c + d*x)/2] + A*b^2*Tan[(c + d*x)/2] + 4*a^2*B*Tan[(c + d*x)/2] + 4*a*b*B*Tan[(c + d*x)/2] - 2*a*A*b*Tan[(c + d*x)/2]^3 - 8*a^2*B*Tan[(c + d*x)/2]^3 + a*A*b*Tan[(c + d*x)/2]^5 - A*b^2*Tan[(c + d*x)/2]^5 + 4*a^2*B*Tan[(c + d*x)/2]^5 - 4*a*b*B*Tan[(c + d*x)/2]^5 + 8*a^2*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a^2*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(A*b + 4*a*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(2*a*A - A*b + 4*b*B)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
355,1,1548,509,20.0398744,"\int \cos ^3(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{1}{12} A \sin (c+d x)+\frac{(A b+6 a B) \sin (2 (c+d x))}{24 a}+\frac{1}{12} A \sin (3 (c+d x))\right)}{d}-\frac{\sqrt{a+b \sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-12 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-12 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+6 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+6 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(16 A a^2+6 b B a-3 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a \left(12 B a^2+2 b (7 A-3 B) a-A b^2\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-12 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 a^2 d \sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\left(16 a^2 A+6 a b B-3 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 a^2 d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2 A+6 a b 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}} 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} (2 a+b) (8 a A+6 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)}{24 a^2 d}-\frac{\sqrt{a+b} \left(8 a^3 B+4 a^2 A b-2 a b^2 B+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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{8 a^3 d}+\frac{(6 a B+A b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a d}+\frac{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((A*Sin[c + d*x])/12 + ((A*b + 6*a*B)*Sin[2*(c + d*x)])/(24*a) + (A*Sin[3*(c + d*x)])/12))/d - (Sqrt[a + b*Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] - 3*a*A*b^2*Tan[(c + d*x)/2] - 3*A*b^3*Tan[(c + d*x)/2] + 6*a^2*b*B*Tan[(c + d*x)/2] + 6*a*b^2*B*Tan[(c + d*x)/2] - 32*a^3*A*Tan[(c + d*x)/2]^3 + 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 12*a^2*b*B*Tan[(c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 - 3*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*A*b^3*Tan[(c + d*x)/2]^5 + 6*a^2*b*B*Tan[(c + d*x)/2]^5 - 6*a*b^2*B*Tan[(c + d*x)/2]^5 + 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 12*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^2*A - 3*A*b^2 + 6*a*b*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(-(A*b^2) + 2*a*b*(7*A - 3*B) + 12*a^2*B)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a^2*d*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
356,1,3766,475,26.4392498,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","-\frac{2 \left(-8 a^2 B+18 a A b-49 b^2 B\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b^2 d}-\frac{2 \left(-8 a^3 B+18 a^2 A b-39 a b^2 B-75 A b^3\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(8 a^3 B-6 a^2 b (3 A-B)-3 a b^2 (57 A-13 B)+3 b^3 (25 A-49 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)}{315 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(-8 a^4 B+18 a^3 A b-33 a^2 b^2 B-246 a A b^3-147 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}} 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 A b-4 a B) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b^2 d}+\frac{2 B \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{5/2}}{9 b d}",1,"(Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((2*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sin[c + d*x])/(315*b^3) + (2*Sec[c + d*x]^3*(9*A*b*Sin[c + d*x] + 10*a*B*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2*(72*a*A*b*Sin[c + d*x] + 3*a^2*B*Sin[c + d*x] + 49*b^2*B*Sin[c + d*x]))/(315*b) + (2*Sec[c + d*x]*(9*a^2*A*b*Sin[c + d*x] + 75*A*b^3*Sin[c + d*x] - 4*a^3*B*Sin[c + d*x] + 88*a*b^2*B*Sin[c + d*x]))/(315*b^2) + (2*b*B*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])) - (2*((2*a^3*A)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (82*a*A*b)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (11*a^2*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*B)/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^2*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a^2*A*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*A*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (5*A*b^2*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*B*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*B*Sqrt[Sec[c + d*x]])/(315*b*Sqrt[b + a*Cos[c + d*x]]) + (13*a*b*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (82*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(315*b^3*Sqrt[b + a*Cos[c + d*x]]) - (11*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*b*Sqrt[b + a*Cos[c + d*x]]) - (7*a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(2*(a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*(-1/315*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(8*a^3*B - 6*a^2*b*(3*A + B) + 3*a*b^2*(57*A + 13*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-18*a^3*A*b + 246*a*A*b^3 + 8*a^4*B + 33*a^2*b^2*B + 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
357,1,3342,388,25.263826,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 \left(-6 a^2 B+21 a A b+25 b^2 B\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 B-a (21 A b-57 b B)+b^2 (63 A-25 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)}{105 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(-6 a^3 B+21 a^2 A b+82 a b^2 B+63 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}} 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 A b-2 a B) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 b d}+\frac{2 B \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 b d}",1,"(Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((-2*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sin[c + d*x])/(105*b^2) + (2*Sec[c + d*x]^2*(7*A*b*Sin[c + d*x] + 8*a*B*Sin[c + d*x]))/35 + (2*Sec[c + d*x]*(42*a*A*b*Sin[c + d*x] + 3*a^2*B*Sin[c + d*x] + 25*b^2*B*Sin[c + d*x]))/(105*b) + (2*b*B*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])) + (2*(-1/5*(a^2*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*A*b^2)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^3*B)/(35*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (82*a*b*B)/(105*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^3*A*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) + (a*A*b*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (31*a^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]) + (5*b^2*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]) - (3*a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (82*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*b^2*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(2*(a + b)*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*B + 3*a*b*(7*A + 19*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*d*(b + a*Cos[c + d*x])^2*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(3/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*B + 3*a*b*(7*A + 19*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*B + 3*a*b*(7*A + 19*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-6*a^2*B + 3*a*b*(7*A + 19*B) + b^2*(63*A + 25*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-6*a^2*B + 3*a*b*(7*A + 19*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-6*a^2*B + 3*a*b*(7*A + 19*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-6*a^2*B + 3*a*b*(7*A + 19*B) + b^2*(63*A + 25*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-21*a^2*A*b - 63*A*b^3 + 6*a^3*B - 82*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
358,1,502,312,19.2586767,"\int \sec (c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \left(\frac{2 \left(3 a^2 B+20 a A b+9 b^2 B\right) \sin (c+d x)}{15 b}+\frac{2}{15} \sec (c+d x) (6 a B \sin (c+d x)+5 A b \sin (c+d x))+\frac{2}{5} b B \tan (c+d x) \sec (c+d x)\right)}{d (a \cos (c+d x)+b) (A \cos (c+d x)+B)}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \left(\left(3 a^2 B+20 a A b+9 b^2 B\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 (a+b) \left(3 a^2 B+20 a A b+9 b^2 B\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 b (a+b) (3 a (5 A+B)+b (5 A+9 B)) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{15 b d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A \cos (c+d x)+B)}","-\frac{2 (a-b) \sqrt{a+b} \left(3 a^2 B+20 a A b+9 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)}{15 b^2 d}+\frac{2 (3 a B+5 A b) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 (a-b) \sqrt{a+b} (15 a A-3 a B-5 A b+9 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)}{15 b d}+\frac{2 B \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"(-2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])*(2*(a + b)*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*(5*A + B) + b*(5*A + 9*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (20*a*A*b + 3*a^2*B + 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b*d*(b + a*Cos[c + d*x])^2*(B + A*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)) + (Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])*((2*(20*a*A*b + 3*a^2*B + 9*b^2*B)*Sin[c + d*x])/(15*b) + (2*Sec[c + d*x]*(5*A*b*Sin[c + d*x] + 6*a*B*Sin[c + d*x]))/15 + (2*b*B*Sec[c + d*x]*Tan[c + d*x])/5))/(d*(b + a*Cos[c + d*x])*(B + A*Cos[c + d*x]))","A",0
359,1,6063,381,24.6213173,"\int (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","-\frac{2 \sqrt{a+b} \left(-3 a^2 B-a (6 A b-4 b B)+b^2 (3 A-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 b d}-\frac{2 (a-b) \sqrt{a+b} (4 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}} 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 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 B \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"Result too large to show","B",0
360,1,971,361,16.7160099,"\int \cos (c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 b B \cos (c+d x) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x))}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a^2 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-a A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a^2 A \tan ^3\left(\frac{1}{2} (c+d x)\right)+4 a b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a^2 A \tan \left(\frac{1}{2} (c+d x)\right)+a A b \tan \left(\frac{1}{2} (c+d x)\right)-2 b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-2 a b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) (a A-2 b B) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(B a^2+2 b (A-B) a-b^2 (A+B)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 a A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 a^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} (a (A+4 B)+2 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)}{d}+\frac{(a-b) \sqrt{a+b} (a A-2 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)}{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{a A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d}",1,"(2*b*B*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])) + ((a + b*Sec[c + d*x])^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(a^2*A*Tan[(c + d*x)/2] + a*A*b*Tan[(c + d*x)/2] - 2*a*b*B*Tan[(c + d*x)/2] - 2*b^2*B*Tan[(c + d*x)/2] - 2*a^2*A*Tan[(c + d*x)/2]^3 + 4*a*b*B*Tan[(c + d*x)/2]^3 + a^2*A*Tan[(c + d*x)/2]^5 - a*A*b*Tan[(c + d*x)/2]^5 - 2*a*b*B*Tan[(c + d*x)/2]^5 + 2*b^2*B*Tan[(c + d*x)/2]^5 + 6*a*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*a*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(a*A - 2*b*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(2*a*b*(A - B) + a^2*B - b^2*(A + B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
361,1,1598,428,19.5710095,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a A \cos (c+d x) (a+b \sec (c+d x))^{3/2} \sin (2 (c+d x))}{4 d (b+a \cos (c+d x))}-\frac{(a+b \sec (c+d x))^{3/2} \left(4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-5 A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+5 a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-10 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-24 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+4 a b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+5 A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+5 a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) (5 A b+4 a B) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i (a-b) (2 a A+b (A+4 B)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-24 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 \sqrt{\frac{b-a}{a+b}} d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","-\frac{\sqrt{a+b} \left(4 a^2 A+12 a b 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}} \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 B+5 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{\sqrt{a+b} (2 a A+4 a B+5 A b+8 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 d}+\frac{(a-b) \sqrt{a+b} (4 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}} 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 A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"(a*A*Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*Sin[2*(c + d*x)])/(4*d*(b + a*Cos[c + d*x])) - ((a + b*Sec[c + d*x])^(3/2)*(5*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 5*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 10*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 8*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 5*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 5*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(5*A*b + 4*a*B)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(2*a*A + b*(A + 4*B))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*Sqrt[(-a + b)/(a + b)]*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(-1 + Tan[(c + d*x)/2]^2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
362,1,1535,520,19.4599472,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos (c+d x) \left(\frac{1}{12} a A \sin (c+d x)+\frac{1}{24} (7 A b+6 a B) \sin (2 (c+d x))+\frac{1}{12} a A \sin (3 (c+d x))\right) (a+b \sec (c+d x))^{3/2}}{d (b+a \cos (c+d x))}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+30 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-6 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-60 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+72 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+36 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+3 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+30 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+30 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(16 A a^2+30 b B a+3 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a \left(12 B a^2+(26 A b-6 b B) a+b^2 (24 B-7 A)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+72 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+36 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{3/2}}{24 a d (b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(16 a^2 A+30 a b B+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+12 a^2 B+14 a A b+30 a b 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 d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2 A+30 a b 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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{24 a b d}-\frac{\sqrt{a+b} \left(8 a^3 B+12 a^2 A b+6 a b^2 B-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}} \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 B+7 A b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"(Cos[c + d*x]*(a + b*Sec[c + d*x])^(3/2)*((a*A*Sin[c + d*x])/12 + ((7*A*b + 6*a*B)*Sin[2*(c + d*x)])/24 + (a*A*Sin[3*(c + d*x)])/12))/(d*(b + a*Cos[c + d*x])) + ((a + b*Sec[c + d*x])^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] + 3*a*A*b^2*Tan[(c + d*x)/2] + 3*A*b^3*Tan[(c + d*x)/2] + 30*a^2*b*B*Tan[(c + d*x)/2] + 30*a*b^2*B*Tan[(c + d*x)/2] - 32*a^3*A*Tan[(c + d*x)/2]^3 - 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 60*a^2*b*B*Tan[(c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 + 3*a*A*b^2*Tan[(c + d*x)/2]^5 - 3*A*b^3*Tan[(c + d*x)/2]^5 + 30*a^2*b*B*Tan[(c + d*x)/2]^5 - 30*a*b^2*B*Tan[(c + d*x)/2]^5 + 72*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 36*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 72*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 36*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^2*A + 3*A*b^2 + 30*a*b*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(12*a^2*B + b^2*(-7*A + 24*B) + a*(26*A*b - 6*b*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
363,1,4227,566,27.3419616,"\int \sec ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","-\frac{2 \left(-8 a^2 B+22 a A b-81 b^2 B\right) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{693 b^2 d}-\frac{2 \left(-40 a^3 B+110 a^2 A b-335 a b^2 B-539 A b^3\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{3465 b^2 d}-\frac{2 \left(-40 a^4 B+110 a^3 A b-285 a^2 b^2 B-1254 a A b^3-675 b^4 B\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3465 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(40 a^4 B-a^3 (110 A b-30 b B)-15 a^2 b^2 (121 A-19 B)+6 a b^3 (209 A-505 B)-3 b^4 (539 A-225 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)}{3465 b^3 d}+\frac{2 (a-b) \sqrt{a+b} \left(-40 a^5 B+110 a^4 A b-255 a^3 b^2 B-3069 a^2 A b^3-3705 a b^4 B-1617 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}} 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 A b-4 a B) \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{99 b^2 d}+\frac{2 B \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^{7/2}}{11 b d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sin[c + d*x])/(3465*b^3) + (2*Sec[c + d*x]^4*(11*A*b^2*Sin[c + d*x] + 23*a*b*B*Sin[c + d*x]))/99 + (2*Sec[c + d*x]^3*(209*a*A*b*Sin[c + d*x] + 113*a^2*B*Sin[c + d*x] + 81*b^2*B*Sin[c + d*x]))/693 + (2*Sec[c + d*x]^2*(825*a^2*A*b*Sin[c + d*x] + 539*A*b^3*Sin[c + d*x] + 15*a^3*B*Sin[c + d*x] + 1145*a*b^2*B*Sin[c + d*x]))/(3465*b) + (2*Sec[c + d*x]*(55*a^3*A*b*Sin[c + d*x] + 1793*a*A*b^3*Sin[c + d*x] - 20*a^4*B*Sin[c + d*x] + 1025*a^2*b^2*B*Sin[c + d*x] + 675*b^4*B*Sin[c + d*x]))/(3465*b^2) + (2*b^2*B*Sec[c + d*x]^4*Tan[c + d*x])/11))/(d*(b + a*Cos[c + d*x])^2) - (2*((2*a^4*A)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a^2*A*b)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*A*b^3)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (17*a^3*B)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*B)/(693*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (247*a*b^2*B)/(231*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (124*a^3*A*Sqrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*A*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) + (38*a*A*b^2*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (8*a^6*B*Sqrt[Sec[c + d*x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (7*a^4*B*Sqrt[Sec[c + d*x]])/(99*b*Sqrt[b + a*Cos[c + d*x]]) - (26*a^2*b*B*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]) + (15*b^3*B*Sqrt[Sec[c + d*x]])/(77*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (7*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (8*a^6*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(693*b^3*Sqrt[b + a*Cos[c + d*x]]) - (17*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*b*Sqrt[b + a*Cos[c + d*x]]) - (247*a^2*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(231*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*(a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*b^3*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*(-1/3465*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3465*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3465*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(40*a^4*B - 10*a^3*b*(11*A + 3*B) + 15*a^2*b^2*(121*A + 19*B) + 3*b^4*(539*A + 225*B) + 6*a*b^3*(209*A + 505*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-110*a^4*A*b + 3069*a^2*A*b^3 + 1617*A*b^5 + 40*a^5*B + 255*a^3*b^2*B + 3705*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3465*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
364,1,3781,469,26.6302835,"\int \sec ^2(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 \left(-10 a^2 B+45 a A b+49 b^2 B\right) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{315 b d}+\frac{2 \left(-10 a^3 B+45 a^2 A b+114 a b^2 B+75 A b^3\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{315 b d}-\frac{2 (a-b) \sqrt{a+b} \left(-10 a^3 B+15 a^2 b (3 A-11 B)-6 a b^2 (60 A-19 B)+3 b^3 (25 A-49 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)}{315 b^2 d}-\frac{2 (a-b) \sqrt{a+b} \left(-10 a^4 B+45 a^3 A b+279 a^2 b^2 B+435 a A b^3+147 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}} 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 A b-2 a B) \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{63 b d}+\frac{2 B \tan (c+d x) (a+b \sec (c+d x))^{7/2}}{9 b d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((2*(45*a^3*A*b + 435*a*A*b^3 - 10*a^4*B + 279*a^2*b^2*B + 147*b^4*B)*Sin[c + d*x])/(315*b^2) + (2*Sec[c + d*x]^3*(9*A*b^2*Sin[c + d*x] + 19*a*b*B*Sin[c + d*x]))/63 + (2*Sec[c + d*x]^2*(135*a*A*b*Sin[c + d*x] + 75*a^2*B*Sin[c + d*x] + 49*b^2*B*Sin[c + d*x]))/315 + (2*Sec[c + d*x]*(135*a^2*A*b*Sin[c + d*x] + 75*A*b^3*Sin[c + d*x] + 5*a^3*B*Sin[c + d*x] + 163*a*b^2*B*Sin[c + d*x]))/(315*b) + (2*b^2*B*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^2) + (2*(-1/7*(a^3*A)/(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a*A*b^2)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^4*B)/(63*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (31*a^2*b*B)/(35*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (7*b^3*B)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^4*A*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*A*b*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (5*A*b^3*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (124*a^3*B*Sqrt[Sec[c + d*x]])/(315*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*B*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) + (38*a*b^2*B*Sqrt[Sec[c + d*x]])/(105*Sqrt[b + a*Cos[c + d*x]]) - (a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (29*a^2*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (31*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(35*Sqrt[b + a*Cos[c + d*x]]) + (2*a^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(63*b^2*Sqrt[b + a*Cos[c + d*x]]) - (7*a*b^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(2*(a + b)*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*B + 15*a^2*b*(3*A + 11*B) + 6*a*b^2*(60*A + 19*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*d*(b + a*Cos[c + d*x])^3*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(5/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*B + 15*a^2*b*(3*A + 11*B) + 6*a*b^2*(60*A + 19*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*B + 15*a^2*b*(3*A + 11*B) + 6*a*b^2*(60*A + 19*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-10*a^3*B + 15*a^2*b*(3*A + 11*B) + 6*a*b^2*(60*A + 19*B) + 3*b^3*(25*A + 49*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-10*a^3*B + 15*a^2*b*(3*A + 11*B) + 6*a*b^2*(60*A + 19*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-10*a^3*B + 15*a^2*b*(3*A + 11*B) + 6*a*b^2*(60*A + 19*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-10*a^3*B + 15*a^2*b*(3*A + 11*B) + 6*a*b^2*(60*A + 19*B) + 3*b^3*(25*A + 49*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-45*a^3*A*b - 435*a*A*b^3 + 10*a^4*B - 279*a^2*b^2*B - 147*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(315*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
365,1,2957,384,23.1669375,"\int \sec (c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 \left(15 a^2 B+56 a A b+25 b^2 B\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 A-B)-8 a b (7 A-15 B)+b^2 (63 A-25 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)}{105 b d}-\frac{2 (a-b) \sqrt{a+b} \left(15 a^3 B+161 a^2 A b+145 a b^2 B+63 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}} 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 B+7 A b) \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{35 d}+\frac{2 B \tan (c+d x) (a+b \sec (c+d x))^{5/2}}{7 d}",1,"(Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])*((2*(161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sin[c + d*x])/(105*b) + (2*Sec[c + d*x]^2*(7*A*b^2*Sin[c + d*x] + 15*a*b*B*Sin[c + d*x]))/35 + (2*Sec[c + d*x]*(77*a*A*b*Sin[c + d*x] + 45*a^2*B*Sin[c + d*x] + 25*b^2*B*Sin[c + d*x]))/105 + (2*b^2*B*Sec[c + d*x]^2*Tan[c + d*x])/7))/(d*(b + a*Cos[c + d*x])^2*(B + A*Cos[c + d*x])) + (2*((-23*a^2*A*b)/(15*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*A*b^3)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (a^3*B)/(7*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (29*a*b^2*B)/(21*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*A*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) + (8*a*A*b^2*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (a^4*B*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*b*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) + (5*b^3*B*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]) - (23*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*Sqrt[b + a*Cos[c + d*x]]) - (3*a*A*b^2*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*Sqrt[b + a*Cos[c + d*x]]) - (a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(7*b*Sqrt[b + a*Cos[c + d*x]]) - (29*a^2*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(21*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x])*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(105*b*d*(b + a*Cos[c + d*x])^2*(B + A*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sec[c + d*x]^(7/2)*(-1/105*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2)))/(105*b*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[b + a*Cos[c + d*x]]*((-2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Tan[(c + d*x)/2]*(-1 + Tan[(c + d*x)/2]^2))*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(105*b*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]) + (2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-3*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + ((Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/((b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x])))^(3/2) + (161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + ((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sec[(c + d*x)/2]^2*(-1 + Tan[(c + d*x)/2]^2))/2 - (2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*Sec[c + d*x]*(-1/2*(b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 - Tan[(c + d*x)/2]^2])))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (2*(Cos[c + d*x]/(1 + Cos[c + d*x]))^(3/2)*((161*a^2*A*b + 63*A*b^3 + 15*a^3*B + 145*a*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - b*(15*a^2*(7*A + B) + 8*a*b*(7*A + 15*B) + b^2*(63*A + 25*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x]*Tan[c + d*x])/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]))/(105*b*Sqrt[Sec[(c + d*x)/2]^2])))","B",0
366,1,7138,442,25.5593164,"\int (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","-\frac{2 (a-b) \sqrt{a+b} \left(23 a^2 B+35 a A b+9 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)}{15 b d}-\frac{2 a^2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{d}+\frac{2 \sqrt{a+b} \left(15 a^3 B+a^2 b (45 A-23 B)-a b^2 (35 A-17 B)+b^3 (5 A-9 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)}{15 b d}+\frac{2 b (8 a B+5 A b) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 b B \tan (c+d x) (a+b \sec (c+d x))^{3/2}}{5 d}",1,"Result too large to show","B",0
367,1,7745,433,26.0773346,"\int \cos (c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{\sqrt{a+b} \left(3 a^2 (A+6 B)+2 a b (9 A-7 B)-2 b^2 (3 A-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 d}+\frac{(a-b) \sqrt{a+b} \left(3 a^2 A-14 a b B-6 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)}{3 b d}-\frac{b (3 a A-2 b B) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 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{a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d}",1,"Result too large to show","B",0
368,1,1326,450,19.7885645,"\int \cos ^2(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^2(c+d x) \left(\frac{1}{4} A \sin (2 (c+d x)) a^2+2 b^2 B \sin (c+d x)\right) (a+b \sec (c+d x))^{5/2}}{d (b+a \cos (c+d x))^2}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-9 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+9 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+4 a^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+8 b^3 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-18 a^2 A b \tan ^3\left(\frac{1}{2} (c+d x)\right)-8 a^3 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+16 a b^2 B \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 a^3 A \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+40 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+9 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+9 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+4 a^3 B \tan \left(\frac{1}{2} (c+d x)\right)-8 b^3 B \tan \left(\frac{1}{2} (c+d x)\right)-8 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+4 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(4 B a^2+9 A b a-8 b^2 B\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(2 A a^3-b (A-12 B) a^2+12 b^2 (A-B) a-4 b^3 (A+B)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 a A b^2 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 a^3 A \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+40 a^2 b B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{5/2}}{4 d (b+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} \left(2 a^2 (A+2 B)+3 a b (3 A+8 B)+8 b^2 (A-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)}{4 d}+\frac{(a-b) \sqrt{a+b} \left(4 a^2 B+9 a A b-8 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)}{4 b d}-\frac{\sqrt{a+b} \left(4 a^2 A+20 a b 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}} \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 B+7 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{a A \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*(2*b^2*B*Sin[c + d*x] + (a^2*A*Sin[2*(c + d*x)])/4))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x])^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(9*a^2*A*b*Tan[(c + d*x)/2] + 9*a*A*b^2*Tan[(c + d*x)/2] + 4*a^3*B*Tan[(c + d*x)/2] + 4*a^2*b*B*Tan[(c + d*x)/2] - 8*a*b^2*B*Tan[(c + d*x)/2] - 8*b^3*B*Tan[(c + d*x)/2] - 18*a^2*A*b*Tan[(c + d*x)/2]^3 - 8*a^3*B*Tan[(c + d*x)/2]^3 + 16*a*b^2*B*Tan[(c + d*x)/2]^3 + 9*a^2*A*b*Tan[(c + d*x)/2]^5 - 9*a*A*b^2*Tan[(c + d*x)/2]^5 + 4*a^3*B*Tan[(c + d*x)/2]^5 - 4*a^2*b*B*Tan[(c + d*x)/2]^5 - 8*a*b^2*B*Tan[(c + d*x)/2]^5 + 8*b^3*B*Tan[(c + d*x)/2]^5 + 8*a^3*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 40*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 8*a^3*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*a*A*b^2*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 40*a^2*b*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(9*a*A*b + 4*a^2*B - 8*b^2*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(2*a^3*A - a^2*b*(A - 12*B) + 12*a*b^2*(A - B) - 4*b^3*(A + B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
369,1,1551,518,19.5349855,"\int \cos ^3(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^3*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^2(c+d x) \left(\frac{1}{12} A \sin (c+d x) a^2+\frac{1}{12} A \sin (3 (c+d x)) a^2+\frac{1}{24} (13 A b+6 a B) \sin (2 (c+d x)) a\right) (a+b \sec (c+d x))^{5/2}}{d (b+a \cos (c+d x))^2}+\frac{\sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(-33 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+33 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-54 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+54 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-66 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-108 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+120 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+180 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+33 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+33 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+54 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+54 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(16 A a^2+54 b B a+33 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 \left(12 B a^3+(38 A b-6 b B) a^2+b^2 (72 B-13 A) a+24 b^3 (A-B)\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+120 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+180 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (a+b \sec (c+d x))^{5/2}}{24 d (b+a \cos (c+d x))^{5/2} \sec ^{\frac{5}{2}}(c+d x) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\left(16 a^2 A+54 a b B+33 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+12 a^2 B+26 a A b+54 a b B+33 A b^2+48 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 d}+\frac{(a-b) \sqrt{a+b} \left(16 a^2 A+54 a b B+33 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 b d}-\frac{\sqrt{a+b} \left(8 a^3 B+20 a^2 A b+30 a b^2 B+5 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}} \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 B+3 A b) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{a A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}",1,"(Cos[c + d*x]^2*(a + b*Sec[c + d*x])^(5/2)*((a^2*A*Sin[c + d*x])/12 + (a*(13*A*b + 6*a*B)*Sin[2*(c + d*x)])/24 + (a^2*A*Sin[3*(c + d*x)])/12))/(d*(b + a*Cos[c + d*x])^2) + ((a + b*Sec[c + d*x])^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] + 33*a*A*b^2*Tan[(c + d*x)/2] + 33*A*b^3*Tan[(c + d*x)/2] + 54*a^2*b*B*Tan[(c + d*x)/2] + 54*a*b^2*B*Tan[(c + d*x)/2] - 32*a^3*A*Tan[(c + d*x)/2]^3 - 66*a*A*b^2*Tan[(c + d*x)/2]^3 - 108*a^2*b*B*Tan[(c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 + 33*a*A*b^2*Tan[(c + d*x)/2]^5 - 33*A*b^3*Tan[(c + d*x)/2]^5 + 54*a^2*b*B*Tan[(c + d*x)/2]^5 - 54*a*b^2*B*Tan[(c + d*x)/2]^5 + 120*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 180*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 120*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 180*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^2*A + 33*A*b^2 + 54*a*b*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*(24*b^3*(A - B) + 12*a^3*B + a*b^2*(-13*A + 72*B) + a^2*(38*A*b - 6*b*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","B",0
370,1,5172,617,24.270925,"\int \cos ^4(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^4*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{\left(36 a^2 A+104 a b B+59 A b^2\right) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\left(128 a^3 B+284 a^2 A b+264 a b^2 B+15 A b^3\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 a d}+\frac{\sqrt{a+b} \left(8 a^3 (9 A+16 B)+4 a^2 b (71 A+52 B)+2 a b^2 (59 A+132 B)+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(128 a^3 B+284 a^2 A b+264 a b^2 B+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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{192 a b d}-\frac{\sqrt{a+b} \left(48 a^4 A+160 a^3 b B+120 a^2 A b^2+40 a b^3 B-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 (8 a B+11 A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{a A \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}",1,"Result too large to show","B",0
371,1,3000,329,22.5815471,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\text{Result too large to show}","\frac{2 (a-b) \sqrt{a+b} \left(-8 a^2 B+10 a A b-9 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)}{15 b^4 d}+\frac{2 \sqrt{a+b} \left(-8 a^2 B+2 a b (5 A+B)+b^2 (5 A-9 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)}{15 b^3 d}+\frac{2 (5 A b-4 a B) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{15 b^2 d}+\frac{2 B \tan (c+d x) \sec (c+d x) \sqrt{a+b \sec (c+d x)}}{5 b d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((2*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sin[c + d*x])/(15*b^3) + (2*Sec[c + d*x]*(5*A*b*Sin[c + d*x] - 4*a*B*Sin[c + d*x]))/(15*b^2) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/(d*Sqrt[a + b*Sec[c + d*x]]) - (2*((2*a*A)/(3*b*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (3*B)/(5*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^2*B)/(15*b^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (A*Sqrt[Sec[c + d*x]])/(3*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*A*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^3*B*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (7*a*B*Sqrt[Sec[c + d*x]])/(15*b*Sqrt[b + a*Cos[c + d*x]]) + (2*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(15*b^3*Sqrt[b + a*Cos[c + d*x]]) - (3*a*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(5*b*Sqrt[b + a*Cos[c + d*x]]))*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*B + 2*a*b*(-5*A + B) + b^2*(5*A + 9*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*A*b + 8*a^2*B + 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]*(-1/15*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*B + 2*a*b*(-5*A + B) + b^2*(5*A + 9*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*A*b + 8*a^2*B + 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*B + 2*a*b*(-5*A + B) + b^2*(5*A + 9*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*A*b + 8*a^2*B + 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-10*a*A*b + 8*a^2*B + 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(8*a^2*B + 2*a*b*(-5*A + B) + b^2*(5*A + 9*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(8*a^2*B + 2*a*b*(-5*A + B) + b^2*(5*A + 9*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-10*a*A*b + 8*a^2*B + 9*b^2*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-10*a*A*b + 8*a^2*B + 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(8*a^2*B + 2*a*b*(-5*A + B) + b^2*(5*A + 9*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-10*a*A*b + 8*a^2*B + 9*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(8*a^2*B + 2*a*b*(-5*A + B) + b^2*(5*A + 9*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-10*a*A*b + 8*a^2*B + 9*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(15*b^3*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
372,1,372,261,15.8351235,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sec (c+d x) (a \cos (c+d x)+b) \left(\frac{2 (3 A b-2 a B) \sin (c+d x)}{3 b^2}+\frac{2 B \tan (c+d x)}{3 b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{2 \sqrt{\sec (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} \left(-\left((3 A b-2 a B) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)+2 b (B (b-2 a)+3 A b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 (a+b) (2 a B-3 A b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b^2 d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 (a-b) \sqrt{a+b} (3 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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d}-\frac{2 \sqrt{a+b} (3 A b-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}} 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 B \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b d}",1,"(2*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-3*A*b + 2*a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(3*A*b + (-2*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (3*A*b - 2*a*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^2*d*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[a + b*Sec[c + d*x]]) + ((b + a*Cos[c + d*x])*Sec[c + d*x]*((2*(3*A*b - 2*a*B)*Sin[c + d*x])/(3*b^2) + (2*B*Tan[c + d*x])/(3*b)))/(d*Sqrt[a + b*Sec[c + d*x]])","A",0
373,1,356,210,14.4022412,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 B \sin (c+d x) (a \cos (c+d x)+b) (A+B \sec (c+d x))}{b d \sqrt{a+b \sec (c+d x)} (A \cos (c+d x)+B)}-\frac{2 \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (A+B \sec (c+d x)) \left(-2 b (A+B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+B \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 B (a+b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{b d \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} (A \cos (c+d x)+B)}","\frac{2 \sqrt{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{2 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^2 d}",1,"(2*B*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x])*Sin[c + d*x])/(b*d*(B + A*Cos[c + d*x])*Sqrt[a + b*Sec[c + d*x]]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x])*(2*(a + b)*B*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(A + B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + B*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b*d*(B + A*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","A",1
374,1,145,208,2.3360505,"\int \frac{A+B \sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/Sqrt[a + b*Sec[c + d*x]],x]","\frac{4 \cos ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sec (c+d x) \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} \left((B-A) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 A \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \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}} 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}",1,"(4*Cos[(c + d*x)/2]^2*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*((-A + B)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*A*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[c + d*x])/(d*Sqrt[a + b*Sec[c + d*x]])","A",1
375,1,1027,348,17.0437349,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(-a A \sqrt{\frac{b-a}{a+b}} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^3\left(\frac{1}{2} (c+d x)\right)+A b \sqrt{\frac{b-a}{a+b}} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 i A b \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 i a B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+a A \sqrt{\frac{b-a}{a+b}} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan \left(\frac{1}{2} (c+d x)\right)+A b \sqrt{\frac{b-a}{a+b}} \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \tan \left(\frac{1}{2} (c+d x)\right)+2 i A b \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 i a B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-i A (a-b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i (A b-a B) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a \sqrt{\frac{b-a}{a+b}} d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)^{3/2} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}}}","\frac{\sqrt{a+b} (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{A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{a d}+\frac{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}} 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{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,"(Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(a*A*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]*Sqrt[1 - Tan[(c + d*x)/2]^2] + A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]*Sqrt[1 - Tan[(c + d*x)/2]^2] - a*A*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3*Sqrt[1 - Tan[(c + d*x)/2]^2] + A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3*Sqrt[1 - Tan[(c + d*x)/2]^2] + (2*I)*A*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*A*b*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (4*I)*a*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*A*(a - b)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*(A*b - a*B)*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*d*Sqrt[a + b*Sec[c + d*x]]*(1 + Tan[(c + d*x)/2]^2)^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)])","C",0
376,1,1639,435,16.2353331,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{A (b+a \cos (c+d x)) \sec (c+d x) \sin (2 (c+d x))}{4 a d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-4 a b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+3 A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)-8 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+8 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^2 \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+4 a b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-3 A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-3 a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)-i (a-b) (4 a B-3 A b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i \left(2 A a^2-b (A+4 B) a+3 A b^2\right) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-8 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+8 i a b B \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{4 a^2 \sqrt{\frac{b-a}{a+b}} d \sqrt{a+b \sec (c+d x)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{(3 A b-4 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 a^2 d}-\frac{\sqrt{a+b} (3 A b-2 a (A+2 B)) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 d}-\frac{(a-b) \sqrt{a+b} (3 A b-4 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^2 b d}-\frac{\sqrt{a+b} \left(4 a^2 A-4 a b 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}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 d}+\frac{A \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{2 a d}",1,"(A*(b + a*Cos[c + d*x])*Sec[c + d*x]*Sin[2*(c + d*x)])/(4*a*d*Sqrt[a + b*Sec[c + d*x]]) + (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-3*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 3*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 6*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 8*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - 3*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 3*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 4*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (8*I)*a*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(-3*A*b + 4*a*B)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(2*a^2*A + 3*A*b^2 - a*b*(A + 4*B))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a^2*Sqrt[(-a + b)/(a + b)]*d*Sqrt[a + b*Sec[c + d*x]]*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",1
377,1,1569,525,19.7887157,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{(b+a \cos (c+d x)) \sec (c+d x) \left(\frac{A \sin (c+d x)}{12 a}+\frac{(6 a B-5 A b) \sin (2 (c+d x))}{24 a^2}+\frac{A \sin (3 (c+d x))}{12 a}\right)}{d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{b+a \cos (c+d x)} \sqrt{\sec (c+d x)} \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-15 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)+15 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-16 a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)+18 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)-18 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)-30 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-32 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)+36 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)-30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+36 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+15 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)+15 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+16 a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+16 a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)-18 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)-18 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(16 A a^2-18 b B a+15 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a \left(12 B a^2+2 b (A-3 B) a+5 A b^2\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-30 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-24 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+48 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+36 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{24 a^3 d \sqrt{a+b \sec (c+d x)} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","-\frac{(5 A b-6 a B) \sin (c+d x) \cos (c+d x) \sqrt{a+b \sec (c+d x)}}{12 a^2 d}+\frac{\left(16 a^2 A-18 a b B+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(16 a^2 A+12 a^2 B-10 a A b-18 a b 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(16 a^2 A-18 a b 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}} 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{\sqrt{a+b} \left(-8 a^3 B+4 a^2 A b-6 a b^2 B+5 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}} \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{A \sin (c+d x) \cos ^2(c+d x) \sqrt{a+b \sec (c+d x)}}{3 a d}",1,"((b + a*Cos[c + d*x])*Sec[c + d*x]*((A*Sin[c + d*x])/(12*a) + ((-5*A*b + 6*a*B)*Sin[2*(c + d*x)])/(24*a^2) + (A*Sin[3*(c + d*x)])/(12*a)))/(d*Sqrt[a + b*Sec[c + d*x]]) - (Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^3*A*Tan[(c + d*x)/2] + 16*a^2*A*b*Tan[(c + d*x)/2] + 15*a*A*b^2*Tan[(c + d*x)/2] + 15*A*b^3*Tan[(c + d*x)/2] - 18*a^2*b*B*Tan[(c + d*x)/2] - 18*a*b^2*B*Tan[(c + d*x)/2] - 32*a^3*A*Tan[(c + d*x)/2]^3 - 30*a*A*b^2*Tan[(c + d*x)/2]^3 + 36*a^2*b*B*Tan[(c + d*x)/2]^3 + 16*a^3*A*Tan[(c + d*x)/2]^5 - 16*a^2*A*b*Tan[(c + d*x)/2]^5 + 15*a*A*b^2*Tan[(c + d*x)/2]^5 - 15*A*b^3*Tan[(c + d*x)/2]^5 - 18*a^2*b*B*Tan[(c + d*x)/2]^5 + 18*a*b^2*B*Tan[(c + d*x)/2]^5 - 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 36*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 24*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 36*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^2*A + 15*A*b^2 - 18*a*b*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(5*A*b^2 + 2*a*b*(A - 3*B) + 12*a^2*B)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a^3*d*Sqrt[a + b*Sec[c + d*x]]*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
378,1,3460,329,24.6439684,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{2 a^2 (A b-a B) \tan (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-8 a^3 B+6 a^2 A b+5 a b^2 B-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}} 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) (3 A b-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}} 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 B \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^2 d}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)) + (2*(a^2*A*b*Sin[c + d*x] - a^3*B*Sin[c + d*x]))/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*B*Tan[c + d*x])/(3*b^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])*((2*a^2*A)/(b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (A*b)/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a*B)/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^3*B)/(3*b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*A*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*A*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*B*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (7*a^2*B*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (b*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (a*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (2*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b^2*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) - (8*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]) + (5*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(3*A*b + (-4*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2)*(-1/3*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(3*A*b + (-4*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(3*A*b + (-4*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(-2*a^2 - a*b + b^2)*(3*A*b + (-4*a + b)*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(-2*a^2 - a*b + b^2)*(3*A*b + (-4*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(-2*a^2 - a*b + b^2)*(3*A*b + (-4*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(-2*a^2 - a*b + b^2)*(3*A*b + (-4*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-6*a^2*A*b + 3*A*b^3 + 8*a^3*B - 5*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(-a^2 + b^2)*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
379,1,467,275,18.0760997,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^2 \left(\frac{2 \left(-2 a^2 B+a A b+b^2 B\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}-\frac{2 \left(a A b \sin (c+d x)-a^2 B \sin (c+d x)\right)}{b \left(b^2-a^2\right) (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{3/2}}+\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b) \left(\left(2 a^2 B-a A b-b^2 B\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 (a+b) \left(2 a^2 B-a A b-b^2 B\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 b (a+b) (B (b-2 a)+A b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{b^2 d \left(b^2-a^2\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{3/2}}","\frac{2 a (A b-a B) \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 B+a A b+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)}{b^3 d \sqrt{a+b}}+\frac{2 (A b-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}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*(a*A*b - 2*a^2*B + b^2*B)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) - (2*(a*A*b*Sin[c + d*x] - a^2*B*Sin[c + d*x]))/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) + (2*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-(a*A*b) + 2*a^2*B - b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(A*b + (-2*a + b)*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-(a*A*b) + 2*a^2*B - b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^2*(-a^2 + b^2)*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2))","A",0
380,1,468,254,15.2460596,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec (c+d x) (a \cos (c+d x)+b)^2 (A+B \sec (c+d x)) \left(\frac{2 (A b \sin (c+d x)-a B \sin (c+d x))}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{2 (A b-a B) \sin (c+d x)}{b \left(b^2-a^2\right)}\right)}{d (a+b \sec (c+d x))^{3/2} (A \cos (c+d x)+B)}-\frac{2 \sqrt{\sec (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(-\left((A b-a B) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)\right)+2 b (a+b) (A-B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 (a+b) (a B-A b) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{d \left(b^3-a^2 b\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{3/2} (A \cos (c+d x)+B)}","-\frac{2 (A b-a B) \tan (c+d x)}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{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}} 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 (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 \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]*(A + B*Sec[c + d*x])*((-2*(A*b - a*B)*Sin[c + d*x])/(b*(-a^2 + b^2)) + (2*(A*b*Sin[c + d*x] - a*B*Sin[c + d*x]))/((-a^2 + b^2)*(b + a*Cos[c + d*x]))))/(d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)) - (2*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x])*(2*(a + b)*(-(A*b) + a*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(A - B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - (A*b - a*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/((-(a^2*b) + b^3)*d*(B + A*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(3/2))","A",0
381,1,1491,376,14.5807885,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 (a B-A b) \sin (c+d x)}{a \left(a^2-b^2\right)}-\frac{2 \left(a b B \sin (c+d x)-A b^2 \sin (c+d x)\right)}{a \left(a^2-b^2\right) (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^2}{d (B+A \cos (c+d x)) (a+b \sec (c+d x))^{3/2}}+\frac{2 \sqrt{\sec (c+d x)} (A+B \sec (c+d x)) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(-a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)+a b \sqrt{\frac{b-a}{a+b}} B \tan ^5\left(\frac{1}{2} (c+d x)\right)-A b^2 \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+a A b \sqrt{\frac{b-a}{a+b}} \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a^2 \sqrt{\frac{b-a}{a+b}} B \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a A b \sqrt{\frac{b-a}{a+b}} \tan ^3\left(\frac{1}{2} (c+d x)\right)+2 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-2 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-a^2 \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)-a b \sqrt{\frac{b-a}{a+b}} B \tan \left(\frac{1}{2} (c+d x)\right)+A b^2 \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+a A b \sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)+i (a-b) (a B-A b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+i (a-b) (2 A b+a (A-B)) F\left(i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+2 i A b^2 \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 i a^2 A \Pi \left(-\frac{a+b}{a-b};i \sinh ^{-1}\left(\sqrt{\frac{b-a}{a+b}} \tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a+b}{a-b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right) (b+a \cos (c+d x))^{3/2}}{a \sqrt{\frac{b-a}{a+b}} \left(a^2-b^2\right) d (B+A \cos (c+d x)) (a+b \sec (c+d x))^{3/2} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right) \sqrt{\frac{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{2 b (A b-a B) \tan (c+d x)}{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 (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)}{a b d \sqrt{a+b}}+\frac{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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a b d \sqrt{a+b}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*(-(A*b) + a*B)*Sin[c + d*x])/(a*(a^2 - b^2)) - (2*(-(A*b^2*Sin[c + d*x]) + a*b*B*Sin[c + d*x]))/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)) + (2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x])*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 2*a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 2*a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + a*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - a^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + a*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (2*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (2*I)*a^2*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(-(A*b) + a*B)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(2*A*b + a*(A - B))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",1
382,1,1597,427,19.6092147,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(b+a \cos (c+d x))^2 \sec ^2(c+d x) \left(\frac{2 \left(a b^2 B \sin (c+d x)-A b^3 \sin (c+d x)\right)}{a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))}-\frac{2 b (A b-a B) \sin (c+d x)}{a^2 \left(b^2-a^2\right)}\right)}{d (a+b \sec (c+d x))^{3/2}}-\frac{(b+a \cos (c+d x))^{3/2} \sec ^{\frac{3}{2}}(c+d x) \sqrt{\frac{1}{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)}} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1}} \left(3 A b^3 \tan ^5\left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan ^5\left(\frac{1}{2} (c+d x)\right)+a^3 A \tan ^5\left(\frac{1}{2} (c+d x)\right)-a^2 A b \tan ^5\left(\frac{1}{2} (c+d x)\right)-2 a b^2 B \tan ^5\left(\frac{1}{2} (c+d x)\right)+2 a^2 b B \tan ^5\left(\frac{1}{2} (c+d x)\right)+6 a A b^2 \tan ^3\left(\frac{1}{2} (c+d x)\right)-2 a^3 A \tan ^3\left(\frac{1}{2} (c+d x)\right)-4 a^2 b B \tan ^3\left(\frac{1}{2} (c+d x)\right)+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-6 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)+4 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-4 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}} \tan ^2\left(\frac{1}{2} (c+d x)\right)-3 A b^3 \tan \left(\frac{1}{2} (c+d x)\right)-3 a A b^2 \tan \left(\frac{1}{2} (c+d x)\right)+a^3 A \tan \left(\frac{1}{2} (c+d x)\right)+a^2 A b \tan \left(\frac{1}{2} (c+d x)\right)+2 a b^2 B \tan \left(\frac{1}{2} (c+d x)\right)+2 a^2 b B \tan \left(\frac{1}{2} (c+d x)\right)+(a+b) \left(A a^2+2 b B a-3 A b^2\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-2 a (a+b) (a B-A b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right) \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+6 A b^3 \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-6 a^2 A b \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}+4 a^3 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}-4 a b^2 B \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{1-\tan ^2\left(\frac{1}{2} (c+d x)\right)} \sqrt{\frac{-a \tan ^2\left(\frac{1}{2} (c+d x)\right)+b \tan ^2\left(\frac{1}{2} (c+d x)\right)+a+b}{a+b}}\right)}{a^2 \left(a^2-b^2\right) d (a+b \sec (c+d x))^{3/2} \sqrt{\tan ^2\left(\frac{1}{2} (c+d x)\right)+1} \left(a \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)-1\right)-b \left(\tan ^2\left(\frac{1}{2} (c+d x)\right)+1\right)\right)}","\frac{\sqrt{a+b} (3 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{b \left(a^2 A+2 a b B-3 A b^2\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 A+2 a b 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}} 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 (A-2 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)}{a^2 d \sqrt{a+b}}+\frac{A \sin (c+d x)}{a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((-2*b*(A*b - a*B)*Sin[c + d*x])/(a^2*(-a^2 + b^2)) + (2*(-(A*b^3*Sin[c + d*x]) + a*b^2*B*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(3/2)) - ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(a^3*A*Tan[(c + d*x)/2] + a^2*A*b*Tan[(c + d*x)/2] - 3*a*A*b^2*Tan[(c + d*x)/2] - 3*A*b^3*Tan[(c + d*x)/2] + 2*a^2*b*B*Tan[(c + d*x)/2] + 2*a*b^2*B*Tan[(c + d*x)/2] - 2*a^3*A*Tan[(c + d*x)/2]^3 + 6*a*A*b^2*Tan[(c + d*x)/2]^3 - 4*a^2*b*B*Tan[(c + d*x)/2]^3 + a^3*A*Tan[(c + d*x)/2]^5 - a^2*A*b*Tan[(c + d*x)/2]^5 - 3*a*A*b^2*Tan[(c + d*x)/2]^5 + 3*A*b^3*Tan[(c + d*x)/2]^5 + 2*a^2*b*B*Tan[(c + d*x)/2]^5 - 2*a*b^2*B*Tan[(c + d*x)/2]^5 - 6*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 6*a^2*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 6*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 4*a^3*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 4*a*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(a^2*A - 3*A*b^2 + 2*a*b*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(a + b)*(-(A*b) + a*B)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(a^2*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
383,1,2667,531,18.2990125,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(5 A b-4 a B) \sin (c+d x)}{4 a^2 d \sqrt{a+b \sec (c+d x)}}-\frac{\sqrt{a+b} \left(4 a^2 A-12 a b 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}} \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{\left(-2 a^2 (A+2 B)+a b (5 A-12 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{b \left(-4 a^3 B+7 a^2 A b+12 a b^2 B-15 A b^3\right) \tan (c+d x)}{4 a^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\left(-4 a^3 B+7 a^2 A b+12 a b^2 B-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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^3 b d \sqrt{a+b}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*((2*b^2*(A*b - a*B)*Sin[c + d*x])/(a^3*(-a^2 + b^2)) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(4*a^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) + ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(-7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 15*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 4*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 4*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 12*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 12*a*b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] + 14*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 30*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 - 8*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 24*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 - 7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 15*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 15*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 4*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 4*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - 12*a^2*b^2*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 12*a*b^3*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (8*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (24*I)*a^3*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b^3*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (8*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (22*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (30*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (24*I)*a^3*b*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (24*I)*a*b^3*B*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - I*(a - b)*(-7*a^2*A*b + 15*A*b^3 + 4*a^3*B - 12*a*b^2*B)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (2*I)*(a - b)*(2*a^3*A + 15*A*b^3 + a^2*b*(A - 8*B) + 2*a*b^2*(5*A - 6*B))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(4*a^3*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
384,1,2319,630,22.291825,"\int \frac{\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\text{Result too large to show}","-\frac{(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{12 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{\left(16 a^2 A-30 a b B+35 A b^2\right) \sin (c+d x)}{24 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{\sqrt{a+b} \left(-8 a^3 B+12 a^2 A b-30 a b^2 B+35 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}} \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^5 d}+\frac{\left(4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)+5 a b^2 (7 A-18 B)+105 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)}{24 a^4 d \sqrt{a+b}}+\frac{b \left(16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right) \tan (c+d x)}{24 a^4 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{\left(16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-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)}{24 a^4 b d \sqrt{a+b}}+\frac{A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(-1/12*((a^4*A - a^2*A*b^2 + 24*A*b^4 - 24*a*b^3*B)*Sin[c + d*x])/(a^4*(-a^2 + b^2)) - (2*(A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x]))/(a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])) + ((-11*A*b + 6*a*B)*Sin[2*(c + d*x)])/(24*a^3) + (A*Sin[3*(c + d*x)])/(12*a^2)))/(d*(a + b*Sec[c + d*x])^(3/2)) - ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^5*A*Tan[(c + d*x)/2] + 16*a^4*A*b*Tan[(c + d*x)/2] + 41*a^3*A*b^2*Tan[(c + d*x)/2] + 41*a^2*A*b^3*Tan[(c + d*x)/2] - 105*a*A*b^4*Tan[(c + d*x)/2] - 105*A*b^5*Tan[(c + d*x)/2] - 42*a^4*b*B*Tan[(c + d*x)/2] - 42*a^3*b^2*B*Tan[(c + d*x)/2] + 90*a^2*b^3*B*Tan[(c + d*x)/2] + 90*a*b^4*B*Tan[(c + d*x)/2] - 32*a^5*A*Tan[(c + d*x)/2]^3 - 82*a^3*A*b^2*Tan[(c + d*x)/2]^3 + 210*a*A*b^4*Tan[(c + d*x)/2]^3 + 84*a^4*b*B*Tan[(c + d*x)/2]^3 - 180*a^2*b^3*B*Tan[(c + d*x)/2]^3 + 16*a^5*A*Tan[(c + d*x)/2]^5 - 16*a^4*A*b*Tan[(c + d*x)/2]^5 + 41*a^3*A*b^2*Tan[(c + d*x)/2]^5 - 41*a^2*A*b^3*Tan[(c + d*x)/2]^5 - 105*a*A*b^4*Tan[(c + d*x)/2]^5 + 105*A*b^5*Tan[(c + d*x)/2]^5 - 42*a^4*b*B*Tan[(c + d*x)/2]^5 + 42*a^3*b^2*B*Tan[(c + d*x)/2]^5 + 90*a^2*b^3*B*Tan[(c + d*x)/2]^5 - 90*a*b^4*B*Tan[(c + d*x)/2]^5 - 72*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 138*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 210*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^5*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 132*a^3*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 180*a*b^4*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 72*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 138*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 210*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 48*a^5*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 132*a^3*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 180*a*b^4*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(a + b)*(-35*A*b^3 + 12*a^3*B - 2*a^2*b*(5*A + 9*B) + 3*a*b^2*(7*A + 10*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(24*a^4*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
385,1,4342,510,27.4200795,"\int \frac{\sec ^4(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^4*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \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 B+a A b+b^2 B\right) \tan (c+d x) \sqrt{a+b \sec (c+d x)}}{3 b^3 d \left(a^2-b^2\right)}-\frac{2 a^2 \left(-6 a^3 B+3 a^2 A b+10 a b^2 B-7 A b^3\right) \tan (c+d x)}{3 b^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(16 a^4 B-a^3 (8 A b-12 b B)-2 a^2 b^2 (3 A+8 B)+9 a b^3 (A-B)+b^4 (3 A-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 b^4 d \sqrt{a+b} \left(a^2-b^2\right)}-\frac{2 \left(-16 a^5 B+8 a^4 A b+28 a^3 b^2 B-15 a^2 A b^3-8 a b^4 B+3 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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^5 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(8*a^4*A*b - 15*a^2*A*b^3 + 3*A*b^5 - 16*a^5*B + 28*a^3*b^2*B - 8*a*b^4*B)*Sin[c + d*x])/(3*b^4*(-a^2 + b^2)^2) + (2*(a^2*A*b*Sin[c + d*x] - a^3*B*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (2*(-4*a^4*A*b*Sin[c + d*x] + 8*a^2*A*b^3*Sin[c + d*x] + 7*a^5*B*Sin[c + d*x] - 11*a^3*b^2*B*Sin[c + d*x]))/(3*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (2*B*Tan[c + d*x])/(3*b^3)))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*((5*a^2*A)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*A)/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (A*b^2)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (16*a^5*B)/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (28*a^3*B)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*a*b*B)/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^5*A*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (17*a^3*A*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (3*a*A*b*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a^2*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (16*a^6*B*Sqrt[Sec[c + d*x]])/(3*b^4*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (32*a^4*B*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (b^2*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (8*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (16*a^6*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^4*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (28*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*B - 9*a*b^3*(A + B) + b^4*(3*A + B) + 4*a^3*b*(2*A + 3*B) + 2*a^2*b^2*(-3*A + 8*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^4*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*B - 9*a*b^3*(A + B) + b^4*(3*A + B) + 4*a^3*b*(2*A + 3*B) + 2*a^2*b^2*(-3*A + 8*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*B - 9*a*b^3*(A + B) + b^4*(3*A + B) + 4*a^3*b*(2*A + 3*B) + 2*a^2*b^2*(-3*A + 8*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + (b*(a + b)*(-16*a^4*B - 9*a*b^3*(A + B) + b^4*(3*A + B) + 4*a^3*b*(2*A + 3*B) + 2*a^2*b^2*(-3*A + 8*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] + (b*(a + b)*(-16*a^4*B - 9*a*b^3*(A + B) + b^4*(3*A + B) + 4*a^3*b*(2*A + 3*B) + 2*a^2*b^2*(-3*A + 8*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 + (b*(a + b)*(-16*a^4*B - 9*a*b^3*(A + B) + b^4*(3*A + B) + 4*a^3*b*(2*A + 3*B) + 2*a^2*b^2*(-3*A + 8*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(-16*a^4*B - 9*a*b^3*(A + B) + b^4*(3*A + B) + 4*a^3*b*(2*A + 3*B) + 2*a^2*b^2*(-3*A + 8*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-8*a^4*A*b + 15*a^2*A*b^3 - 3*A*b^5 + 16*a^5*B - 28*a^3*b^2*B + 8*a*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
386,1,3920,417,26.7706974,"\int \frac{\sec ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 a^2 (A b-a B) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 a \left(-5 a^3 B+2 a^2 A b+9 a b^2 B-6 A b^3\right) \tan (c+d x)}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-8 a^3 B+2 a^2 b (A-3 B)+3 a b^2 (A+3 B)-3 b^3 (A-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 b^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(-8 a^4 B+2 a^3 A b+15 a^2 b^2 B-6 a A b^3-3 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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^4 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sin[c + d*x])/(3*b^3*(-a^2 + b^2)^2) - (2*(a*A*b*Sin[c + d*x] - a^2*B*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (2*(-(a^3*A*b*Sin[c + d*x]) + 5*a*A*b^3*Sin[c + d*x] + 4*a^4*B*Sin[c + d*x] - 8*a^2*b^2*B*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) - (2*(b + a*Cos[c + d*x])^2*((2*a^3*A)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*A*b)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (5*a^2*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a^4*B)/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (b^2*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (5*a^2*A*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*A*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (A*b^2*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*B*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (17*a^3*B*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (3*a*b*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (8*a^5*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (5*a^3*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*b*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(A - 3*B) + 8*a^3*B + 3*b^3*(A + B) - 2*a^2*b*(A + 3*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*(-1/3*(a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(A - 3*B) + 8*a^3*B + 3*b^3*(A + B) - 2*a^2*b*(A + 3*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(b^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) + (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(A - 3*B) + 8*a^3*B + 3*b^3*(A + B) - 2*a^2*b*(A + 3*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(3*a*b^2*(A - 3*B) + 8*a^3*B + 3*b^3*(A + B) - 2*a^2*b*(A + 3*B))*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(3*a*b^2*(A - 3*B) + 8*a^3*B + 3*b^3*(A + B) - 2*a^2*b*(A + 3*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(3*a*b^2*(A - 3*B) + 8*a^3*B + 3*b^3*(A + B) - 2*a^2*b*(A + 3*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) - ((2*(a + b)*(-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(3*a*b^2*(A - 3*B) + 8*a^3*B + 3*b^3*(A + B) - 2*a^2*b*(A + 3*B))*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-2*a^3*A*b + 6*a*A*b^3 + 8*a^4*B - 15*a^2*b^2*B + 3*b^4*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*b^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
387,1,3514,387,24.4594015,"\int \frac{\sec ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \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 B+a b (A+3 B)-3 b^2 (A+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 b^2 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{2 \left(2 a^3 B+a^2 A b-6 a b^2 B+3 A b^3\right) \tan (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(2 a^3 B+a^2 A b-6 a b^2 B+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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 b^3 d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-2*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(3*b^2*(-a^2 + b^2)^2) + (2*(A*b*Sin[c + d*x] - a*B*Sin[c + d*x]))/(3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (2*(2*a^2*A*b*Sin[c + d*x] + 2*A*b^3*Sin[c + d*x] + a^3*B*Sin[c + d*x] - 5*a*b^2*B*Sin[c + d*x]))/(3*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*((a^2*A)/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (A*b^2)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (2*a^3*B)/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (2*a*b*B)/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^3*A*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (a*A*b*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (5*a^2*B*Sqrt[Sec[c + d*x]])/(3*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (b^2*B*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (a^3*A*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (a*A*b*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a^2*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/((-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*a^4*B*Cos[2*(c + d*x)]*Sqrt[Sec[c + d*x]])/(3*b^2*(-a^2 + b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(2*(a + b)*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(A - 3*B) + 3*b^2*(A - B) + 2*a^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*d*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2)*((a*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Sin[c + d*x]*(2*(a + b)*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(A - 3*B) + 3*b^2*(A - B) + 2*a^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*(b + a*Cos[c + d*x])^(3/2)*Sqrt[Sec[(c + d*x)/2]^2]) - (Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*Tan[(c + d*x)/2]*(2*(a + b)*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(A - 3*B) + 3*b^2*(A - B) + 2*a^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + (2*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(((a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^4)/2 + ((a + b)*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] - (b*(a + b)*(a*b*(A - 3*B) + 3*b^2*(A - B) + 2*a^2*B)*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*((Cos[c + d*x]*Sin[c + d*x])/(1 + Cos[c + d*x])^2 - Sin[c + d*x]/(1 + Cos[c + d*x])))/Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])] + ((a + b)*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - (b*(a + b)*(a*b*(A - 3*B) + 3*b^2*(A - B) + 2*a^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*(-((a*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x]))) + ((b + a*Cos[c + d*x])*Sin[c + d*x])/((a + b)*(1 + Cos[c + d*x])^2)))/Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))] - a*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] - (a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Sin[c + d*x]*Tan[(c + d*x)/2] + (a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]^2 - (b*(a + b)*(a*b*(A - 3*B) + 3*b^2*(A - B) + 2*a^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2)/(Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*Sec[(c + d*x)/2]^2*Sqrt[1 - ((a - b)*Tan[(c + d*x)/2]^2)/(a + b)])/Sqrt[1 - Tan[(c + d*x)/2]^2]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]) + ((2*(a + b)*(a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*b*(a + b)*(a*b*(A - 3*B) + 3*b^2*(A - B) + 2*a^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (a^2*A*b + 3*A*b^3 + 2*a^3*B - 6*a*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(3*(-(a^2*b) + b^3)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[(c + d*x)/2]^2]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]])))","B",0
388,1,603,353,18.8731009,"\int \frac{\sec (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \sec ^{\frac{3}{2}}(c+d x) \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (a \cos (c+d x)+b)^2 (A+B \sec (c+d x)) \left(\left(a^2 B-4 a A b+3 b^2 B\right) \cos (c+d x) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)+2 (a+b) \left(a^2 B-4 a A b+3 b^2 B\right) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)+2 b (a+b) (3 a A-a B+A b-3 b B) \sqrt{\frac{\cos (c+d x)}{\cos (c+d x)+1}} \sqrt{\frac{a \cos (c+d x)+b}{(a+b) (\cos (c+d x)+1)}} F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a+b \sec (c+d x))^{5/2} (A \cos (c+d x)+B)}+\frac{\sec ^2(c+d x) (a \cos (c+d x)+b)^3 (A+B \sec (c+d x)) \left(-\frac{2 \left(a^2 B-4 a A b+3 b^2 B\right) \sin (c+d x)}{3 b \left(b^2-a^2\right)^2}+\frac{2 \left(A b^2 \sin (c+d x)-a b B \sin (c+d x)\right)}{3 a \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{2 \left(2 a^3 B \sin (c+d x)-5 a^2 A b \sin (c+d x)+2 a b^2 B \sin (c+d x)+A b^3 \sin (c+d x)\right)}{3 a \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{5/2} (A \cos (c+d x)+B)}","-\frac{2 \left(a^2 (-B)+4 a A b-3 b^2 B\right) \tan (c+d x)}{3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 (A b-a B) \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 (-B)+4 a A b-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)}{3 b^2 d (a-b) (a+b)^{3/2}}+\frac{2 (3 a A+a B-A b-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)}{3 b d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((-2*(-4*a*A*b + a^2*B + 3*b^2*B)*Sin[c + d*x])/(3*b*(-a^2 + b^2)^2) + (2*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x]))/(3*a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (2*(-5*a^2*A*b*Sin[c + d*x] + A*b^3*Sin[c + d*x] + 2*a^3*B*Sin[c + d*x] + 2*a*b^2*B*Sin[c + d*x]))/(3*a*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x])*(2*(a + b)*(-4*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + 2*b*(a + b)*(3*a*A + A*b - a*B - 3*b*B)*Sqrt[Cos[c + d*x]/(1 + Cos[c + d*x])]*Sqrt[(b + a*Cos[c + d*x])/((a + b)*(1 + Cos[c + d*x]))]*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] + (-4*a*A*b + a^2*B + 3*b^2*B)*Cos[c + d*x]*(b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2]))/(3*b*(a^2 - b^2)^2*d*(B + A*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*(a + b*Sec[c + d*x])^(5/2))","A",0
389,1,2083,495,16.8064047,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","-\frac{2 A \sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^3 d}+\frac{2 b (A b-a B) \tan (c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(-4 a^3 B+7 a^2 A b-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}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b d (a-b) (a+b)^{3/2}}+\frac{2 b \left(-4 a^3 B+7 a^2 A b-3 A b^3\right) \tan (c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^3 B+6 a^2 A b+a^2 b B-a A b^2-3 A b^3\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{3 a^2 b d (a-b) (a+b)^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*(-7*a^2*A*b + 3*A*b^3 + 4*a^3*B)*Sin[c + d*x])/(3*a^2*(a^2 - b^2)^2) - (2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-8*a^2*A*b^2*Sin[c + d*x] + 4*A*b^4*Sin[c + d*x] + 5*a^3*b*B*Sin[c + d*x] - a*b^3*B*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(5/2)) + (2*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] + 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 3*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 3*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2] - 4*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 4*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2] - 14*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 6*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^3 + 8*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^3 + 7*a^3*A*b*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 7*a^2*A*b^2*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 3*a*A*b^3*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 + 3*A*b^4*Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]^5 - 4*a^4*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 + 4*a^3*b*Sqrt[(-a + b)/(a + b)]*B*Tan[(c + d*x)/2]^5 - (6*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (12*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*a^4*A*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (12*I)*a^2*A*b^2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - (6*I)*A*b^4*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(-7*a^2*A*b + 3*A*b^3 + 4*a^3*B)*EllipticE[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + I*(a - b)*(-4*a*A*b^2 - 6*A*b^3 + 3*a^3*(A - B) + a^2*b*(9*A + B))*EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(c + d*x)/2]], (a + b)/(a - b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a^2*Sqrt[(-a + b)/(a + b)]*(a^2 - b^2)^2*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(5/2)*(-1 + Tan[(c + d*x)/2]^2)*Sqrt[(1 + Tan[(c + d*x)/2]^2)/(1 - Tan[(c + d*x)/2]^2)]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","C",0
390,1,2366,582,21.9866768,"\int \frac{\cos (c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{\sqrt{a+b} (5 A b-2 a B) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{a^4 d}+\frac{b \left(3 a^2 A+2 a b B-5 A b^2\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(-3 a^3 (A-4 B)-a^2 b (21 A+2 B)+a b^2 (5 A-6 B)+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)}{3 a^3 d \sqrt{a+b} \left(a^2-b^2\right)}+\frac{b \left(3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+15 A b^4\right) \tan (c+d x)}{3 a^3 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^4 A+14 a^3 b B-26 a^2 A b^2-6 a b^3 B+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{A \sin (c+d x)}{a d (a+b \sec (c+d x))^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((-2*b*(-10*a^2*A*b + 6*A*b^3 + 7*a^3*B - 3*a*b^2*B)*Sin[c + d*x])/(3*a^3*(-a^2 + b^2)^2) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (2*(-11*a^2*A*b^3*Sin[c + d*x] + 7*A*b^5*Sin[c + d*x] + 8*a^3*b^2*B*Sin[c + d*x] - 4*a*b^4*B*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(3*a^5*A*Tan[(c + d*x)/2] + 3*a^4*A*b*Tan[(c + d*x)/2] - 26*a^3*A*b^2*Tan[(c + d*x)/2] - 26*a^2*A*b^3*Tan[(c + d*x)/2] + 15*a*A*b^4*Tan[(c + d*x)/2] + 15*A*b^5*Tan[(c + d*x)/2] + 14*a^4*b*B*Tan[(c + d*x)/2] + 14*a^3*b^2*B*Tan[(c + d*x)/2] - 6*a^2*b^3*B*Tan[(c + d*x)/2] - 6*a*b^4*B*Tan[(c + d*x)/2] - 6*a^5*A*Tan[(c + d*x)/2]^3 + 52*a^3*A*b^2*Tan[(c + d*x)/2]^3 - 30*a*A*b^4*Tan[(c + d*x)/2]^3 - 28*a^4*b*B*Tan[(c + d*x)/2]^3 + 12*a^2*b^3*B*Tan[(c + d*x)/2]^3 + 3*a^5*A*Tan[(c + d*x)/2]^5 - 3*a^4*A*b*Tan[(c + d*x)/2]^5 - 26*a^3*A*b^2*Tan[(c + d*x)/2]^5 + 26*a^2*A*b^3*Tan[(c + d*x)/2]^5 + 15*a*A*b^4*Tan[(c + d*x)/2]^5 - 15*A*b^5*Tan[(c + d*x)/2]^5 + 14*a^4*b*B*Tan[(c + d*x)/2]^5 - 14*a^3*b^2*B*Tan[(c + d*x)/2]^5 - 6*a^2*b^3*B*Tan[(c + d*x)/2]^5 + 6*a*b^4*B*Tan[(c + d*x)/2]^5 - 30*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 12*a^5*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 24*a^3*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 12*a*b^4*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 60*a^2*A*b^3*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 30*A*b^5*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 12*a^5*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 24*a^3*b^2*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + 12*a*b^4*B*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Tan[(c + d*x)/2]^2*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] + (a + b)*(3*a^4*A - 26*a^2*A*b^2 + 15*A*b^4 + 14*a^3*b*B - 6*a*b^3*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 2*a*(a + b)*(5*A*b^3 + 3*a^3*B + 3*a^2*b*(-2*A + B) - a*b^2*(3*A + 2*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*(1 + Tan[(c + d*x)/2]^2)*Sqrt[(a + b - a*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)]))/(3*a*(a^3 - a*b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*Sqrt[1 + Tan[(c + d*x)/2]^2]*(a*(-1 + Tan[(c + d*x)/2]^2) - b*(1 + Tan[(c + d*x)/2]^2)))","B",0
391,1,821,686,15.2672016,"\int \frac{\cos ^2(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^2*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(b+a \cos (c+d x))^3 \sec ^3(c+d x) \left(\frac{2 \left(10 B a^3-13 A b a^2-6 b^2 B a+9 A b^3\right) \sin (c+d x) b^2}{3 a^4 \left(b^2-a^2\right)^2}-\frac{2 \left(A b^5 \sin (c+d x)-a b^4 B \sin (c+d x)\right)}{3 a^4 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}-\frac{2 \left(10 A \sin (c+d x) b^6-7 a B \sin (c+d x) b^5-14 a^2 A \sin (c+d x) b^4+11 a^3 B \sin (c+d x) b^3\right)}{3 a^4 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}+\frac{A \sin (2 (c+d x))}{4 a^3}\right)}{d (a+b \sec (c+d x))^{5/2}}-\frac{(b+a \cos (c+d x))^2 \sec (c+d x) \left(-a (a+b) \left(12 B a^5-33 A b a^4-104 b^2 B a^3+170 A b^3 a^2+60 b^4 B a-105 A b^5\right) E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}} \sec ^2\left(\frac{1}{2} (c+d x)\right)+b (a+b) \left(6 (A+2 B) a^5-3 b (13 A+48 B) a^4+4 b^2 (57 A+10 B) a^3+2 b^3 (60 B-29 A) a^2-30 b^4 (7 A+2 B) a+105 A b^5\right) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}} \sec ^2\left(\frac{1}{2} (c+d x)\right)+3 (a-b)^2 (a+b)^2 \left(4 A a^2-20 b B a+35 A b^2\right) \left((a-b) F\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)-2 a \Pi \left(-1;\sin ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{a-b}{a+b}\right)\right) \sqrt{\frac{(b+a \cos (c+d x)) \sec ^2\left(\frac{1}{2} (c+d x)\right)}{a+b}} \sec ^2\left(\frac{1}{2} (c+d x)\right)-a \left(12 B a^5-33 A b a^4-104 b^2 B a^3+170 A b^3 a^2+60 b^4 B a-105 A b^5\right) (b+a \cos (c+d x)) \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} \sec (c+d x) \tan \left(\frac{1}{2} (c+d x)\right)\right)}{12 a^5 \left(a^2-b^2\right)^2 d \left(\cos (c+d x) \sec ^2\left(\frac{1}{2} (c+d x)\right)\right)^{3/2} (a+b \sec (c+d x))^{5/2}}","-\frac{(7 A b-4 a B) \sin (c+d x)}{4 a^2 d (a+b \sec (c+d x))^{3/2}}-\frac{\sqrt{a+b} \left(4 a^2 A-20 a b B+35 A b^2\right) \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \Pi \left(\frac{a+b}{a};\sin ^{-1}\left(\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{4 a^5 d}-\frac{b \left(-12 a^3 B+27 a^2 A b+20 a b^2 B-35 A b^3\right) \tan (c+d x)}{12 a^3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{\left(6 a^4 (A+2 B)-a^3 (27 A b-84 b B)-5 a^2 b^2 (27 A+4 B)+5 a b^3 (7 A-12 B)+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{b \left(-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 a b^4 B+105 A b^5\right) \tan (c+d x)}{12 a^4 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{\left(-12 a^5 B+33 a^4 A b+104 a^3 b^2 B-170 a^2 A b^3-60 a b^4 B+105 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}} 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 b d (a-b) (a+b)^{3/2}}+\frac{A \sin (c+d x) \cos (c+d x)}{2 a d (a+b \sec (c+d x))^{3/2}}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^3*((2*b^2*(-13*a^2*A*b + 9*A*b^3 + 10*a^3*B - 6*a*b^2*B)*Sin[c + d*x])/(3*a^4*(-a^2 + b^2)^2) - (2*(A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-14*a^2*A*b^4*Sin[c + d*x] + 10*A*b^6*Sin[c + d*x] + 11*a^3*b^3*B*Sin[c + d*x] - 7*a*b^5*B*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(4*a^3)))/(d*(a + b*Sec[c + d*x])^(5/2)) - ((b + a*Cos[c + d*x])^2*Sec[c + d*x]*(-(a*(a + b)*(-33*a^4*A*b + 170*a^2*A*b^3 - 105*A*b^5 + 12*a^5*B - 104*a^3*b^2*B + 60*a*b^4*B)*EllipticE[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]) + b*(a + b)*(105*A*b^5 + 6*a^5*(A + 2*B) - 30*a*b^4*(7*A + 2*B) + 4*a^3*b^2*(57*A + 10*B) - 3*a^4*b*(13*A + 48*B) + 2*a^2*b^3*(-29*A + 60*B))*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + 3*(a - b)^2*(a + b)^2*(4*a^2*A + 35*A*b^2 - 20*a*b*B)*((a - b)*EllipticF[ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)] - 2*a*EllipticPi[-1, ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - a*(-33*a^4*A*b + 170*a^2*A*b^3 - 105*A*b^5 + 12*a^5*B - 104*a^3*b^2*B + 60*a*b^4*B)*(b + a*Cos[c + d*x])*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*Sec[c + d*x]*Tan[(c + d*x)/2]))/(12*a^5*(a^2 - b^2)^2*d*(Cos[c + d*x]*Sec[(c + d*x)/2]^2)^(3/2)*(a + b*Sec[c + d*x])^(5/2))","A",0
392,1,248,105,10.6920394,"\int \frac{\sec (e+f x) (A+A \sec (e+f x))}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(A + A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],x]","\frac{A (\sec (e+f x)+1) \left(2 \tan \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)+\frac{\left(\tan ^2\left(\frac{1}{2} (e+f x)\right)-1\right) \sqrt{\sec ^2\left(\frac{1}{2} (e+f x)\right)} \sqrt{\cos ^2\left(\frac{1}{2} (e+f x)\right) \sec (e+f x)} \left(\tan \left(\frac{1}{2} (e+f x)\right) (a \cos (e+f x)+b)+\frac{\sqrt{\frac{a-b}{a+b}} (a+b) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} E\left(\sin ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a+b}{a-b}\right)}{\sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}}}\right)}{\sqrt{\sec (e+f x)}}\right)}{b f \sqrt{a+b \sec (e+f x)}}","-\frac{2 A (a-b) \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right)|\frac{a+b}{a-b}\right)}{b^2 f}",1,"(A*(1 + Sec[e + f*x])*(2*(b + a*Cos[e + f*x])*Tan[(e + f*x)/2] + (Sqrt[Sec[(e + f*x)/2]^2]*Sqrt[Cos[(e + f*x)/2]^2*Sec[e + f*x]]*((Sqrt[(a - b)/(a + b)]*(a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*EllipticE[ArcSin[Sqrt[(a - b)/(a + b)]*Tan[(e + f*x)/2]], (a + b)/(a - b)])/Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])] + (b + a*Cos[e + f*x])*Tan[(e + f*x)/2])*(-1 + Tan[(e + f*x)/2]^2))/Sqrt[Sec[e + f*x]]))/(b*f*Sqrt[a + b*Sec[e + f*x]])","B",0
393,1,211,107,7.9395146,"\int \frac{\sec (e+f x) (A-A \sec (e+f x))}{\sqrt{a+b \sec (e+f x)}} \, dx","Integrate[(Sec[e + f*x]*(A - A*Sec[e + f*x]))/Sqrt[a + b*Sec[e + f*x]],x]","\frac{A (a+b) \sec ^2\left(\frac{1}{2} (e+f x)\right) \sqrt{\sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \left(\sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)+1} E\left(\sin ^{-1}\left(\tan \left(\frac{1}{2} (e+f x)\right)\right)|\frac{a-b}{a+b}\right)-\sin (e+f x) \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}}\right)}{b f \left(\frac{1}{\cos (e+f x)+1}\right)^{3/2} \sqrt{a+b \sec (e+f x)}}","\frac{2 A \sqrt{a-b} (a+b) \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} E\left(\sin ^{-1}\left(\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a-b}}\right)|\frac{a-b}{a+b}\right)}{b^2 f}",1,"(A*(a + b)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*Sec[(e + f*x)/2]^2*Sqrt[Sec[e + f*x]]*(Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*EllipticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sqrt[1 + Sec[e + f*x]] - Sqrt[(1 + Cos[e + f*x])^(-1)]*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*Sqrt[Sec[e + f*x]]*Sin[e + f*x]))/(b*f*((1 + Cos[e + f*x])^(-1))^(3/2)*Sqrt[a + b*Sec[e + f*x]])","A",0
394,1,132,180,1.8570192,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(20 (a B+A b) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-12 (5 a A+3 b B) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) (10 (a B+A b) \cos (c+d x)+3 (5 a A+3 b B) \cos (2 (c+d x))+15 (a A+b B))\right)}{30 d}","\frac{2 (a B+A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 (5 a A+3 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\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)}{3 d}-\frac{2 (5 a A+3 b 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 b B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 d}",1,"(Sec[c + d*x]^(5/2)*(-12*(5*a*A + 3*b*B)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*(A*b + a*B)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(a*A + b*B) + 10*(A*b + a*B)*Cos[c + d*x] + 3*(5*a*A + 3*b*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
395,1,104,143,0.8862204,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left((3 a A+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) (3 (a B+A b) \cos (c+d x)+b B)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 (a B+A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{d}+\frac{2 (3 a A+b B) \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 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)}{d}+\frac{2 b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(-3*(A*b + a*B)*EllipticE[(c + d*x)/2, 2] + (3*a*A + b*B)*EllipticF[(c + d*x)/2, 2] + ((b*B + 3*(A*b + a*B)*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
396,1,84,111,0.2940626,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \left((a B+A b) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+(a A-b B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+b B \sin (c+d x)\right)}{d}","\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)}{d}+\frac{2 (a A-b B) \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 b B \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(2*Sqrt[Sec[c + d*x]]*((a*A - b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + b*B*Sin[c + d*x]))/d","A",1
397,1,90,115,0.2623203,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 (a A+3 b B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (a B+A b) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a A \sin (2 (c+d x))\right)}{3 d}","\frac{2 (a A+3 b B) \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 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)}{d}+\frac{2 a A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(6*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(a*A + 3*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*A*Sin[2*(c + d*x)]))/(3*d)","A",1
398,1,108,148,0.6941138,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (3 a A \cos (c+d x)+5 a B+5 A b)+10 (a B+A b) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 (3 a A+5 b B) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{15 d}","\frac{2 (a B+A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}+\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)}{3 d}+\frac{2 (3 a A+5 b 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 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(6*(3*a*A + 5*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (5*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
399,1,125,180,1.0809516,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) (42 (a B+A b) \cos (c+d x)+15 a A \cos (2 (c+d x))+65 a A+70 b B)+20 (5 a A+7 b B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+252 (a B+A b) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 (a B+A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 a A+7 b B) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 (5 a A+7 b B) \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{6 (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)}{5 d}+\frac{2 a A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(252*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(5*a*A + 7*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (65*a*A + 70*b*B + 42*(A*b + a*B)*Cos[c + d*x] + 15*a*A*Cos[2*(c + d*x)])*Sin[2*(c + d*x)]))/(210*d)","A",1
400,1,221,263,4.4864391,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\sec ^{\frac{7}{2}}(c+d x) \left(40 \left(7 a^2 B+14 a A b+5 b^2 B\right) \cos ^{\frac{7}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-168 \left(5 a^2 A+6 a b B+3 A b^2\right) \cos ^{\frac{7}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(21 \left(15 a^2 A+26 a b B+13 A b^2\right) \cos (c+d x)+10 \left(7 a^2 B+14 a A b+5 b^2 B\right) \cos (2 (c+d x))+105 a^2 A \cos (3 (c+d x))+70 a^2 B+140 a A b+126 a b B \cos (3 (c+d x))+63 A b^2 \cos (3 (c+d x))+110 b^2 B\right)\right)}{420 d}","\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) \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 \left(7 a (a B+2 A b)+5 b^2 B\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(7 a (a B+2 A b)+5 b^2 B\right) \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 b (9 a B+7 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))}{7 d}",1,"(Sec[c + d*x]^(7/2)*(-168*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Cos[c + d*x]^(7/2)*EllipticE[(c + d*x)/2, 2] + 40*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Cos[c + d*x]^(7/2)*EllipticF[(c + d*x)/2, 2] + 2*(140*a*A*b + 70*a^2*B + 110*b^2*B + 21*(15*a^2*A + 13*A*b^2 + 26*a*b*B)*Cos[c + d*x] + 10*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Cos[2*(c + d*x)] + 105*a^2*A*Cos[3*(c + d*x)] + 63*A*b^2*Cos[3*(c + d*x)] + 126*a*b*B*Cos[3*(c + d*x)])*Sin[c + d*x]))/(420*d)","A",1
401,1,171,221,2.7213371,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(20 \left(3 a^2 A+2 a b B+A b^2\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-12 \left(5 a^2 B+10 a A b+3 b^2 B\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+2 \sin (c+d x) \left(3 \left(5 a^2 B+10 a A b+3 b^2 B\right) \cos (2 (c+d x))+15 \left(a^2 B+2 a A b+b^2 B\right)+10 b (2 a B+A b) \cos (c+d x)\right)\right)}{30 d}","\frac{2 \left(3 a^2 A+2 a b B+A b^2\right) \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 \left(5 a (a B+2 A b)+3 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}-\frac{2 \left(5 a (a B+2 A b)+3 b^2 B\right) \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 b (7 a B+5 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))}{5 d}",1,"(Sec[c + d*x]^(5/2)*(-12*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*(3*a^2*A + A*b^2 + 2*a*b*B)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*(15*(2*a*A*b + a^2*B + b^2*B) + 10*b*(A*b + 2*a*B)*Cos[c + d*x] + 3*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
402,1,125,177,1.2814041,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \left(a^2 A-2 a b B-A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b \sin (c+d x) (3 (2 a B+A b) \cos (c+d x)+b B)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\right) \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 \left(a^2 A-2 a b B-A b^2\right) \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 b (5 a B+3 A b) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 b B \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}{3 d}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(3*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE[(c + d*x)/2, 2] + (6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(c + d*x)/2, 2] + (b*(b*B + 3*(A*b + 2*a*B)*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
403,1,124,161,0.7584718,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (c+d x) \left(a^2 A \cos (c+d x)+3 b^2 B\right)+2 \left(a^2 A+6 a b B+3 A b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \left(a^2 B+2 a A b-b^2 B\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","\frac{2 \left(a^2 A+6 a b B+3 A b^2\right) \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^2 A \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}-\frac{2 \left(b^2 B-a (a B+2 A b)\right) \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 b^2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{d}",1,"(Sqrt[Sec[c + d*x]]*(6*(2*a*A*b + a^2*B - b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 2*(a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(3*b^2*B + a^2*A*Cos[c + d*x])*Sin[c + d*x]))/(3*d)","A",1
404,1,128,171,0.9600296,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(10 \left(a^2 B+2 a A b+3 b^2 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \left(3 a^2 A+10 a b B+5 A b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \sin (2 (c+d x)) (3 a A \cos (c+d x)+5 a B+10 A b)\right)}{15 d}","\frac{2 \left(a^2 B+2 a A b+3 b^2 B\right) \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 \left(3 a^2 A+10 a b B+5 A b^2\right) \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^2 A \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Sec[c + d*x]]*(6*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 10*(2*a*A*b + a^2*B + 3*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*(10*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[2*(c + d*x)]))/(15*d)","A",1
405,1,161,213,1.5836005,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(5 \left(3 a^2 A \cos (2 (c+d x))+13 a^2 A+28 a b B+14 A b^2\right)+42 a (a B+2 A b) \cos (c+d x)\right)+20 \left(5 a^2 A+14 a b B+7 A b^2\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(3 a^2 B+6 a A b+5 b^2 B\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 \left(5 a^2 A+14 a b B+7 A b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a^2 A+14 a b B+7 A b^2\right) \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 \left(3 a^2 B+6 a A b+5 b^2 B\right) \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^2 A \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a (a B+2 A b) \sin (c+d x)}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*(6*a*A*b + 3*a^2*B + 5*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (42*a*(2*A*b + a*B)*Cos[c + d*x] + 5*(13*a^2*A + 14*A*b^2 + 28*a*b*B + 3*a^2*A*Cos[2*(c + d*x)]))*Sin[2*(c + d*x)]))/(210*d)","A",1
406,1,189,254,1.8704871,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(7 \left(43 a^2 A+72 a b B+36 A b^2\right) \cos (c+d x)+5 \left(7 a^2 A \cos (3 (c+d x))+78 a^2 B+18 a (a B+2 A b) \cos (2 (c+d x))+156 a A b+84 b^2 B\right)\right)+120 \left(5 a^2 B+10 a A b+7 b^2 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 \left(7 a^2 A+18 a b B+9 A b^2\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 \left(7 a^2 A+18 a b B+9 A b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(7 a^2 A+18 a b B+9 A b^2\right) \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^2 A \sin (c+d x)}{9 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(5 a (a B+2 A b)+7 b^2 B\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a (a B+2 A b)+7 b^2 B\right) \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 (a B+2 A b) \sin (c+d x)}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*(7*a^2*A + 9*A*b^2 + 18*a*b*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*(10*a*A*b + 5*a^2*B + 7*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*(43*a^2*A + 36*A*b^2 + 72*a*b*B)*Cos[c + d*x] + 5*(156*a*A*b + 78*a^2*B + 84*b^2*B + 18*a*(2*A*b + a*B)*Cos[2*(c + d*x)] + 7*a^2*A*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
407,1,452,345,6.5650232,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{\cos ^4(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \left(2 \left(35 a^3 B+105 a^2 A b+75 a b^2 B+25 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{2 \left(-105 a^3 A-189 a^2 b B-189 a A b^2-49 b^3 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)}}\right)}{105 d (a \cos (c+d x)+b)^3 (A \cos (c+d x)+B)}+\frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \left(\frac{2}{45} \sec ^2(c+d x) \left(27 a^2 b B \sin (c+d x)+27 a A b^2 \sin (c+d x)+7 b^3 B \sin (c+d x)\right)+\frac{2}{15} \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) \sin (c+d x)+\frac{2}{21} \sec (c+d x) \left(7 a^3 B \sin (c+d x)+21 a^2 A b \sin (c+d x)+15 a b^2 B \sin (c+d x)+5 A b^3 \sin (c+d x)\right)+\frac{2}{7} \sec ^3(c+d x) \left(3 a b^2 B \sin (c+d x)+A b^3 \sin (c+d x)\right)+\frac{2}{9} b^3 B \tan (c+d x) \sec ^3(c+d x)\right)}{d \sec ^{\frac{7}{2}}(c+d x) (a \cos (c+d x)+b)^3 (A \cos (c+d x)+B)}","\frac{2 b \left(22 a^2 B+27 a A b+7 b^2 B\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{45 d}+\frac{2 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{15 d}+\frac{2 \left(7 a^3 B+21 a^2 A b+15 a b^2 B+5 A b^3\right) \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 \left(15 a^3 A+27 a^2 b B+27 a A b^2+7 b^3 B\right) \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 b^2 (13 a B+9 A b) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{63 d}+\frac{2 b B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2}{9 d}",1,"(Cos[c + d*x]^4*((2*(-105*a^3*A - 189*a*A*b^2 - 189*a^2*b*B - 49*b^3*B)*EllipticE[(c + d*x)/2, 2])/(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + 2*(105*a^2*A*b + 25*A*b^3 + 35*a^3*B + 75*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2]*Sqrt[Sec[c + d*x]])*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/(105*d*(b + a*Cos[c + d*x])^3*(B + A*Cos[c + d*x])) + ((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*((2*(15*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 7*b^3*B)*Sin[c + d*x])/15 + (2*Sec[c + d*x]^3*(A*b^3*Sin[c + d*x] + 3*a*b^2*B*Sin[c + d*x]))/7 + (2*Sec[c + d*x]*(21*a^2*A*b*Sin[c + d*x] + 5*A*b^3*Sin[c + d*x] + 7*a^3*B*Sin[c + d*x] + 15*a*b^2*B*Sin[c + d*x]))/21 + (2*Sec[c + d*x]^2*(27*a*A*b^2*Sin[c + d*x] + 27*a^2*b*B*Sin[c + d*x] + 7*b^3*B*Sin[c + d*x]))/45 + (2*b^3*B*Sec[c + d*x]^3*Tan[c + d*x])/9))/(d*(b + a*Cos[c + d*x])^3*(B + A*Cos[c + d*x])*Sec[c + d*x]^(7/2))","A",1
408,1,225,295,3.8759659,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{\sec (c+d x)} \left(5 b \left(21 a^2 B+21 a A b+5 b^2 B\right) \tan (c+d x)+21 \left(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right) \sin (c+d x)+5 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-21 \left(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+21 b^2 (3 a B+A b) \tan (c+d x) \sec (c+d x)+15 b^3 B \tan (c+d x) \sec ^2(c+d x)\right)}{105 d}","\frac{2 b \left(18 a^2 B+21 a A b+5 b^2 B\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{21 d}+\frac{2 \left(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(21 a^3 A+21 a^2 b B+21 a A b^2+5 b^3 B\right) \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 \left(5 a^3 B+15 a^2 A b+9 a b^2 B+3 A b^3\right) \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 b^2 (11 a B+7 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{35 d}+\frac{2 b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2}{7 d}",1,"(2*Sqrt[Sec[c + d*x]]*(-21*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 5*(21*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 21*(15*a^2*A*b + 3*A*b^3 + 5*a^3*B + 9*a*b^2*B)*Sin[c + d*x] + 5*b*(21*a*A*b + 21*a^2*B + 5*b^2*B)*Tan[c + d*x] + 21*b^2*(A*b + 3*a*B)*Sec[c + d*x]*Tan[c + d*x] + 15*b^3*B*Sec[c + d*x]^2*Tan[c + d*x]))/(105*d)","A",1
409,1,190,244,2.5482088,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(2 b \sin (c+d x) \left(9 \left(5 a^2 B+5 a A b+b^2 B\right) \cos (2 (c+d x))+15 \left(3 a^2 B+3 a A b+b^2 B\right)+10 b (3 a B+A b) \cos (c+d x)\right)+20 \left(3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right) \cos ^{\frac{5}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+12 \left(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right) \cos ^{\frac{5}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 b \left(14 a^2 B+15 a A b+3 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 \left(3 a^3 B+9 a^2 A b+3 a b^2 B+A b^3\right) \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 \left(5 a^3 A-15 a^2 b B-15 a A b^2-3 b^3 B\right) \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 b^2 (9 a B+5 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{15 d}+\frac{2 b B \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}{5 d}",1,"(Sec[c + d*x]^(5/2)*(12*(5*a^3*A - 15*a*A*b^2 - 15*a^2*b*B - 3*b^3*B)*Cos[c + d*x]^(5/2)*EllipticE[(c + d*x)/2, 2] + 20*(9*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B)*Cos[c + d*x]^(5/2)*EllipticF[(c + d*x)/2, 2] + 2*b*(15*(3*a*A*b + 3*a^2*B + b^2*B) + 10*b*(A*b + 3*a*B)*Cos[c + d*x] + 9*(5*a*A*b + 5*a^2*B + b^2*B)*Cos[2*(c + d*x)])*Sin[c + d*x]))/(30*d)","A",1
410,1,166,239,1.9525852,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \left(\frac{\sin (c+d x) \left(a^3 A \cos (2 (c+d x))+a^3 A+6 b^2 (3 a B+A b) \cos (c+d x)+2 b^3 B\right)}{\cos ^{\frac{3}{2}}(c+d x)}+2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+6 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{3 d}","-\frac{2 b \left(2 a^2 A-9 a b B-3 A b^2\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d}+\frac{2 \left(a^3 A+9 a^2 b B+9 a A b^2+b^3 B\right) \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 \left(a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right) \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 b^2 (a A-b B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 d}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{3 d \sqrt{\sec (c+d x)}}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(6*(3*a^2*A*b - A*b^3 + a^3*B - 3*a*b^2*B)*EllipticE[(c + d*x)/2, 2] + 2*(a^3*A + 9*a*A*b^2 + 9*a^2*b*B + b^3*B)*EllipticF[(c + d*x)/2, 2] + ((a^3*A + 2*b^3*B + 6*b^2*(A*b + 3*a*B)*Cos[c + d*x] + a^3*A*Cos[2*(c + d*x)])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
411,1,172,236,1.6893411,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{\sqrt{\sec (c+d x)} \left(2 \sin (c+d x) \left(3 \left(a^3 A \cos (2 (c+d x))+a^3 A+10 b^3 B\right)+10 a^2 (a B+3 A b) \cos (c+d x)\right)+20 \left(a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+12 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{30 d}","\frac{2 a^2 (5 a B+9 A b) \sin (c+d x)}{15 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^3 B+3 a^2 A b+9 a b^2 B+3 A b^3\right) \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 \left(3 a^3 A+15 a^2 b B+15 a A b^2-5 b^3 B\right) \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 b^2 (a A-5 b B) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 d}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^2}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(12*(3*a^3*A + 15*a*A*b^2 + 15*a^2*b*B - 5*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(3*a^2*A*b + 3*A*b^3 + a^3*B + 9*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 2*(10*a^2*(3*A*b + a*B)*Cos[c + d*x] + 3*(a^3*A + 10*b^3*B + a^3*A*Cos[2*(c + d*x)]))*Sin[c + d*x]))/(30*d)","A",1
412,1,180,245,1.3569842,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{\sec (c+d x)} \left(a \sin (2 (c+d x)) \left(5 \left(3 a^2 A \cos (2 (c+d x))+13 a^2 A+42 a b B+42 A b^2\right)+42 a (a B+3 A b) \cos (c+d x)\right)+20 \left(5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+84 \left(3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{210 d}","\frac{2 a \left(5 a^2 A+21 a b B+18 A b^2\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 a^2 (7 a B+11 A b) \sin (c+d x)}{35 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(5 a^3 A+21 a^2 b B+21 a A b^2+21 b^3 B\right) \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 \left(3 a^3 B+9 a^2 A b+15 a b^2 B+5 A b^3\right) \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) (a+b \sec (c+d x))^2}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(84*(9*a^2*A*b + 5*A*b^3 + 3*a^3*B + 15*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 20*(5*a^3*A + 21*a*A*b^2 + 21*a^2*b*B + 21*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + a*(42*a*(3*A*b + a*B)*Cos[c + d*x] + 5*(13*a^2*A + 42*A*b^2 + 42*a*b*B + 3*a^2*A*Cos[2*(c + d*x)]))*Sin[2*(c + d*x)]))/(210*d)","A",1
413,1,219,295,2.0345351,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(7 a \left(43 a^2 A+108 a b B+108 A b^2\right) \cos (c+d x)+5 \left(7 a^3 A \cos (3 (c+d x))+78 a^3 B+18 a^2 (a B+3 A b) \cos (2 (c+d x))+234 a^2 A b+252 a b^2 B+84 A b^3\right)\right)+120 \left(5 a^3 B+15 a^2 A b+21 a b^2 B+7 A b^3\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+168 \left(7 a^3 A+27 a^2 b B+27 a A b^2+15 b^3 B\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{1260 d}","\frac{2 a \left(7 a^2 A+27 a b B+22 A b^2\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (9 a B+13 A b) \sin (c+d x)}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(5 a^3 B+15 a^2 A b+21 a b^2 B+7 A b^3\right) \sin (c+d x)}{21 d \sqrt{\sec (c+d x)}}+\frac{2 \left(5 a^3 B+15 a^2 A b+21 a b^2 B+7 A b^3\right) \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 \left(7 a^3 A+27 a^2 b B+27 a A b^2+15 b^3 B\right) \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) (a+b \sec (c+d x))^2}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(168*(7*a^3*A + 27*a*A*b^2 + 27*a^2*b*B + 15*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 120*(15*a^2*A*b + 7*A*b^3 + 5*a^3*B + 21*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (7*a*(43*a^2*A + 108*A*b^2 + 108*a*b*B)*Cos[c + d*x] + 5*(234*a^2*A*b + 84*A*b^3 + 78*a^3*B + 252*a*b^2*B + 18*a^2*(3*A*b + a*B)*Cos[2*(c + d*x)] + 7*a^3*A*Cos[3*(c + d*x)]))*Sin[2*(c + d*x)]))/(1260*d)","A",1
414,1,256,345,3.2316379,"\int \frac{(a+b \sec (c+d x))^3 (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]","\frac{\sqrt{\sec (c+d x)} \left(\sin (2 (c+d x)) \left(180 a \left(16 a^2 A+33 a b B+33 A b^2\right) \cos (2 (c+d x))+770 a^2 (a B+3 A b) \cos (3 (c+d x))+154 \left(43 a^3 B+129 a^2 A b+108 a b^2 B+36 A b^3\right) \cos (c+d x)+15 \left(21 a^3 A \cos (4 (c+d x))+531 a^3 A+1716 a^2 b B+1716 a A b^2+616 b^3 B\right)\right)+240 \left(45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3696 \left(7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{27720 d}","\frac{2 a \left(9 a^2 A+33 a b B+26 A b^2\right) \sin (c+d x)}{77 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a^2 (11 a B+15 A b) \sin (c+d x)}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 \left(7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right) \sin (c+d x)}{45 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right) \sin (c+d x)}{231 d \sqrt{\sec (c+d x)}}+\frac{2 \left(45 a^3 A+165 a^2 b B+165 a A b^2+77 b^3 B\right) \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{2 \left(7 a^3 B+21 a^2 A b+27 a b^2 B+9 A b^3\right) \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) (a+b \sec (c+d x))^2}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"(Sqrt[Sec[c + d*x]]*(3696*(21*a^2*A*b + 9*A*b^3 + 7*a^3*B + 27*a*b^2*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 240*(45*a^3*A + 165*a*A*b^2 + 165*a^2*b*B + 77*b^3*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*(129*a^2*A*b + 36*A*b^3 + 43*a^3*B + 108*a*b^2*B)*Cos[c + d*x] + 180*a*(16*a^2*A + 33*A*b^2 + 33*a*b*B)*Cos[2*(c + d*x)] + 770*a^2*(3*A*b + a*B)*Cos[3*(c + d*x)] + 15*(531*a^3*A + 1716*a*A*b^2 + 1716*a^2*b*B + 616*b^3*B + 21*a^3*A*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]))/(27720*d)","A",1
415,1,664,277,6.9381128,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2 \left(5 a^2 B-5 a A b+3 b^2 B\right) \sin (c+d x)}{5 b^3}+\frac{2 \sec (c+d x) (A b \sin (c+d x)-a B \sin (c+d x))}{3 b^2}+\frac{2 B \tan (c+d x) \sec (c+d x)}{5 b}\right)}{d}-\frac{\frac{2 \left(40 a^2 b B-40 a A b^2+18 b^3 B\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(15 a^3 B-15 a^2 A b+9 a b^2 B\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(45 a^3 B-45 a^2 A b+19 a b^2 B-10 A b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{30 b^3 d}","\frac{2 a^2 (A b-a B) \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)}{b^3 d (a+b)}-\frac{2 \left(-5 a^2 B+5 a A b-3 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{5 b^3 d}+\frac{2 \left(-5 a^2 B+5 a A b-3 b^2 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}+\frac{2 (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d}+\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)}{3 b^2 d}+\frac{2 B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{5 b d}",1,"-1/30*((2*(-45*a^2*A*b - 10*A*b^3 + 45*a^3*B + 19*a*b^2*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-40*a*A*b^2 + 40*a^2*b*B + 18*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-15*a^2*A*b + 15*a^3*B + 9*a*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(b^3*d) + (Sqrt[Sec[c + d*x]]*((2*(-5*a*A*b + 5*a^2*B + 3*b^2*B)*Sin[c + d*x])/(5*b^3) + (2*Sec[c + d*x]*(A*b*Sin[c + d*x] - a*B*Sin[c + d*x]))/(3*b^2) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(5*b)))/d","B",0
416,1,225,210,3.6596761,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\frac{\cot (c+d x) \left(-2 \left(3 a^2 B+3 a b (B-A)+b^2 (B-3 A)\right) \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+6 a^2 B \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-6 b (A b-a B) \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-6 a A b \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-b^2 B \sec ^{\frac{5}{2}}(c+d x)+b^2 B \cos (2 (c+d x)) \sec ^{\frac{5}{2}}(c+d x)\right)}{3 b^3 d}","\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d}-\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)}{b^2 d}-\frac{2 a (A b-a B) \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)}{b^2 d (a+b)}+\frac{2 B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d}+\frac{2 B \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}",1,"-1/3*(Cot[c + d*x]*(-(b^2*B*Sec[c + d*x]^(5/2)) + b^2*B*Cos[2*(c + d*x)]*Sec[c + d*x]^(5/2) - 6*b*(A*b - a*B)*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*(3*a^2*B + b^2*(-3*A + B) + 3*a*b*(-A + B))*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 6*a*A*b*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 6*a^2*B*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(b^3*d)","A",1
417,1,123,126,1.3682869,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","-\frac{2 \cos (2 (c+d x)) \sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec (c+d x) \left((A b-B (a+b)) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+(a B-A b) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+b B E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b^2 d \left(\sec ^2(c+d x)-2\right)}","\frac{2 (A b-a B) \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)}{b d (a+b)}+\frac{2 B \sin (c+d x) \sqrt{\sec (c+d x)}}{b d}-\frac{2 B \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*Cos[2*(c + d*x)]*Csc[c + d*x]*(b*B*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (A*b - (a + b)*B)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (-(A*b) + a*B)*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sec[c + d*x]*Sqrt[-Tan[c + d*x]^2])/(b^2*d*(-2 + Sec[c + d*x]^2))","A",1
418,1,76,101,0.581632,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{2 \sqrt{-\tan ^2(c+d x)} \cot (c+d x) \left((A b-a B) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a B F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a b d}","\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 (A b-a B) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}",1,"(2*Cot[c + d*x]*(a*B*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] + (A*b - a*B)*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*Sqrt[-Tan[c + d*x]^2])/(a*b*d)","A",1
419,1,220,149,7.0227331,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{\cot (c+d x) \left(-2 A b \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+a A \sec ^{\frac{7}{2}}(c+d x)-a A \sec ^{\frac{3}{2}}(c+d x)+a A \cos (2 (c+d x)) \sec ^{\frac{7}{2}}(c+d x)-a A \cos (2 (c+d x)) \sec ^{\frac{3}{2}}(c+d x)+2 a A \sqrt{-\tan ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-2 a A \sqrt{-\tan ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+2 a B \sqrt{-\tan ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{a^2 d}","-\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 b (A b-a B) \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 \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,"(Cot[c + d*x]*(-(a*A*Sec[c + d*x]^(3/2)) - a*A*Cos[2*(c + d*x)]*Sec[c + d*x]^(3/2) + a*A*Sec[c + d*x]^(7/2) + a*A*Cos[2*(c + d*x)]*Sec[c + d*x]^(7/2) - 2*a*A*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*a*A*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] - 2*A*b*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2] + 2*a*B*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[-Tan[c + d*x]^2]))/(a^2*d)","A",1
420,1,540,196,6.7399969,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\frac{A \sin (2 (c+d x)) \sqrt{\sec (c+d x)}}{3 a d}-\frac{\frac{(3 A b-3 a B) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 (A b-3 a B) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}-\frac{4 A \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{\left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{6 a d}","-\frac{2 b^2 (A b-a B) \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^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 \left(a^2 A-3 a b B+3 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d}+\frac{2 A \sin (c+d x)}{3 a d \sqrt{\sec (c+d x)}}",1,"-1/6*((2*(A*b - 3*a*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) - (4*A*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/((b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((3*A*b - 3*a*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(a*d) + (A*Sqrt[Sec[c + d*x]]*Sin[2*(c + d*x)])/(3*a*d)","B",0
421,1,612,242,6.9127378,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),x]","\frac{\frac{2 \left(9 a^2 A-5 a b B+5 A b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(9 a^2 A-15 a b B+15 A b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(10 a^2 B+8 a A b\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{30 a^2 d}+\frac{\sqrt{\sec (c+d x)} \left(\frac{(a B-A b) \sin (2 (c+d x))}{3 a^2}+\frac{A \sin (c+d x)}{10 a}+\frac{A \sin (3 (c+d x))}{10 a}\right)}{d}","\frac{2 b^3 (A b-a B) \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^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 \left(a^2+3 b^2\right) (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)}{3 a^4 d}+\frac{2 \left(3 a^2 A-5 a b B+5 A b^2\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}+\frac{2 A \sin (c+d x)}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"((2*(9*a^2*A + 5*A*b^2 - 5*a*b*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(8*a*A*b + 10*a^2*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^2*A + 15*A*b^2 - 15*a*b*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(30*a^2*d) + (Sqrt[Sec[c + d*x]]*((A*Sin[c + d*x])/(10*a) + ((-(A*b) + a*B)*Sin[2*(c + d*x)])/(3*a^2) + (A*Sin[3*(c + d*x)])/(10*a)))/d","B",0
422,1,733,406,7.3177336,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{a^2 A b \sin (c+d x)-a^3 B \sin (c+d x)}{b^2 \left(b^2-a^2\right) (a \cos (c+d x)+b)}+\frac{\left(5 a^3 B-3 a^2 A b-4 a b^2 B+2 A b^3\right) \sin (c+d x)}{b^3 \left(b^2-a^2\right)}+\frac{2 B \tan (c+d x)}{3 b^2}\right)}{d}+\frac{\frac{2 \left(40 a^3 b B-24 a^2 A b^2-28 a b^3 B+12 A b^4\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(15 a^4 B-9 a^3 A b-12 a^2 b^2 B+6 a A b^3\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(45 a^4 B-27 a^3 A b-44 a^2 b^2 B+30 a A b^3-4 b^4 B\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{12 b^3 d (a-b) (a+b)}","\frac{a (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{\left(-5 a^2 B+3 a A b+2 b^2 B\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b^2 d \left(a^2-b^2\right)}-\frac{\left(-5 a^2 B+3 a A b+2 b^2 B\right) \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 \left(a^2-b^2\right)}+\frac{\left(-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^3 d \left(a^2-b^2\right)}-\frac{\left(-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{a \left(-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\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)}{b^3 d (a-b) (a+b)^2}",1,"((2*(-27*a^3*A*b + 30*a*A*b^3 + 45*a^4*B - 44*a^2*b^2*B - 4*b^4*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a^2*A*b^2 + 12*A*b^4 + 40*a^3*b*B - 28*a*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-9*a^3*A*b + 6*a*A*b^3 + 15*a^4*B - 12*a^2*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*(a - b)*b^3*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((-3*a^2*A*b + 2*A*b^3 + 5*a^3*B - 4*a*b^2*B)*Sin[c + d*x])/(b^3*(-a^2 + b^2)) + (a^2*A*b*Sin[c + d*x] - a^3*B*Sin[c + d*x])/(b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*B*Tan[c + d*x])/(3*b^2)))/d","A",0
423,1,680,315,6.9982717,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \sin (c+d x)}{b^2 \left(b^2-a^2\right)}+\frac{a^2 B \sin (c+d x)-a A b \sin (c+d x)}{b \left(b^2-a^2\right) (a \cos (c+d x)+b)}\right)}{d}-\frac{\frac{2 \left(8 a^2 b B-4 a A b^2-4 b^3 B\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(3 a^3 B-a^2 A b-2 a b^2 B\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(9 a^3 B-3 a^2 A b-10 a b^2 B+4 A b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{4 b^2 d (a-b) (a+b)}","\frac{a (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}+\frac{(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)}{b d \left(a^2-b^2\right)}+\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \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 \left(a^2-b^2\right)}+\frac{\left(-3 a^3 B+a^2 A b+5 a b^2 B-3 A b^3\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)}{b^2 d (a-b) (a+b)^2}",1,"-1/4*((2*(-3*a^2*A*b + 4*A*b^3 + 9*a^3*B - 10*a*b^2*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-4*a*A*b^2 + 8*a^2*b*B - 4*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-(a^2*A*b) + 3*a^3*B - 2*a*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)*b^2*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(((a*A*b - 3*a^2*B + 2*b^2*B)*Sin[c + d*x])/(b^2*(-a^2 + b^2)) + (-(a*A*b*Sin[c + d*x]) + a^2*B*Sin[c + d*x])/(b*(-a^2 + b^2)*(b + a*Cos[c + d*x]))))/d","B",0
424,1,638,257,7.0028228,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{A b \sin (c+d x)-a B \sin (c+d x)}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}-\frac{(A b-a B) \sin (c+d x)}{b \left(b^2-a^2\right)}\right)}{d}+\frac{\frac{2 \left(-3 a^2 B-a A b+4 b^2 B\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(a A b-a^2 B\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(4 A b^2-4 a b B\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{4 b d (b-a) (a+b)}","\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) (a+b \sec (c+d x))}-\frac{(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 d \left(a^2-b^2\right)}-\frac{(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)}{b d \left(a^2-b^2\right)}+\frac{\left(a^3 B+a^2 A b-3 a b^2 B+A b^3\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 b d (a-b) (a+b)^2}",1,"((2*(-(a*A*b) - 3*a^2*B + 4*b^2*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*A*b^2 - 4*a*b*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((a*A*b - a^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*b*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-(((A*b - a*B)*Sin[c + d*x])/(b*(-a^2 + b^2))) + (A*b*Sin[c + d*x] - a*B*Sin[c + d*x])/((-a^2 + b^2)*(b + a*Cos[c + d*x]))))/d","B",0
425,1,722,263,6.9554577,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2 (A+B \sec (c+d x)) \left(\frac{(a B-A b) \sin (c+d x)}{a \left(a^2-b^2\right)}+\frac{A b^2 \sin (c+d x)-a b B \sin (c+d x)}{a \left(a^2-b^2\right) (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^2 (A \cos (c+d x)+B)}+\frac{\sec (c+d x) (a \cos (c+d x)+b)^2 (A+B \sec (c+d x)) \left(\frac{(A b-a B) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 (4 a A-4 b B) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 (a B-A b) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}\right)}{4 d (a-b) (a+b) (a+b \sec (c+d x))^2 (A \cos (c+d x)+B)}","-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(2 a^2 A-a b B-A b^2\right) \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 \left(a^2-b^2\right)}+\frac{(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 d \left(a^2-b^2\right)}-\frac{\left(a^3 (-B)+3 a^2 A b-a b^2 B-A b^3\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) (a+b)^2}",1,"((b + a*Cos[c + d*x])^2*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*(-(A*b) + a*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A - 4*b*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((A*b - a*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(4*(a - b)*(a + b)*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^2) + ((b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x])*(((-(A*b) + a*B)*Sin[c + d*x])/(a*(a^2 - b^2)) + (A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x])/(a*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^2)","B",0
426,1,652,283,6.9255203,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{2 \left(-2 a^2 A+a b B+A b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(-2 a^2 A-a b B+3 A b^2\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(4 a A b-4 a^2 B\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}}{4 a d (b-a) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(\frac{a b^2 B \sin (c+d x)-A b^3 \sin (c+d x)}{a^2 \left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{b (A b-a B) \sin (c+d x)}{a^2 \left(b^2-a^2\right)}\right)}{d}","\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) (a+b \sec (c+d x))}+\frac{\left(2 a^2 A+a b B-3 A b^2\right) \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 \left(a^2-b^2\right)}-\frac{\left(-2 a^3 B+4 a^2 A b+a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\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^3 d (a-b) (a+b)^2}",1,"((2*(-2*a^2*A + A*b^2 + a*b*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a*A*b - 4*a^2*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-2*a^2*A + 3*A*b^2 - a*b*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(4*a*(-a + b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*(-((b*(A*b - a*B)*Sin[c + d*x])/(a^2*(-a^2 + b^2))) + (-(A*b^3*Sin[c + d*x]) + a*b^2*B*Sin[c + d*x])/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/d","B",0
427,1,699,365,7.0214226,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{2 \left(4 a^3 A-12 a^2 b B+8 a A b^2\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)}{a \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{2 \left(6 a^3 B-8 a^2 A b-3 a b^2 B+5 A b^3\right) \sin (c+d x) \cos ^2(c+d x) \sqrt{1-\sec ^2(c+d x)} (a+b \sec (c+d x)) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right)}{b \left(1-\cos ^2(c+d x)\right) (a \cos (c+d x)+b)}+\frac{\left(6 a^3 B-12 a^2 A b-9 a b^2 B+15 A b^3\right) \sin (c+d x) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 a^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 b^2 \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)+4 a b \sec ^2(c+d x)-2 a (a-2 b) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-4 a b\right)}{a^2 b \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right) (a \cos (c+d x)+b)}}{12 a^2 d (a-b) (a+b)}+\frac{\sqrt{\sec (c+d x)} \left(\frac{A \sin (2 (c+d x))}{3 a^2}+\frac{b^2 (A b-a B) \sin (c+d x)}{a^3 \left(b^2-a^2\right)}-\frac{a b^3 B \sin (c+d x)-A b^4 \sin (c+d x)}{a^3 \left(a^2-b^2\right) (a \cos (c+d x)+b)}\right)}{d}","\frac{b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\left(2 a^2 A+3 a b B-5 A b^2\right) \sin (c+d x)}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{\left(-2 a^3 B+4 a^2 A b+3 a b^2 B-5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}-\frac{b^2 \left(-5 a^3 B+7 a^2 A b+3 a b^2 B-5 A b^3\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^4 d (a-b) (a+b)^2}+\frac{\left(2 a^4 A-12 a^3 b B+16 a^2 A b^2+9 a b^3 B-15 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}",1,"((2*(-8*a^2*A*b + 5*A*b^3 + 6*a^3*B - 3*a*b^2*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(4*a^3*A + 8*a*A*b^2 - 12*a^2*b*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-12*a^2*A*b + 15*A*b^3 + 6*a^3*B - 9*a*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(12*a^2*(a - b)*(a + b)*d) + (Sqrt[Sec[c + d*x]]*((b^2*(A*b - a*B)*Sin[c + d*x])/(a^3*(-a^2 + b^2)) - (-(A*b^4*Sin[c + d*x]) + a*b^3*B*Sin[c + d*x])/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(3*a^2)))/d","A",0
428,1,897,583,7.5015794,"\int \frac{\sec ^{\frac{9}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(9/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{2 \left(315 B a^6-135 A b a^5-641 b^2 B a^4+285 A b^3 a^3+328 b^4 B a^2-168 A b^5 a+16 b^6 B\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(-48 A b^6+160 a B b^5+240 a^2 A b^4-512 a^3 B b^3-120 a^4 A b^2+280 a^5 B b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(105 B a^6-45 A b a^5-195 b^2 B a^4+87 A b^3 a^3+72 b^4 B a^2-24 A b^5 a\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{48 (a-b)^2 b^4 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(-35 B a^5+15 A b a^4+65 b^2 B a^3-29 A b^3 a^2-24 b^4 B a+8 A b^5\right) \sin (c+d x)}{4 b^4 \left(b^2-a^2\right)^2}+\frac{a^2 A b \sin (c+d x)-a^3 B \sin (c+d x)}{2 b^2 \left(b^2-a^2\right) (b+a \cos (c+d x))^2}+\frac{9 B \sin (c+d x) a^5-5 A b \sin (c+d x) a^4-15 b^2 B \sin (c+d x) a^3+11 A b^3 \sin (c+d x) a^2}{4 b^3 \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}+\frac{2 B \tan (c+d x)}{3 b^3}\right)}{d}","\frac{a (A b-a B) \sin (c+d x) \sec ^{\frac{7}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{12 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{\left(-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\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)}{4 b^4 d (a-b)^2 (a+b)^3}",1,"((2*(-135*a^5*A*b + 285*a^3*A*b^3 - 168*a*A*b^5 + 315*a^6*B - 641*a^4*b^2*B + 328*a^2*b^4*B + 16*b^6*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-120*a^4*A*b^2 + 240*a^2*A*b^4 - 48*A*b^6 + 280*a^5*b*B - 512*a^3*b^3*B + 160*a*b^5*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-45*a^5*A*b + 87*a^3*A*b^3 - 24*a*A*b^5 + 105*a^6*B - 195*a^4*b^2*B + 72*a^2*b^4*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*(a - b)^2*b^4*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((15*a^4*A*b - 29*a^2*A*b^3 + 8*A*b^5 - 35*a^5*B + 65*a^3*b^2*B - 24*a*b^4*B)*Sin[c + d*x])/(4*b^4*(-a^2 + b^2)^2) + (a^2*A*b*Sin[c + d*x] - a^3*B*Sin[c + d*x])/(2*b^2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (-5*a^4*A*b*Sin[c + d*x] + 11*a^2*A*b^3*Sin[c + d*x] + 9*a^5*B*Sin[c + d*x] - 15*a^3*b^2*B*Sin[c + d*x])/(4*b^3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (2*B*Tan[c + d*x])/(3*b^3)))/d","A",0
429,1,842,480,7.2496863,"\int \frac{\sec ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(15 B a^4-3 A b a^3-29 b^2 B a^2+9 A b^3 a+8 b^4 B\right) \sin (c+d x)}{4 b^3 \left(b^2-a^2\right)^2}+\frac{a^2 B \sin (c+d x)-a A b \sin (c+d x)}{2 b \left(b^2-a^2\right) (b+a \cos (c+d x))^2}+\frac{-5 B \sin (c+d x) a^4+A b \sin (c+d x) a^3+11 b^2 B \sin (c+d x) a^2-7 A b^3 \sin (c+d x) a}{4 b^2 \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}\right)}{d}-\frac{\frac{2 \left(45 B a^5-9 A b a^4-95 b^2 B a^3+19 A b^3 a^2+56 b^4 B a-16 A b^5\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 B b^5+32 a A b^4-80 a^2 B b^3-8 a^3 A b^2+40 a^4 B b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(15 B a^5-3 A b a^4-29 b^2 B a^3+9 A b^3 a^2+8 b^4 B a\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 (a-b)^2 b^3 (a+b)^2 d}","\frac{a (A b-a B) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{a \left(-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{\left(-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}+\frac{\left(-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\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)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"-1/16*((2*(-9*a^4*A*b + 19*a^2*A*b^3 - 16*A*b^5 + 45*a^5*B - 95*a^3*b^2*B + 56*a*b^4*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-8*a^3*A*b^2 + 32*a*A*b^4 + 40*a^4*b*B - 80*a^2*b^3*B + 16*b^5*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-3*a^4*A*b + 9*a^2*A*b^3 + 15*a^5*B - 29*a^3*b^2*B + 8*a*b^4*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/((a - b)^2*b^3*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(((-3*a^3*A*b + 9*a*A*b^3 + 15*a^4*B - 29*a^2*b^2*B + 8*b^4*B)*Sin[c + d*x])/(4*b^3*(-a^2 + b^2)^2) + (-(a*A*b*Sin[c + d*x]) + a^2*B*Sin[c + d*x])/(2*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (a^3*A*b*Sin[c + d*x] - 7*a*A*b^3*Sin[c + d*x] - 5*a^4*B*Sin[c + d*x] + 11*a^2*b^2*B*Sin[c + d*x])/(4*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/d","A",0
430,1,795,402,6.9643775,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{2 \left(9 B a^4+3 A b a^3-19 b^2 B a^2-9 A b^3 a+16 b^4 B\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 A b^4-32 a B b^3+8 a^2 A b^2+8 a^3 B b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(3 B a^4+A b a^3-9 b^2 B a^2+5 A b^3 a\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 (a-b)^2 b^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{\left(3 B a^3+A b a^2-9 b^2 B a+5 A b^3\right) \sin (c+d x)}{4 b^2 \left(b^2-a^2\right)^2}+\frac{A b \sin (c+d x)-a B \sin (c+d x)}{2 \left(b^2-a^2\right) (b+a \cos (c+d x))^2}+\frac{B \sin (c+d x) a^3+3 A b \sin (c+d x) a^2-7 b^2 B \sin (c+d x) a+3 A b^3 \sin (c+d x)}{4 b \left(b^2-a^2\right)^2 (b+a \cos (c+d x))}\right)}{d}","\frac{a (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{a \left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(a^3 B+3 a^2 A b-7 a b^2 B+3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}+\frac{\left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{\left(3 a^5 B+a^4 A b-6 a^3 b^2 B-10 a^2 A b^3+15 a b^4 B-3 A b^5\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)}{4 a b^2 d (a-b)^2 (a+b)^3}",1,"((2*(3*a^3*A*b - 9*a*A*b^3 + 9*a^4*B - 19*a^2*b^2*B + 16*b^4*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(8*a^2*A*b^2 + 16*A*b^4 + 8*a^3*b*B - 32*a*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((a^3*A*b + 5*a*A*b^3 + 3*a^4*B - 9*a^2*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*(a - b)^2*b^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Sin[c + d*x])/(b^2*(-a^2 + b^2)^2) + (A*b*Sin[c + d*x] - a*B*Sin[c + d*x])/(2*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (3*a^2*A*b*Sin[c + d*x] + 3*A*b^3*Sin[c + d*x] + a^3*B*Sin[c + d*x] - 7*a*b^2*B*Sin[c + d*x])/(4*b*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/d","A",0
431,1,882,402,6.9861222,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \left(3 B a^3+A b a^2-9 b^2 B a+5 A b^3\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 B b^3-24 a A b^2+8 a^2 B b\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(B a^3-5 A b a^2+5 b^2 B a-A b^3\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^3}{16 (a-b)^2 b (a+b)^2 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{\left(-B a^3+5 A b a^2-5 b^2 B a+A b^3\right) \sin (c+d x)}{4 a b \left(b^2-a^2\right)^2}-\frac{a b B \sin (c+d x)-A b^2 \sin (c+d x)}{2 a \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{3 B \sin (c+d x) a^3-7 A b \sin (c+d x) a^2+3 b^2 B \sin (c+d x) a+A b^3 \sin (c+d x)}{4 a \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^3}{d (B+A \cos (c+d x)) (a+b \sec (c+d x))^3}","\frac{a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{\left(a^3 B+3 a^2 A b-7 a b^2 B+3 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}-\frac{\left(-3 a^3 B+7 a^2 A b-3 a b^2 B-A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(a^3 (-B)+5 a^2 A b-5 a b^2 B+A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}+\frac{\left(a^5 B+3 a^4 A b-10 a^3 b^2 B+10 a^2 A b^3-3 a b^4 B-A b^5\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)}{4 a^2 b d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*(a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-24*a*A*b^2 + 8*a^2*b*B + 16*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(16*(a - b)^2*b*(a + b)^2*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*(((5*a^2*A*b + A*b^3 - a^3*B - 5*a*b^2*B)*Sin[c + d*x])/(4*a*b*(-a^2 + b^2)^2) - (-(A*b^2*Sin[c + d*x]) + a*b*B*Sin[c + d*x])/(2*a*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (-7*a^2*A*b*Sin[c + d*x] + A*b^3*Sin[c + d*x] + 3*a^3*B*Sin[c + d*x] + 3*a*b^2*B*Sin[c + d*x])/(4*a*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^3)","B",0
432,1,885,402,6.9826144,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 \left(B a^3-5 A b a^2+5 b^2 B a-A b^3\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 A a^3-24 b B a^2+8 A b^2 a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(-5 B a^3+9 A b a^2-b^2 B a-3 A b^3\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}\right) (b+a \cos (c+d x))^3}{16 a (a-b)^2 (a+b)^2 d (B+A \cos (c+d x)) (a+b \sec (c+d x))^3}+\frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x)) \left(\frac{\left(5 B a^3-9 A b a^2+b^2 B a+3 A b^3\right) \sin (c+d x)}{4 a^2 \left(b^2-a^2\right)^2}-\frac{A b^3 \sin (c+d x)-a b^2 B \sin (c+d x)}{2 a^2 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{-5 A \sin (c+d x) b^4+a B \sin (c+d x) b^3+11 a^2 A \sin (c+d x) b^2-7 a^3 B \sin (c+d x) b}{4 a^2 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right) (b+a \cos (c+d x))^3}{d (B+A \cos (c+d x)) (a+b \sec (c+d x))^3}","-\frac{(A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}-\frac{\left(-3 a^3 B+7 a^2 A b-3 a b^2 B-A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(-5 a^3 B+9 a^2 A b-a b^2 B-3 A b^3\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}+\frac{\left(8 a^4 A-7 a^3 b B-5 a^2 A b^2+a b^3 B+3 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^5 B+15 a^4 A b-10 a^3 b^2 B-6 a^2 A b^3+a b^4 B+3 A b^5\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)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"((b + a*Cos[c + d*x])^3*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*(-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^3*A + 8*a*A*b^2 - 24*a^2*b*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((9*a^2*A*b - 3*A*b^3 - 5*a^3*B - a*b^2*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2))))/(16*a*(a - b)^2*(a + b)^2*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^3) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x])*(((-9*a^2*A*b + 3*A*b^3 + 5*a^3*B + a*b^2*B)*Sin[c + d*x])/(4*a^2*(-a^2 + b^2)^2) - (A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x])/(2*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (11*a^2*A*b^2*Sin[c + d*x] - 5*A*b^4*Sin[c + d*x] - 7*a^3*b*B*Sin[c + d*x] + a*b^3*B*Sin[c + d*x])/(4*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^3)","B",0
433,1,818,427,7.2264727,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{2 \left(8 A a^4-5 b B a^3-7 A b^2 a^2-b^3 B a+5 A b^4\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 B a^4-32 A b a^3+8 b^2 B a^2+8 A b^3 a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(8 A a^4+9 b B a^3-29 A b^2 a^2-3 b^3 B a+15 A b^4\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{16 a^2 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(-\frac{b \left(9 B a^3-13 A b a^2-3 b^2 B a+7 A b^3\right) \sin (c+d x)}{4 a^3 \left(b^2-a^2\right)^2}-\frac{a b^3 B \sin (c+d x)-A b^4 \sin (c+d x)}{2 a^3 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{9 A \sin (c+d x) b^5-5 a B \sin (c+d x) b^4-15 a^2 A \sin (c+d x) b^3+11 a^3 B \sin (c+d x) b^2}{4 a^3 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}\right)}{d}","\frac{b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{2 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^2}+\frac{b \left(-7 a^3 B+11 a^2 A b+a b^2 B-5 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))}+\frac{\left(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+15 A b^4\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-8 a^5 B+24 a^4 A b+5 a^3 b^2 B-33 a^2 A b^3-3 a b^4 B+15 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b \left(-15 a^5 B+35 a^4 A b+6 a^3 b^2 B-38 a^2 A b^3-3 a b^4 B+15 A b^5\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)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((2*(8*a^4*A - 7*a^2*A*b^2 + 5*A*b^4 - 5*a^3*b*B - a*b^3*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(-32*a^3*A*b + 8*a*A*b^3 + 16*a^4*B + 8*a^2*b^2*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(16*a^2*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*(-1/4*(b*(-13*a^2*A*b + 7*A*b^3 + 9*a^3*B - 3*a*b^2*B)*Sin[c + d*x])/(a^3*(-a^2 + b^2)^2) - (-(A*b^4*Sin[c + d*x]) + a*b^3*B*Sin[c + d*x])/(2*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (-15*a^2*A*b^3*Sin[c + d*x] + 9*A*b^5*Sin[c + d*x] + 11*a^3*b^2*B*Sin[c + d*x] - 5*a*b^4*B*Sin[c + d*x])/(4*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/d","A",0
434,1,863,521,7.5265961,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{2 \left(24 B a^5-56 A b a^4-21 b^2 B a^3+73 A b^3 a^2+15 b^4 B a-35 A b^5\right) \left(F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)-\Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right)\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{2 \left(16 A a^5-96 b B a^4+112 A b^2 a^3+24 b^3 B a^2-56 A b^4 a\right) \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) (a+b \sec (c+d x)) \sqrt{1-\sec ^2(c+d x)} \sin (c+d x) \cos ^2(c+d x)}{a (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right)}+\frac{\left(24 B a^5-72 A b a^4-87 b^2 B a^3+195 A b^3 a^2+45 b^4 B a-105 A b^5\right) \cos (2 (c+d x)) (a+b \sec (c+d x)) \left(2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a^2+4 b \sec ^2(c+d x) a-4 b a-4 b E\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-2 (a-2 b) F\left(\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)} a-4 b^2 \Pi \left(-\frac{b}{a};\left.\sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right|-1\right) \sqrt{\sec (c+d x)} \sqrt{1-\sec ^2(c+d x)}\right) \sin (c+d x)}{a^2 b (b+a \cos (c+d x)) \left(1-\cos ^2(c+d x)\right) \sqrt{\sec (c+d x)} \left(2-\sec ^2(c+d x)\right)}}{48 a^3 (a-b)^2 (a+b)^2 d}+\frac{\sqrt{\sec (c+d x)} \left(\frac{\left(13 B a^3-17 A b a^2-7 b^2 B a+11 A b^3\right) \sin (c+d x) b^2}{4 a^4 \left(b^2-a^2\right)^2}-\frac{A b^5 \sin (c+d x)-a b^4 B \sin (c+d x)}{2 a^4 \left(a^2-b^2\right) (b+a \cos (c+d x))^2}+\frac{-13 A \sin (c+d x) b^6+9 a B \sin (c+d x) b^5+19 a^2 A \sin (c+d x) b^4-15 a^3 B \sin (c+d x) b^3}{4 a^4 \left(a^2-b^2\right)^2 (b+a \cos (c+d x))}+\frac{A \sin (2 (c+d x))}{3 a^3}\right)}{d}","\frac{b (A b-a B) \sin (c+d x)}{2 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^2}+\frac{b \left(-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right) \sin (c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)} (a+b \sec (c+d x))}+\frac{\left(8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right) \sin (c+d x)}{12 a^3 d \left(a^2-b^2\right)^2 \sqrt{\sec (c+d x)}}-\frac{\left(-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b^2 \left(-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\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)}{4 a^5 d (a-b)^2 (a+b)^3}+\frac{\left(8 a^6 A-72 a^5 b B+128 a^4 A b^2+99 a^3 b^3 B-223 a^2 A b^4-45 a b^5 B+105 A b^6\right) \sqrt{\cos (c+d x)} \sqrt{\sec (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}",1,"((2*(-56*a^4*A*b + 73*a^2*A*b^3 - 35*A*b^5 + 24*a^5*B - 21*a^3*b^2*B + 15*a*b^4*B)*Cos[c + d*x]^2*(EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1] - EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1])*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + (2*(16*a^5*A + 112*a^3*A*b^2 - 56*a*A*b^4 - 96*a^4*b*B + 24*a^2*b^3*B)*Cos[c + d*x]^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*(a + b*Sec[c + d*x])*Sqrt[1 - Sec[c + d*x]^2]*Sin[c + d*x])/(a*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)) + ((-72*a^4*A*b + 195*a^2*A*b^3 - 105*A*b^5 + 24*a^5*B - 87*a^3*b^2*B + 45*a*b^4*B)*Cos[2*(c + d*x)]*(a + b*Sec[c + d*x])*(-4*a*b + 4*a*b*Sec[c + d*x]^2 - 4*a*b*EllipticE[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 2*a*(a - 2*b)*EllipticF[ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] + 2*a^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2] - 4*b^2*EllipticPi[-(b/a), ArcSin[Sqrt[Sec[c + d*x]]], -1]*Sqrt[Sec[c + d*x]]*Sqrt[1 - Sec[c + d*x]^2])*Sin[c + d*x])/(a^2*b*(b + a*Cos[c + d*x])*(1 - Cos[c + d*x]^2)*Sqrt[Sec[c + d*x]]*(2 - Sec[c + d*x]^2)))/(48*a^3*(a - b)^2*(a + b)^2*d) + (Sqrt[Sec[c + d*x]]*((b^2*(-17*a^2*A*b + 11*A*b^3 + 13*a^3*B - 7*a*b^2*B)*Sin[c + d*x])/(4*a^4*(-a^2 + b^2)^2) - (A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x])/(2*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (19*a^2*A*b^4*Sin[c + d*x] - 13*A*b^6*Sin[c + d*x] - 15*a^3*b^3*B*Sin[c + d*x] + 9*a*b^5*B*Sin[c + d*x])/(4*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(3*a^3)))/d","A",0
435,1,422,336,5.5212063,"\int \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 \left(-3 a^2 B+4 a A b+8 b^2 B\right) \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b (a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 i (a B+4 A b) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b^2 \sqrt{\frac{1}{a-b}} \sqrt{a \cos (c+d x)+b}}+\frac{4 (a B+4 A b) \tan (c+d x)}{b}+\frac{8 a B F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+8 B \tan (c+d x) \sec (c+d x)\right)}{16 d \sqrt{\sec (c+d x)}}","\frac{\left(a^2 (-B)+4 a A b+4 b^2 B\right) \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)}{4 b d \sqrt{a+b \sec (c+d x)}}+\frac{(a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 b d}+\frac{(3 a B+4 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)}{4 d \sqrt{a+b \sec (c+d x)}}-\frac{(a B+4 A 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{B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"(Sqrt[a + b*Sec[c + d*x]]*((8*a*B*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(4*a*A*b - 3*a^2*B + 8*b^2*B)*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b*(a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - ((2*I)*(4*A*b + a*B)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b^2*Sqrt[b + a*Cos[c + d*x]]) + (4*(4*A*b + a*B)*Tan[c + d*x])/b + 8*B*Sec[c + d*x]*Tan[c + d*x]))/(16*d*Sqrt[Sec[c + d*x]])","C",0
436,1,377,253,6.3660752,"\int \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(\frac{2 (a B+4 A b) \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{8 a A F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a+b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 i B \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}} \sqrt{a \cos (c+d x)+b}}+4 B \tan (c+d x)\right)}{4 d \sqrt{\sec (c+d x)}}","\frac{(2 a A+b B) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{(a B+2 A b) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}-\frac{B \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"(Sqrt[a + b*Sec[c + d*x]]*((8*a*A*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) + (2*(4*A*b + a*B)*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/((a + b)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]) - ((2*I)*B*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b*Sqrt[b + a*Cos[c + d*x]]) + 4*B*Tan[c + d*x]))/(4*d*Sqrt[Sec[c + d*x]])","C",1
437,1,122,208,2.634935,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{2 \sqrt{a+b \sec (c+d x)} \left(A (a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+B \left(a F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+b \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)}{d (a+b) \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}","\frac{2 A \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 \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 \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*(a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + B*(a*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + b*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)]))*Sqrt[a + b*Sec[c + d*x]])/((a + b)*d*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*Sqrt[Sec[c + d*x]])","A",1
438,1,165,201,0.8149396,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{2 \sqrt{a+b \sec (c+d x)} \left(A \left(a^2-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)+(a+b) (3 a B+A b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+a A \sin (c+d x) (a \cos (c+d x)+b)\right)}{3 a d \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}","\frac{2 A \left(a^2-b^2\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)}}",1,"(2*Sqrt[a + b*Sec[c + d*x]]*((a + b)*(A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + A*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*A*(b + a*Cos[c + d*x])*Sin[c + d*x]))/(3*a*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",1
439,1,200,267,1.2059136,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{2 \sqrt{a+b \sec (c+d x)} \left(\left(a^2-b^2\right) (5 a B-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)+(a+b) \left(9 a^2 A+5 a b B-2 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+a \sin (c+d x) (a \cos (c+d x)+b) (3 a A \cos (c+d x)+5 a B+A b)\right)}{15 a^2 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+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 \left(9 a^2 A+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 (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*Sqrt[a + b*Sec[c + d*x]]*((a + b)*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (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)] + a*(b + a*Cos[c + d*x])*(A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x]))/(15*a^2*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","A",1
440,1,208,343,1.3643431,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{\sqrt{a+b \sec (c+d x)} \left(a \left(\left(115 a^2 A+28 a b B-16 A b^2\right) \sin (c+d x)+3 a (2 (7 a B+A b) \sin (2 (c+d x))+5 a A \sin (3 (c+d x)))\right)+\frac{4 \left((a-b) \left(25 a^2 A-14 a b B+8 A b^2\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(63 a^3 B+19 a^2 A b-14 a b^2 B+8 A b^3\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{\sqrt{\frac{a \cos (c+d x)+b}{a+b}}}\right)}{210 a^3 d \sqrt{\sec (c+d x)}}","\frac{2 \left(25 a^2 A+7 a b B-4 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A-14 a b B+8 A b^2\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)}{105 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(63 a^3 B+19 a^2 A 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,"(Sqrt[a + b*Sec[c + d*x]]*((4*((19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a - b)*(25*a^2*A + 8*A*b^2 - 14*a*b*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/Sqrt[(b + a*Cos[c + d*x])/(a + b)] + a*((115*a^2*A - 16*A*b^2 + 28*a*b*B)*Sin[c + d*x] + 3*a*(2*(A*b + 7*a*B)*Sin[2*(c + d*x)] + 5*a*A*Sin[3*(c + d*x)]))))/(210*a^3*d*Sqrt[Sec[c + d*x]])","A",1
441,1,673,421,6.8797915,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{\sec (c+d x) \left(3 a^2 B \sin (c+d x)+30 a A b \sin (c+d x)+16 b^2 B \sin (c+d x)\right)}{24 b}+\frac{1}{12} \sec ^2(c+d x) (7 a B \sin (c+d x)+6 A b \sin (c+d x))+\frac{1}{3} b B \tan (c+d x) \sec ^2(c+d x)\right)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{2 \left(-28 a^2 b B-24 a A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(3 a^3 B+30 a^2 A b+16 a b^2 B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(9 a^3 B-6 a^2 A b-56 a b^2 B-48 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{96 b d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2}}","\frac{\left(3 a^2 B+30 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{24 b d}+\frac{\left(17 a^2 B+42 a A b+16 b^2 B\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)}{24 d \sqrt{a+b \sec (c+d x)}}-\frac{\left(3 a^2 B+30 a A b+16 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)}{24 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\right) \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)}{8 b d \sqrt{a+b \sec (c+d x)}}+\frac{(7 a B+6 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{12 d}+\frac{b B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3 d}",1,"-1/96*((a + b*Sec[c + d*x])^(3/2)*((2*(-24*a*A*b^2 - 28*a^2*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(-6*a^2*A*b - 48*A*b^3 + 9*a^3*B - 56*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(30*a^2*A*b + 3*a^3*B + 16*a*b^2*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(b*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((a + b*Sec[c + d*x])^(3/2)*((Sec[c + d*x]^2*(6*A*b*Sin[c + d*x] + 7*a*B*Sin[c + d*x]))/12 + (Sec[c + d*x]*(30*a*A*b*Sin[c + d*x] + 3*a^2*B*Sin[c + d*x] + 16*b^2*B*Sin[c + d*x]))/(24*b) + (b*B*Sec[c + d*x]^2*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2))","C",0
442,1,595,339,6.8825495,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{1}{4} \sec (c+d x) (5 a B \sin (c+d x)+4 A b \sin (c+d x))+\frac{1}{2} b B \tan (c+d x) \sec (c+d x)\right)}{d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}+\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{2 \left(a^2 B+20 a A b+8 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(-5 a^2 B-4 a A b\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(16 a^2 A+4 a b B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{16 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2}}","\frac{\left(8 a^2 A+7 a b B+4 A b^2\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)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2 B+12 a A b+4 b^2 B\right) \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)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{(5 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}-\frac{(5 a B+4 A 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{b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 d}",1,"((a + b*Sec[c + d*x])^(3/2)*((2*(16*a^2*A + 4*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(20*a*A*b + a^2*B + 8*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(-4*a*A*b - 5*a^2*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(16*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2)) + ((a + b*Sec[c + d*x])^(3/2)*((Sec[c + d*x]*(4*A*b*Sin[c + d*x] + 5*a*B*Sin[c + d*x]))/4 + (b*B*Sec[c + d*x]*Tan[c + d*x])/2))/(d*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2))","C",1
443,1,554,272,6.7057652,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{b B \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{d \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}+\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{2 \left(2 a^2 A+5 a b B+4 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(2 a^2 A-a b B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(4 a^2 B+8 a A b\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{4 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^{3/2}}","\frac{\left(2 a^2 B+2 a A b+b^2 B\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)}{d \sqrt{a+b \sec (c+d x)}}+\frac{(2 a A-b 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{b (3 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 B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{d}",1,"(b*B*(a + b*Sec[c + d*x])^(3/2)*Sin[c + d*x])/(d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]) + ((a + b*Sec[c + d*x])^(3/2)*((2*(8*a*A*b + 4*a^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(2*a^2*A + 4*A*b^2 + 5*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(2*a^2*A - a*b*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(4*d*(b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^(3/2))","C",0
444,1,437,276,4.5320944,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(\frac{4 \left(a^2 A+6 a b B+3 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)}{(a \cos (c+d x)+b)^2}+\frac{2 \left(3 a^2 B+4 a A b+6 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{(a \cos (c+d x)+b)^2}+\frac{2 i (3 a B+4 A b) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}} (a \cos (c+d x)+b)^{3/2}}+\frac{4 a A \sin (c+d x)}{a \cos (c+d x)+b}\right)}{6 d \sec ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(a^2 A+3 a b B-A b^2\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 d \sqrt{a+b \sec (c+d x)}}+\frac{2 (3 a B+4 A b) \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{2 a A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\sec (c+d x)}}+\frac{2 b^2 B \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(3/2)*((4*(a^2*A + 3*A*b^2 + 6*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/(b + a*Cos[c + d*x])^2 + (2*(4*a*A*b + 3*a^2*B + 6*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/(b + a*Cos[c + d*x])^2 + ((2*I)*(4*A*b + 3*a*B)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b*(b + a*Cos[c + d*x])^(3/2)) + (4*a*A*Sin[c + d*x])/(b + a*Cos[c + d*x])))/(6*d*Sec[c + d*x]^(3/2))","C",0
445,1,201,266,1.8383401,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{2 (a+b \sec (c+d x))^{3/2} \left(\left(a^2-b^2\right) (5 a B+3 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+b) \left(9 a^2 A+20 a b B+3 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+a \sin (c+d x) (a \cos (c+d x)+b) (3 a A \cos (c+d x)+5 a B+6 A b)\right)}{15 a d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(a^2-b^2\right) (5 a B+3 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 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2 A+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+6 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*(a + b*Sec[c + d*x])^(3/2)*((a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a^2 - b^2)*(3*A*b + 5*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*(b + a*Cos[c + d*x])*(6*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x]))/(15*a*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","A",1
446,1,255,342,2.0149501,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(a (a \cos (c+d x)+b) \left(\left(115 a^2 A+168 a b B+12 A b^2\right) \sin (c+d x)+3 a (2 (7 a B+8 A b) \sin (2 (c+d x))+5 a A \sin (3 (c+d x)))\right)+4 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(a^2 \left(25 a^2 A+84 a b B+51 A b^2\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(63 a^3 B+82 a^2 A b+21 a b^2 B-6 A b^3\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)\right)}{210 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(25 a^2 A+42 a b B+3 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 a d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A+21 a b B-6 A b^2\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)}{105 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(63 a^3 B+82 a^2 A 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+8 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 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(3/2)*(4*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(25*a^2*A + 51*A*b^2 + 84*a*b*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*((115*a^2*A + 12*A*b^2 + 168*a*b*B)*Sin[c + d*x] + 3*a*(2*(8*A*b + 7*a*B)*Sin[2*(c + d*x)] + 5*a*A*Sin[3*(c + d*x)]))))/(210*a^2*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","A",1
447,1,313,427,2.3098916,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{(a+b \sec (c+d x))^{3/2} \left(a (a \cos (c+d x)+b) \left(a \left(2 \left(133 a^2 A+144 a b B+6 A b^2\right) \sin (2 (c+d x))+5 a (2 (9 a B+10 A b) \sin (3 (c+d x))+7 a A \sin (4 (c+d x)))\right)+\left(690 a^3 B+804 a^2 A b+72 a b^2 B-32 A b^3\right) \sin (c+d x)\right)+8 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(a^2 \left(75 a^3 B+186 a^2 A b+153 a b^2 B+2 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)\right)}{1260 a^3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(49 a^2 A+72 a b B+3 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \left(75 a^3 B+39 a^2 A b-18 a b^2 B+8 A b^3\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)}{315 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 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 (9 a B+10 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{63 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(3/2)*(8*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(186*a^2*A*b + 2*A*b^3 + 75*a^3*B + 153*a*b^2*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*((804*a^2*A*b - 32*A*b^3 + 690*a^3*B + 72*a*b^2*B)*Sin[c + d*x] + a*(2*(133*a^2*A + 6*A*b^2 + 144*a*b*B)*Sin[2*(c + d*x)] + 5*a*(2*(10*A*b + 9*a*B)*Sin[3*(c + d*x)] + 7*a*A*Sin[4*(c + d*x)])))))/(1260*a^3*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","A",1
448,1,768,513,6.9622677,"\int \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{1}{96} \sec ^2(c+d x) \left(59 a^2 B \sin (c+d x)+104 a A b \sin (c+d x)+36 b^2 B \sin (c+d x)\right)+\frac{\sec (c+d x) \left(15 a^3 B \sin (c+d x)+264 a^2 A b \sin (c+d x)+284 a b^2 B \sin (c+d x)+128 A b^3 \sin (c+d x)\right)}{192 b}+\frac{1}{24} \sec ^3(c+d x) \left(17 a b B \sin (c+d x)+8 A b^2 \sin (c+d x)\right)+\frac{1}{4} b^2 B \tan (c+d x) \sec ^3(c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}-\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2 \left(-236 a^3 b B-416 a^2 A b^2-144 a 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)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(15 a^4 B+264 a^3 A b+284 a^2 b^2 B+128 a A b^3\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(45 a^4 B+24 a^3 A b-436 a^2 b^2 B-832 a A b^3-288 b^4 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{768 b d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{\left(59 a^2 B+104 a A b+36 b^2 B\right) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{96 d}+\frac{\left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{192 b d}+\frac{\left(133 a^3 B+472 a^2 A b+356 a b^2 B+128 A b^3\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)}{192 d \sqrt{a+b \sec (c+d x)}}-\frac{\left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 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)}{192 b d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 B\right) \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)}{64 b d \sqrt{a+b \sec (c+d x)}}+\frac{b (11 a B+8 A b) \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{24 d}+\frac{b B \sin (c+d x) \sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{4 d}",1,"-1/768*((a + b*Sec[c + d*x])^(5/2)*((2*(-416*a^2*A*b^2 - 236*a^3*b*B - 144*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(24*a^3*A*b - 832*a*A*b^3 + 45*a^4*B - 436*a^2*b^2*B - 288*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(264*a^3*A*b + 128*a*A*b^3 + 15*a^4*B + 284*a^2*b^2*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(b*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((Sec[c + d*x]^3*(8*A*b^2*Sin[c + d*x] + 17*a*b*B*Sin[c + d*x]))/24 + (Sec[c + d*x]^2*(104*a*A*b*Sin[c + d*x] + 59*a^2*B*Sin[c + d*x] + 36*b^2*B*Sin[c + d*x]))/96 + (Sec[c + d*x]*(264*a^2*A*b*Sin[c + d*x] + 128*A*b^3*Sin[c + d*x] + 15*a^3*B*Sin[c + d*x] + 284*a*b^2*B*Sin[c + d*x]))/(192*b) + (b^2*B*Sec[c + d*x]^3*Tan[c + d*x])/4))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2))","C",0
449,1,678,422,7.219581,"\int \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{1}{24} \sec (c+d x) \left(33 a^2 B \sin (c+d x)+54 a A b \sin (c+d x)+16 b^2 B \sin (c+d x)\right)+\frac{1}{12} \sec ^2(c+d x) \left(13 a b B \sin (c+d x)+6 A b^2 \sin (c+d x)\right)+\frac{1}{3} b^2 B \tan (c+d x) \sec ^2(c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2 \left(96 a^3 A+52 a^2 b B+24 a A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(-33 a^3 B-54 a^2 A b-16 a b^2 B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(-3 a^3 B+126 a^2 A b+104 a b^2 B+48 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{96 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{\left(33 a^2 B+54 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{24 d}-\frac{\left(33 a^2 B+54 a A b+16 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)}{24 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(48 a^3 A+59 a^2 b B+66 a A b^2+16 b^3 B\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)}{24 d \sqrt{a+b \sec (c+d x)}}+\frac{\left(5 a^3 B+30 a^2 A b+20 a b^2 B+8 A b^3\right) \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)}{8 d \sqrt{a+b \sec (c+d x)}}+\frac{b (3 a B+2 A b) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{b B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{3 d}",1,"((a + b*Sec[c + d*x])^(5/2)*((2*(96*a^3*A + 24*a*A*b^2 + 52*a^2*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(126*a^2*A*b + 48*A*b^3 - 3*a^3*B + 104*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(-54*a^2*A*b - 33*a^3*B - 16*a*b^2*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(96*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((Sec[c + d*x]^2*(6*A*b^2*Sin[c + d*x] + 13*a*b*B*Sin[c + d*x]))/12 + (Sec[c + d*x]*(54*a*A*b*Sin[c + d*x] + 33*a^2*B*Sin[c + d*x] + 16*b^2*B*Sin[c + d*x]))/24 + (b^2*B*Sec[c + d*x]^2*Tan[c + d*x])/3))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2))","C",0
450,1,628,359,6.873405,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\sec (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Sec[c + d*x]],x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{1}{4} \sec (c+d x) \left(9 a b B \sin (c+d x)+4 A b^2 \sin (c+d x)\right)+\frac{1}{2} b^2 B \tan (c+d x) \sec (c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2 \left(16 a^3 B+48 a^2 A b+4 a b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(8 a^3 A-9 a^2 b B-4 a A b^2\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(8 a^3 A+21 a^2 b B+36 a A b^2+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)}{\sqrt{a \cos (c+d x)+b}}\right)}{16 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{\left(8 a^2 A-9 a b B-4 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)}{4 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b \left(15 a^2 B+20 a A b+4 b^2 B\right) \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)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{\left(8 a^3 B+16 a^2 A b+11 a b^2 B+4 A b^3\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)}{4 d \sqrt{a+b \sec (c+d x)}}+\frac{b (7 a B+4 A b) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 d}+\frac{b B \sin (c+d x) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}{2 d}",1,"((a + b*Sec[c + d*x])^(5/2)*((2*(48*a^2*A*b + 16*a^3*B + 4*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(8*a^3*A + 36*a*A*b^2 + 21*a^2*b*B + 8*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(8*a^3*A - 4*a*A*b^2 - 9*a^2*b*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(16*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((Sec[c + d*x]*(4*A*b^2*Sin[c + d*x] + 9*a*b*B*Sin[c + d*x]))/4 + (b^2*B*Sec[c + d*x]*Tan[c + d*x])/2))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2))","C",0
451,1,599,349,7.0243642,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(3/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2}{3} a^2 A \sin (c+d x)+b^2 B \tan (c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2 \left(4 a^3 A+36 a^2 b B+36 a A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 i \left(6 a^3 B+14 a^2 A b-3 a b^2 B\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(6 a^3 B+14 a^2 A b+27 a b^2 B+12 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{12 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{\left(6 a^2 B+14 a A b-3 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)}{3 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(2 a^3 A+12 a^2 b B+4 a A b^2+3 b^3 B\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 d \sqrt{a+b \sec (c+d x)}}+\frac{b^2 (5 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 (2 a A-3 b B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \sqrt{\sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(5/2)*((2*(4*a^3*A + 36*a*A*b^2 + 36*a^2*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(14*a^2*A*b + 12*A*b^3 + 6*a^3*B + 27*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(14*a^2*A*b + 6*a^3*B - 3*a*b^2*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(12*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((2*a^2*A*Sin[c + d*x])/3 + b^2*B*Tan[c + d*x]))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2))","C",0
452,1,616,342,7.0018901,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(5/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{1}{5} a^2 A \sin (2 (c+d x))+\frac{2}{15} a (5 a B+11 A b) \sin (c+d x)\right)}{d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac{(a+b \sec (c+d x))^{5/2} \left(\frac{2 i \left(9 a^3 A+35 a^2 b B+23 a A b^2\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(10 a^3 B+34 a^2 A b+90 a b^2 B+30 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 \left(9 a^3 A+35 a^2 b B+23 a A b^2+30 b^3 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}\right)}{30 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2}}","\frac{2 \left(9 a^2 A+35 a b B+23 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 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(5 a^3 B+8 a^2 A b+10 a b^2 B-8 A b^3\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)}{15 d \sqrt{a+b \sec (c+d x)}}+\frac{2 a (5 a B+8 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 d \sqrt{\sec (c+d x)}}+\frac{2 a 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^3 B \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"((a + b*Sec[c + d*x])^(5/2)*((2*(34*a^2*A*b + 30*A*b^3 + 10*a^3*B + 90*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(9*a^3*A + 23*a*A*b^2 + 35*a^2*b*B + 30*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(9*a^3*A + 23*a*A*b^2 + 35*a^2*b*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(30*d*(b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)) + ((a + b*Sec[c + d*x])^(5/2)*((2*a*(11*A*b + 5*a*B)*Sin[c + d*x])/15 + (a^2*A*Sin[2*(c + d*x)])/5))/(d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2))","C",0
453,1,257,340,1.8334066,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{7}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(7/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(a \sin (c+d x) (a \cos (c+d x)+b) \left(15 a^2 A \cos (2 (c+d x))+65 a^2 A+6 a (7 a B+15 A b) \cos (c+d x)+154 a b B+90 A b^2\right)+2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(a \left(25 a^3 A+119 a^2 b B+135 a A b^2+105 b^3 B\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(63 a^3 B+145 a^2 A b+161 a b^2 B+15 A b^3\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)\right)}{105 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(25 a^2 A+77 a b B+45 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{105 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A+56 a b B+15 A b^2\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)}{105 a d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(63 a^3 B+145 a^2 A 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 a (7 a B+10 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 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{7 d \sec ^{\frac{5}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(5/2)*(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a*(25*a^3*A + 135*a*A*b^2 + 119*a^2*b*B + 105*b^3*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*(65*a^2*A + 90*A*b^2 + 154*a*b*B + 6*a*(15*A*b + 7*a*B)*Cos[c + d*x] + 15*a^2*A*Cos[2*(c + d*x)])*Sin[c + d*x]))/(105*a*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","A",1
454,1,313,425,2.7050815,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{9}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(9/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(a (a \cos (c+d x)+b) \left(a \left(\left(266 a^2 A+540 a b B+300 A b^2\right) \sin (2 (c+d x))+5 a (2 (9 a B+19 A b) \sin (3 (c+d x))+7 a A \sin (4 (c+d x)))\right)+2 \left(345 a^3 B+747 a^2 A b+540 a b^2 B+20 A b^3\right) \sin (c+d x)\right)+8 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(a^2 \left(75 a^3 B+261 a^2 A b+405 a b^2 B+155 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)\right)}{1260 a^2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(49 a^2 A+135 a b B+75 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{315 a d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \left(75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\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)}{315 a^2 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(147 a^4 A+435 a^3 b B+279 a^2 A b^2+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 a (3 a B+4 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 A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{9 d \sec ^{\frac{7}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(5/2)*(8*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(261*a^2*A*b + 155*A*b^3 + 75*a^3*B + 405*a*b^2*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*(2*(747*a^2*A*b + 20*A*b^3 + 345*a^3*B + 540*a*b^2*B)*Sin[c + d*x] + a*((266*a^2*A + 300*A*b^2 + 540*a*b*B)*Sin[2*(c + d*x)] + 5*a*(2*(19*A*b + 9*a*B)*Sin[3*(c + d*x)] + 7*a*A*Sin[4*(c + d*x)])))))/(1260*a^2*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","A",1
455,1,380,519,3.8035082,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac{11}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sec[c + d*x]^(11/2),x]","\frac{(a+b \sec (c+d x))^{5/2} \left(a (a \cos (c+d x)+b) \left(a \left(5 a \left(\left(513 a^2 A+836 a b B+452 A b^2\right) \sin (3 (c+d x))+7 a ((22 a B+46 A b) \sin (4 (c+d x))+9 a A \sin (5 (c+d x)))\right)+4 \left(1463 a^3 B+3095 a^2 A b+1650 a b^2 B+30 A b^3\right) \sin (2 (c+d x))\right)+2 \left(6525 a^4 A+16434 a^3 b B+9330 a^2 A b^2+440 a b^3 B-160 A b^4\right) \sin (c+d x)\right)+16 \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(a^2 \left(675 a^4 A+2871 a^3 b B+3315 a^2 A b^2+1705 a b^3 B+10 A b^4\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)\right)}{27720 a^3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{693 d \sec ^{\frac{5}{2}}(c+d x)}+\frac{2 \left(539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3465 a d \sec ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3465 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(a^2-b^2\right) \left(675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 a b^3 B+40 A b^4\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)}{3465 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 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 a (11 a B+14 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{99 d \sec ^{\frac{7}{2}}(c+d x)}+\frac{2 a A \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{11 d \sec ^{\frac{9}{2}}(c+d x)}",1,"((a + b*Sec[c + d*x])^(5/2)*(16*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(675*a^4*A + 3315*a^2*A*b^2 + 10*A*b^4 + 2871*a^3*b*B + 1705*a*b^3*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(b + a*Cos[c + d*x])*(2*(6525*a^4*A + 9330*a^2*A*b^2 - 160*A*b^4 + 16434*a^3*b*B + 440*a*b^3*B)*Sin[c + d*x] + a*(4*(3095*a^2*A*b + 30*A*b^3 + 1463*a^3*B + 1650*a*b^2*B)*Sin[2*(c + d*x)] + 5*a*((513*a^2*A + 452*A*b^2 + 836*a*b*B)*Sin[3*(c + d*x)] + 7*a*((46*A*b + 22*a*B)*Sin[4*(c + d*x)] + 9*a*A*Sin[5*(c + d*x)]))))))/(27720*a^3*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","A",1
456,1,451,344,4.1534947,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(\frac{2 \left(9 a^2 B-12 a A b+8 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2}+\frac{2 i (3 a B-4 A b) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \sqrt{a \cos (c+d x)+b} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b^3 \sqrt{\frac{1}{a-b}}}-\frac{4 a (3 a B-4 A b) \sin (c+d x)}{b^2}+\frac{4 (4 A b-3 a B) \tan (c+d x)}{b}+\frac{8 a B \tan (c+d x)}{b}+\frac{8 a B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b}+8 B \tan (c+d x) \sec (c+d x)\right)}{16 d \sqrt{a+b \sec (c+d x)}}","-\frac{\left(-3 a^2 B+4 a A b-4 b^2 B\right) \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)}{4 b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{(4 A b-3 a B) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{4 b^2 d}-\frac{(4 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)}{4 b^2 d \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(4 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)}{4 b d \sqrt{a+b \sec (c+d x)}}+\frac{B \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d}",1,"(Sqrt[Sec[c + d*x]]*((8*a*B*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/b + (2*(-12*a*A*b + 9*a^2*B + 8*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/b^2 + ((2*I)*(-4*A*b + 3*a*B)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Sqrt[b + a*Cos[c + d*x]]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b^3) - (4*a*(-4*A*b + 3*a*B)*Sin[c + d*x])/b^2 + (8*a*B*Tan[c + d*x])/b + (4*(4*A*b - 3*a*B)*Tan[c + d*x])/b + 8*B*Sec[c + d*x]*Tan[c + d*x]))/(16*d*Sqrt[a + b*Sec[c + d*x]])","C",0
457,1,339,256,7.1853232,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{\sqrt{\sec (c+d x)} \left(2 (4 A b-3 a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+4 B \tan (c+d x) (a \cos (c+d x)+b)-\frac{2 i B \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \sqrt{a \cos (c+d x)+b} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}}}\right)}{4 b 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}} \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{B \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b d}+\frac{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{B \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,"(Sqrt[Sec[c + d*x]]*(2*(4*A*b - 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)] - ((2*I)*B*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Sqrt[b + a*Cos[c + d*x]]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b) + 4*B*(b + a*Cos[c + d*x])*Tan[c + d*x]))/(4*b*d*Sqrt[a + b*Sec[c + d*x]])","C",1
458,1,91,138,0.2718604,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(A F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+B \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{d \sqrt{a+b \sec (c+d x)}}","\frac{2 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)}{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}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{a+b \sec (c+d x)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(A*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + 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",1
459,1,103,150,3.8906223,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{\sec (c+d x)} \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left((a B-A b) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+A (a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\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 (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)}}",1,"(2*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(A*(a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (-(A*b) + 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]])","A",1
460,1,161,212,0.8826372,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{\sec (c+d x)} \left(\left(a^2 A-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)+(a+b) (3 a B-2 A b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+a A \sin (c+d x) (a \cos (c+d x)+b)\right)}{3 a^2 d \sqrt{a+b \sec (c+d x)}}","\frac{2 \left(a^2 A-3 a b B+2 A b^2\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^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*Sqrt[Sec[c + d*x]]*((a + b)*(-2*A*b + 3*a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (a^2*A + 2*A*b^2 - 3*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*A*(b + a*Cos[c + d*x])*Sin[c + d*x]))/(3*a^2*d*Sqrt[a + b*Sec[c + d*x]])","A",1
461,1,198,280,1.3884057,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\frac{2 \sqrt{\sec (c+d x)} \left((a+b) \left(9 a^2 A-10 a b B+8 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(5 a^3 B-7 a^2 A 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)+a \sin (c+d x) (a \cos (c+d x)+b) (3 a A \cos (c+d x)+5 a B-4 A b)\right)}{15 a^3 d \sqrt{a+b \sec (c+d x)}}","-\frac{2 (4 A b-5 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^2 d \sqrt{\sec (c+d x)}}+\frac{2 \left(9 a^2 A-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 \left(-5 a^3 B+7 a^2 A b-10 a b^2 B+8 A b^3\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)}{15 a^3 d \sqrt{a+b \sec (c+d x)}}+\frac{2 A \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d \sec ^{\frac{3}{2}}(c+d x)}",1,"(2*Sqrt[Sec[c + d*x]]*((a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)] + (-7*a^2*A*b - 8*A*b^3 + 5*a^3*B + 10*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*(b + a*Cos[c + d*x])*(-4*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x]))/(15*a^3*d*Sqrt[a + b*Sec[c + d*x]])","A",1
462,1,518,371,5.9713599,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(\frac{4 \tan (c+d x) (a \cos (c+d x)+b) \left(a \left(-3 a^2 B+2 a A b+b^2 B\right) \cos (c+d x)+b B \left(b^2-a^2\right)\right)}{b^4-a^2 b^2}-\frac{(a \cos (c+d x)+b)^{3/2} \left(\frac{2 i \left(3 a^2 B-2 a A b-b^2 B\right) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}}}+\frac{2 \left(9 a^3 B-6 a^2 A b-7 a b^2 B+4 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{8 a b (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)}{\sqrt{a \cos (c+d x)+b}}\right)}{b^2 (a-b) (a+b)}\right)}{4 d (a+b \sec (c+d x))^{3/2}}","\frac{2 a (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) \sin (c+d x) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right)}+\frac{\left(-3 a^2 B+2 a A b+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)}{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 A b-3 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)}{b^2 d \sqrt{a+b \sec (c+d x)}}+\frac{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)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(Sec[c + d*x]^(3/2)*(-(((b + a*Cos[c + d*x])^(3/2)*((8*a*b*(-(A*b) + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(-6*a^2*A*b + 4*A*b^3 + 9*a^3*B - 7*a*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(-2*a*A*b + 3*a^2*B - b^2*B)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b)))/((a - b)*b^2*(a + b))) + (4*(b + a*Cos[c + d*x])*(b*(-a^2 + b^2)*B + a*(2*a*A*b - 3*a^2*B + b^2*B)*Cos[c + d*x])*Tan[c + d*x])/(-(a^2*b^2) + b^4)))/(4*d*(a + b*Sec[c + d*x])^(3/2))","C",1
463,1,464,220,4.8924663,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{\sec ^{\frac{3}{2}}(c+d x) \left(\frac{4 a (A b-a B) \sin (c+d x) (a \cos (c+d x)+b)}{a^2-b^2}+\frac{(a \cos (c+d x)+b)^{3/2} \left(\frac{2 \left(-3 a^2 B+a A b+2 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{4 b (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)}{\sqrt{a \cos (c+d x)+b}}-\frac{2 i (a B-A b) \csc (c+d x) \sqrt{-\frac{a (\cos (c+d x)-1)}{a+b}} \sqrt{\frac{a (\cos (c+d x)+1)}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{a b \sqrt{\frac{1}{a-b}}}\right)}{(b-a) (a+b)}\right)}{2 b d (a+b \sec (c+d x))^{3/2}}","\frac{2 a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 (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)}{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 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)}{b d \sqrt{a+b \sec (c+d x)}}",1,"(Sec[c + d*x]^(3/2)*(((b + a*Cos[c + d*x])^(3/2)*((4*b*(A*b - a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(a*A*b - 3*a^2*B + 2*b^2*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] - ((2*I)*(-(A*b) + a*B)*Sqrt[-((a*(-1 + Cos[c + d*x]))/(a + b))]*Sqrt[(a*(1 + Cos[c + d*x]))/(a - b)]*Csc[c + d*x]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)])))/(a*Sqrt[(a - b)^(-1)]*b)))/((-a + b)*(a + b)) + (4*a*(A*b - a*B)*(b + a*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)))/(2*b*d*(a + b*Sec[c + d*x])^(3/2))","C",1
464,1,161,215,0.8180733,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 \sqrt{\sec (c+d x)} \left(A \left(a^2-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)+a (a B-A b) \sin (c+d x)-\left((a+b) (a B-A b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)}{a d (a-b) (a+b) \sqrt{a+b \sec (c+d x)}}","-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 (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)}{a 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)}}",1,"(2*Sqrt[Sec[c + d*x]]*(-((a + b)*(-(A*b) + a*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]) + A*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + a*(-(A*b) + a*B)*Sin[c + d*x]))/(a*(a - b)*(a + b)*d*Sqrt[a + b*Sec[c + d*x]])","A",1
465,1,178,235,1.0652815,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),x]","-\frac{2 \sqrt{\sec (c+d x)} \left(-\left(a^2-b^2\right) (a B-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)-\left((a+b) \left(a^2 A+a b B-2 A b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)+a b (a B-A b) \sin (c+d x)\right)}{a^2 d (a-b) (a+b) \sqrt{a+b \sec (c+d x)}}","\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2 A+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*Sqrt[Sec[c + d*x]]*(-((a + b)*(a^2*A - 2*A*b^2 + a*b*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticE[(c + d*x)/2, (2*a)/(a + b)]) - (a^2 - b^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*b*(-(A*b) + a*B)*Sin[c + d*x]))/(a^2*(a - b)*(a + b)*d*Sqrt[a + b*Sec[c + d*x]])","A",1
466,1,252,326,1.6084382,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{2 \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b) \left(a (a-b) (a+b) \sin (c+d x) \left(b \left(a^2 A+3 a b B-4 A b^2\right)+a A \left(a^2-b^2\right) \cos (c+d x)\right)+\left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(a^2 \left(a^2 A-3 a b B+2 A b^2\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(3 a^3 B-5 a^2 A b-6 a b^2 B+8 A b^3\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)\right)}{3 a^3 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}","\frac{2 \left(a^2 A+3 a b B-4 A b^2\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}+\frac{2 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2 A-6 a b B+8 A b^2\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^3 d \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^3 B+5 a^2 A 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*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*((a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(a^2*A + 2*A*b^2 - 3*a*b*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(a - b)*(a + b)*(b*(a^2*A - 4*A*b^2 + 3*a*b*B) + a*A*(a^2 - b^2)*Cos[c + d*x])*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2))","A",1
467,1,316,423,2.4025349,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","-\frac{\sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b) \left(a (a-b) (a+b) \left(2 \left(a^2-b^2\right) (9 A b-5 a B) \sin (c+d x) (a \cos (c+d x)+b)-3 a A \left(a^2-b^2\right) \sin (2 (c+d x)) (a \cos (c+d x)+b)+30 b^3 (a B-A b) \sin (c+d x)\right)+2 \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \left(a^2 \left(-5 a^3 B+3 a^2 A b-10 a b^2 B+12 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-\left(9 a^4 A-25 a^3 b B+24 a^2 A b^2+40 a b^3 B-48 A b^4\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)\right)}{15 a^4 d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{3/2}}","\frac{2 \left(a^2 A+5 a b B-6 A b^2\right) \sin (c+d x) \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 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sec ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-5 a^3 B+9 a^2 A b+20 a b^2 B-24 A b^3\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right) \sqrt{\sec (c+d x)}}-\frac{2 \left(-5 a^3 B+12 a^2 A b-40 a b^2 B+48 A b^3\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)}{15 a^4 d \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^4 A-25 a^3 b B+24 a^2 A b^2+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,"-1/15*((b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*(2*(a^2 - b^2)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*(a^2*(3*a^2*A*b + 12*A*b^3 - 5*a^3*B - 10*a*b^2*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] - (9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])) + a*(a - b)*(a + b)*(30*b^3*(-(A*b) + a*B)*Sin[c + d*x] + 2*(a^2 - b^2)*(9*A*b - 5*a*B)*(b + a*Cos[c + d*x])*Sin[c + d*x] - 3*a*A*(a^2 - b^2)*(b + a*Cos[c + d*x])*Sin[2*(c + d*x)])))/(a^4*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(3/2))","A",1
468,1,726,399,6.8608833,"\int \frac{\sec ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3 \left(-\frac{2 \left(a A b \sin (c+d x)-a^2 B \sin (c+d x)\right)}{3 b \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}-\frac{2 \left(3 a^4 B \sin (c+d x)-7 a^2 b^2 B \sin (c+d x)+4 a A b^3 \sin (c+d x)\right)}{3 b^2 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d (a+b \sec (c+d x))^{5/2}}+\frac{\sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^{5/2} \left(\frac{2 i \left(3 a^4 B-7 a^2 b^2 B+4 a A b^3\right) \sin (c+d x) \cos (2 (c+d x)) \sqrt{\frac{a-a \cos (c+d x)}{a+b}} \sqrt{\frac{a \cos (c+d x)+a}{a-b}} \left(a \left(2 b F\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)+a \Pi \left(1-\frac{a}{b};i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)-2 b (a+b) E\left(i \sinh ^{-1}\left(\sqrt{\frac{1}{a-b}} \sqrt{b+a \cos (c+d x)}\right)|\frac{b-a}{a+b}\right)\right)}{b \sqrt{\frac{1}{a-b}} \sqrt{1-\cos ^2(c+d x)} \sqrt{\frac{a^2-a^2 \cos ^2(c+d x)}{a^2}} \left(-a^2+2 (a \cos (c+d x)+b)^2-4 b (a \cos (c+d x)+b)+2 b^2\right)}+\frac{2 \left(9 a^4 B-19 a^2 b^2 B+4 a A b^3+6 b^4 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{\sqrt{a \cos (c+d x)+b}}+\frac{2 \left(4 a^3 b B+2 a^2 A b^2-12 a b^3 B+6 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)}{\sqrt{a \cos (c+d x)+b}}\right)}{6 b^2 d (a-b)^2 (a+b)^2 (a+b \sec (c+d x))^{5/2}}","\frac{2 a (A b-a B) \sin (c+d x) \sec ^{\frac{3}{2}}(c+d x)}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\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)}{3 b d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 a \left(3 a^3 B-7 a b^2 B+4 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^3 B-7 a b^2 B+4 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 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 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)}{b^2 d \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^(5/2)*Sec[c + d*x]^(5/2)*((2*(2*a^2*A*b^2 + 6*A*b^4 + 4*a^3*b*B - 12*a*b^3*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticF[(c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + (2*(4*a*A*b^3 + 9*a^4*B - 19*a^2*b^2*B + 6*b^4*B)*Sqrt[(b + a*Cos[c + d*x])/(a + b)]*EllipticPi[2, (c + d*x)/2, (2*a)/(a + b)])/Sqrt[b + a*Cos[c + d*x]] + ((2*I)*(4*a*A*b^3 + 3*a^4*B - 7*a^2*b^2*B)*Sqrt[(a - a*Cos[c + d*x])/(a + b)]*Sqrt[(a + a*Cos[c + d*x])/(a - b)]*Cos[2*(c + d*x)]*(-2*b*(a + b)*EllipticE[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*(2*b*EllipticF[I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)] + a*EllipticPi[1 - a/b, I*ArcSinh[Sqrt[(a - b)^(-1)]*Sqrt[b + a*Cos[c + d*x]]], (-a + b)/(a + b)]))*Sin[c + d*x])/(Sqrt[(a - b)^(-1)]*b*Sqrt[1 - Cos[c + d*x]^2]*Sqrt[(a^2 - a^2*Cos[c + d*x]^2)/a^2]*(-a^2 + 2*b^2 - 4*b*(b + a*Cos[c + d*x]) + 2*(b + a*Cos[c + d*x])^2))))/(6*(a - b)^2*b^2*(a + b)^2*d*(a + b*Sec[c + d*x])^(5/2)) + ((b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2)*((-2*(a*A*b*Sin[c + d*x] - a^2*B*Sin[c + d*x]))/(3*b*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) - (2*(4*a*A*b^3*Sin[c + d*x] + 3*a^4*B*Sin[c + d*x] - 7*a^2*b^2*B*Sin[c + d*x]))/(3*b^2*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*(a + b*Sec[c + d*x])^(5/2))","C",0
469,1,217,329,2.3051237,"\int \frac{\sec ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sec[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) \left(\frac{2 \sin (c+d x) (a \cos (c+d x)+b) \left(a^3 B+a \left(3 a^2 A-4 a b B+A b^2\right) \cos (c+d x)+2 a^2 A b-5 a b^2 B+2 A b^3\right)}{\left(a^2-b^2\right)^2}-\frac{2 (a+b) \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{5/2} \left(\left(3 a^2 A-4 a b B+A b^2\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-(a-b) (a B-A b) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{a (a-b)^2}\right)}{3 d (a+b \sec (c+d x))^{5/2}}","\frac{2 a (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}-\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)}{3 a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2 A-4 a b B+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)}{3 a 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 \left(a^3 B+2 a^2 A b-5 a b^2 B+2 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 b d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}",1,"(Sec[c + d*x]^(5/2)*((-2*(a + b)*((b + a*Cos[c + d*x])/(a + b))^(5/2)*((3*a^2*A + A*b^2 - 4*a*b*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - (a - b)*(-(A*b) + a*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a*(a - b)^2) + (2*(b + a*Cos[c + d*x])*(2*a^2*A*b + 2*A*b^3 + a^3*B - 5*a*b^2*B + a*(3*a^2*A + A*b^2 - 4*a*b*B)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*d*(a + b*Sec[c + d*x])^(5/2))","A",1
470,1,245,346,2.2149485,"\int \frac{\sqrt{\sec (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Sec[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(\frac{a \sin (c+d x) \left(a \left(3 a^3 B-6 a^2 A b+a b^2 B+2 A b^3\right) \cos (c+d x)+b \left(2 a^3 B-5 a^2 A b+2 a b^2 B+A b^3\right)\right)}{\left(a^2-b^2\right)^2}-\frac{\left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(\left(3 a^3 B-6 a^2 A b+a b^2 B+2 A b^3\right) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-(a-b) \left(3 a^2 A-a b B-2 A b^2\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)}{(a-b)^2}\right)}{3 a^2 d (a+b \sec (c+d x))^{5/2}}","-\frac{2 (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2 A-a b B-2 A b^2\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^2 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-2 a^3 B+5 a^2 A b-2 a b^2 B-A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-3 a^3 B+6 a^2 A 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*(b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*(-((((b + a*Cos[c + d*x])/(a + b))^(3/2)*((-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - (a - b)*(3*a^2*A - 2*A*b^2 - a*b*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]))/(a - b)^2) + (a*(b*(-5*a^2*A*b + A*b^3 + 2*a^3*B + 2*a*b^2*B) + a*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*a^2*d*(a + b*Sec[c + d*x])^(5/2))","A",1
471,1,297,368,2.6909558,"\int \frac{A+B \sec (c+d x)}{\sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Sec[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(-\frac{a b \sin (c+d x) \left(a \left(6 a^3 B-9 a^2 A b-2 a b^2 B+5 A b^3\right) \cos (c+d x)+b \left(5 a^3 B-8 a^2 A b-a b^2 B+4 A b^3\right)\right)}{\left(a^2-b^2\right)^2}-\frac{\left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(-\left(a^2 \left(3 a^3 B-6 a^2 A b+a b^2 B+2 A b^3\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)-\left(3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 a b^3 B+8 A b^4\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}\right)}{3 a^3 d (a+b \sec (c+d x))^{5/2}}","\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left(-5 a^3 B+8 a^2 A b+a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{\sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^3 B+9 a^2 A b+2 a b^2 B-8 A b^3\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^3 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 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*(b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*(-((((b + a*Cos[c + d*x])/(a + b))^(3/2)*(-(a^2*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)]) - (3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])))/((a - b)^2*(a + b))) - (a*b*(b*(-8*a^2*A*b + 4*A*b^3 + 5*a^3*B - a*b^2*B) + a*(-9*a^2*A*b + 5*A*b^3 + 6*a^3*B - 2*a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2))/(3*a^3*d*(a + b*Sec[c + d*x])^(5/2))","A",1
472,1,353,472,3.1131387,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{2 \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(\frac{a \sin (c+d x) \left(a^6 A+A \left(a^3-a b^2\right)^2 \cos (2 (c+d x))+16 a^3 b^3 B-25 a^2 A b^4+2 a b \left(2 a^4 A+9 a^3 b B-16 a^2 A b^2-5 a b^3 B+10 A b^4\right) \cos (c+d x)-8 a b^5 B+16 A b^6\right)}{2 \left(a^2-b^2\right)^2}+\frac{\left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(a^2 \left(a^4 A-6 a^3 b B+7 a^2 A b^2+2 a b^3 B-4 A b^4\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)+\left(3 a^5 B-8 a^4 A b-15 a^3 b^2 B+28 a^2 A b^3+8 a b^4 B-16 A b^5\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}\right)}{3 a^4 d (a+b \sec (c+d x))^{5/2}}","\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\sec (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left(-7 a^3 B+10 a^2 A b+3 a b^2 B-6 A b^3\right) \sin (c+d x)}{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 \left(a^4 A+8 a^3 b B-13 a^2 A b^2-4 a b^3 B+8 A b^4\right) \sin (c+d x) \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 \left(a^4 A-9 a^3 b B+16 a^2 A b^2+8 a b^3 B-16 A b^4\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^4 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-3 a^5 B+8 a^4 A b+15 a^3 b^2 B-28 a^2 A b^3-8 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*(b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((((b + a*Cos[c + d*x])/(a + b))^(3/2)*(a^2*(a^4*A + 7*a^2*A*b^2 - 4*A*b^4 - 6*a^3*b*B + 2*a*b^3*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] + (-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])))/((a - b)^2*(a + b)) + (a*(a^6*A - 25*a^2*A*b^4 + 16*A*b^6 + 16*a^3*b^3*B - 8*a*b^5*B + 2*a*b*(2*a^4*A - 16*a^2*A*b^2 + 10*A*b^4 + 9*a^3*b*B - 5*a*b^3*B)*Cos[c + d*x] + A*(a^3 - a*b^2)^2*Cos[2*(c + d*x)])*Sin[c + d*x])/(2*(a^2 - b^2)^2)))/(3*a^4*d*(a + b*Sec[c + d*x])^(5/2))","A",1
473,1,392,588,4.2982607,"\int \frac{A+B \sec (c+d x)}{\sec ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sec[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{\sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b) \left(a \left(\frac{10 b^4 (A b-a B) \sin (c+d x)}{b^2-a^2}-\frac{10 b^3 \left(12 a^3 B-15 a^2 A b-8 a b^2 B+11 A b^3\right) \sin (c+d x) (a \cos (c+d x)+b)}{\left(a^2-b^2\right)^2}-2 (14 A b-5 a B) \sin (c+d x) (a \cos (c+d x)+b)^2+3 a A \sin (2 (c+d x)) (a \cos (c+d x)+b)^2\right)-\frac{2 \left(\frac{a \cos (c+d x)+b}{a+b}\right)^{3/2} \left(a^2 \left(-5 a^5 B+8 a^4 A b-35 a^3 b^2 B+44 a^2 A b^3+20 a b^4 B-32 A b^5\right) F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-\left(9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right) \left((a+b) E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)-b F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)\right)\right)}{(a-b)^2 (a+b)}\right)}{15 a^5 d (a+b \sec (c+d x))^{5/2}}","\frac{2 b (A b-a B) \sin (c+d x)}{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 b \left(-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right) \sin (c+d x)}{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 \left(3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right) \sin (c+d x) \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 \left(-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right) \sin (c+d x) \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 \left(-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+80 a b^4 B-128 A b^5\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)}{15 a^5 d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 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,"((b + a*Cos[c + d*x])*Sec[c + d*x]^(5/2)*((-2*((b + a*Cos[c + d*x])/(a + b))^(3/2)*(a^2*(8*a^4*A*b + 44*a^2*A*b^3 - 32*A*b^5 - 5*a^5*B - 35*a^3*b^2*B + 20*a*b^4*B)*EllipticF[(c + d*x)/2, (2*a)/(a + b)] - (9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*((a + b)*EllipticE[(c + d*x)/2, (2*a)/(a + b)] - b*EllipticF[(c + d*x)/2, (2*a)/(a + b)])))/((a - b)^2*(a + b)) + a*((10*b^4*(A*b - a*B)*Sin[c + d*x])/(-a^2 + b^2) - (10*b^3*(-15*a^2*A*b + 11*A*b^3 + 12*a^3*B - 8*a*b^2*B)*(b + a*Cos[c + d*x])*Sin[c + d*x])/(a^2 - b^2)^2 - 2*(14*A*b - 5*a*B)*(b + a*Cos[c + d*x])^2*Sin[c + d*x] + 3*a*A*(b + a*Cos[c + d*x])^2*Sin[2*(c + d*x)])))/(15*a^5*d*(a + b*Sec[c + d*x])^(5/2))","A",1
474,0,0,126,22.7119102,"\int (a+b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]),x]","\int (a+b \sec (c+d x))^{2/3} (A+B \sec (c+d x)) \, dx","A \text{Int}\left((a+b \sec (c+d x))^{2/3},x\right)+\frac{\sqrt{2} B \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)}{d \sqrt{\sec (c+d x)+1} \left(\frac{a+b \sec (c+d x)}{a+b}\right)^{2/3}}",0,"Integrate[(a + b*Sec[c + d*x])^(2/3)*(A + B*Sec[c + d*x]), x]","A",-1
475,0,0,126,19.3234973,"\int \sqrt[3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[(a + b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x]),x]","\int \sqrt[3]{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","A \text{Int}\left(\sqrt[3]{a+b \sec (c+d x)},x\right)+\frac{\sqrt{2} B \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)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{\frac{a+b \sec (c+d x)}{a+b}}}",0,"Integrate[(a + b*Sec[c + d*x])^(1/3)*(A + B*Sec[c + d*x]), x]","A",-1
476,0,0,126,3.5327785,"\int \frac{A+B \sec (c+d x)}{\sqrt[3]{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(1/3),x]","\int \frac{A+B \sec (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 \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)}{d \sqrt{\sec (c+d x)+1} \sqrt[3]{a+b \sec (c+d x)}}",0,"Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(1/3), x]","A",-1
477,0,0,126,3.460192,"\int \frac{A+B \sec (c+d x)}{(a+b \sec (c+d x))^{2/3}} \, dx","Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(2/3),x]","\int \frac{A+B \sec (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 \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)}{d \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^{2/3}}",0,"Integrate[(A + B*Sec[c + d*x])/(a + b*Sec[c + d*x])^(2/3), x]","A",-1
478,0,0,36,4.777801,"\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx","Integrate[(c*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m*(A + B*Sec[e + f*x]),x]","\int (c \sec (e+f x))^n (a+b \sec (e+f x))^m (A+B \sec (e+f x)) \, dx","\text{Int}\left((A+B \sec (e+f x)) (c \sec (e+f x))^n (a+b \sec (e+f x))^m,x\right)",0,"Integrate[(c*Sec[e + f*x])^n*(a + b*Sec[e + f*x])^m*(A + B*Sec[e + f*x]), x]","A",-1
479,1,365,544,4.9245786,"\int \sec ^m(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-1}(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \left(\frac{a^4 A \cos ^5(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right)}{m}+\frac{a^3 (a B+4 A b) \cos ^4(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sec ^2(c+d x)\right)}{m+1}+b \left(\frac{2 a^2 (2 a B+3 A b) \cos ^3(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sec ^2(c+d x)\right)}{m+2}+b \left(\frac{2 a (3 a B+2 A b) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\sec ^2(c+d x)\right)}{m+3}+b \left(\frac{(4 a B+A b) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\sec ^2(c+d x)\right)}{m+4}+\frac{b B \, _2F_1\left(\frac{1}{2},\frac{m+5}{2};\frac{m+7}{2};\sec ^2(c+d x)\right)}{m+5}\right)\right)\right)\right)}{d (a \cos (c+d x)+b)^4 (A \cos (c+d x)+B)}","\frac{b^2 \sin (c+d x) \left(a^2 B \left(m^2+9 m+26\right)+2 a A b (m+4)^2+b^2 B (m+3)^2\right) \sec ^{m+2}(c+d x)}{d (m+2) (m+3) (m+4)}+\frac{b \sin (c+d x) \left(2 a^3 B \left(m^2+8 m+19\right)+a^2 A b \left(5 m^2+37 m+68\right)+4 a b^2 B \left(m^2+6 m+8\right)+A b^3 \left(m^2+6 m+8\right)\right) \sec ^{m+1}(c+d x)}{d (m+1) (m+3) (m+4)}-\frac{\sin (c+d x) \left(a^4 A \left(m^2+4 m+3\right)+4 a^3 b B m (m+3)+6 a^2 A b^2 m (m+3)+4 a b^3 B m (m+2)+A b^4 m (m+2)\right) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right)}{d (1-m) (m+1) (m+3) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(a^4 B \left(m^2+6 m+8\right)+4 a^3 A b \left(m^2+6 m+8\right)+6 a^2 b^2 B \left(m^2+5 m+4\right)+4 a A b^3 \left(m^2+5 m+4\right)+b^4 B \left(m^2+4 m+3\right)\right) \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},-\frac{m}{2};\frac{2-m}{2};\cos ^2(c+d x)\right)}{d m (m+2) (m+4) \sqrt{\sin ^2(c+d x)}}+\frac{b \sin (c+d x) (a B (m+7)+A b (m+4)) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))^2}{d (m+3) (m+4)}+\frac{b B \sin (c+d x) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))^3}{d (m+4)}",1,"(Csc[c + d*x]*((a^4*A*Cos[c + d*x]^5*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2])/m + (a^3*(4*A*b + a*B)*Cos[c + d*x]^4*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sec[c + d*x]^2])/(1 + m) + b*((2*a^2*(3*A*b + 2*a*B)*Cos[c + d*x]^3*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sec[c + d*x]^2])/(2 + m) + b*((2*a*(2*A*b + 3*a*B)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Sec[c + d*x]^2])/(3 + m) + b*(((A*b + 4*a*B)*Cos[c + d*x]*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Sec[c + d*x]^2])/(4 + m) + (b*B*Hypergeometric2F1[1/2, (5 + m)/2, (7 + m)/2, Sec[c + d*x]^2])/(5 + m)))))*Sec[c + d*x]^(-1 + m)*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x])*Sqrt[-Tan[c + d*x]^2])/(d*(b + a*Cos[c + d*x])^4*(B + A*Cos[c + d*x]))","A",1
480,1,307,366,2.5577682,"\int \sec ^m(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m-1}(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \left(\frac{a^3 A \cos ^4(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right)}{m}+\frac{a^2 (a B+3 A b) \cos ^3(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sec ^2(c+d x)\right)}{m+1}+b \left(\frac{3 a (a B+A b) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sec ^2(c+d x)\right)}{m+2}+b \left(\frac{(3 a B+A b) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\sec ^2(c+d x)\right)}{m+3}+\frac{b B \, _2F_1\left(\frac{1}{2},\frac{m+4}{2};\frac{m+6}{2};\sec ^2(c+d x)\right)}{m+4}\right)\right)\right)}{d (a \cos (c+d x)+b)^3 (A \cos (c+d x)+B)}","\frac{b \sin (c+d x) \left(2 a^2 B (m+4)+3 a A b (m+3)+b^2 B (m+2)\right) \sec ^{m+1}(c+d x)}{d (m+1) (m+3)}-\frac{\sin (c+d x) \left(a^3 A \left(m^2+4 m+3\right)+3 a^2 b B m (m+3)+3 a A b^2 m (m+3)+b^3 B m (m+2)\right) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right)}{d (m+3) \left(1-m^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(a^3 B (m+2)+3 a^2 A b (m+2)+3 a b^2 B (m+1)+A b^3 (m+1)\right) \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},-\frac{m}{2};\frac{2-m}{2};\cos ^2(c+d x)\right)}{d m (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{b^2 \sin (c+d x) (a B (m+5)+A b (m+3)) \sec ^{m+2}(c+d x)}{d (m+2) (m+3)}+\frac{b B \sin (c+d x) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))^2}{d (m+3)}",1,"(Csc[c + d*x]*((a^3*A*Cos[c + d*x]^4*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2])/m + (a^2*(3*A*b + a*B)*Cos[c + d*x]^3*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sec[c + d*x]^2])/(1 + m) + b*((3*a*(A*b + a*B)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sec[c + d*x]^2])/(2 + m) + b*(((A*b + 3*a*B)*Cos[c + d*x]*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Sec[c + d*x]^2])/(3 + m) + (b*B*Hypergeometric2F1[1/2, (4 + m)/2, (6 + m)/2, Sec[c + d*x]^2])/(4 + m))))*Sec[c + d*x]^(-1 + m)*(a + b*Sec[c + d*x])^3*(A + B*Sec[c + d*x])*Sqrt[-Tan[c + d*x]^2])/(d*(b + a*Cos[c + d*x])^3*(B + A*Cos[c + d*x]))","A",1
481,1,239,261,1.0759485,"\int \sec ^m(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m+2}(c+d x) \left(a^2 A \left(m^3+6 m^2+11 m+6\right) \cos ^3(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right)+a m \left(m^2+5 m+6\right) (a B+2 A b) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sec ^2(c+d x)\right)+b m (m+1) \left((m+3) (2 a B+A b) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sec ^2(c+d x)\right)+b B (m+2) \, _2F_1\left(\frac{1}{2},\frac{m+3}{2};\frac{m+5}{2};\sec ^2(c+d x)\right)\right)\right)}{d m (m+1) (m+2) (m+3)}","-\frac{\sin (c+d x) \left(a^2 A (m+1)+2 a b B m+A b^2 m\right) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right)}{d \left(1-m^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{\sin (c+d x) \left(a (m+2) (a B+2 A b)+b^2 B (m+1)\right) \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},-\frac{m}{2};\frac{2-m}{2};\cos ^2(c+d x)\right)}{d m (m+2) \sqrt{\sin ^2(c+d x)}}+\frac{b \sin (c+d x) (a B (m+3)+A b (m+2)) \sec ^{m+1}(c+d x)}{d (m+1) (m+2)}+\frac{b B \sin (c+d x) \sec ^{m+1}(c+d x) (a+b \sec (c+d x))}{d (m+2)}",1,"(Csc[c + d*x]*(a^2*A*(6 + 11*m + 6*m^2 + m^3)*Cos[c + d*x]^3*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2] + a*(2*A*b + a*B)*m*(6 + 5*m + m^2)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sec[c + d*x]^2] + b*m*(1 + m)*((A*b + 2*a*B)*(3 + m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sec[c + d*x]^2] + b*B*(2 + m)*Hypergeometric2F1[1/2, (3 + m)/2, (5 + m)/2, Sec[c + d*x]^2]))*Sec[c + d*x]^(2 + m)*Sqrt[-Tan[c + d*x]^2])/(d*m*(1 + m)*(2 + m)*(3 + m))","A",1
482,1,168,177,0.4622755,"\int \sec ^m(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sec[c + d*x]^m*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{-\tan ^2(c+d x)} \csc (c+d x) \sec ^{m+1}(c+d x) \left(m (m+2) (a B+A b) \cos (c+d x) \, _2F_1\left(\frac{1}{2},\frac{m+1}{2};\frac{m+3}{2};\sec ^2(c+d x)\right)+a A \left(m^2+3 m+2\right) \cos ^2(c+d x) \, _2F_1\left(\frac{1}{2},\frac{m}{2};\frac{m+2}{2};\sec ^2(c+d x)\right)+b B m (m+1) \, _2F_1\left(\frac{1}{2},\frac{m+2}{2};\frac{m+4}{2};\sec ^2(c+d x)\right)\right)}{d m (m+1) (m+2)}","-\frac{\sin (c+d x) (a A (m+1)+b B m) \sec ^{m-1}(c+d x) \, _2F_1\left(\frac{1}{2},\frac{1-m}{2};\frac{3-m}{2};\cos ^2(c+d x)\right)}{d \left(1-m^2\right) \sqrt{\sin ^2(c+d x)}}+\frac{(a B+A b) \sin (c+d x) \sec ^m(c+d x) \, _2F_1\left(\frac{1}{2},-\frac{m}{2};\frac{2-m}{2};\cos ^2(c+d x)\right)}{d m \sqrt{\sin ^2(c+d x)}}+\frac{b B \sin (c+d x) \sec ^{m+1}(c+d x)}{d (m+1)}",1,"(Csc[c + d*x]*(a*A*(2 + 3*m + m^2)*Cos[c + d*x]^2*Hypergeometric2F1[1/2, m/2, (2 + m)/2, Sec[c + d*x]^2] + (A*b + a*B)*m*(2 + m)*Cos[c + d*x]*Hypergeometric2F1[1/2, (1 + m)/2, (3 + m)/2, Sec[c + d*x]^2] + b*B*m*(1 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, Sec[c + d*x]^2])*Sec[c + d*x]^(1 + m)*Sqrt[-Tan[c + d*x]^2])/(d*m*(1 + m)*(2 + m))","A",1
483,1,872,132,6.2949455,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{3 (A+B) \cot (c)}{5 d}+\frac{(23 A+28 B) \cos (d x) \sin (c)}{84 d}+\frac{(A+B) \cos (2 d x) \sin (2 c)}{10 d}+\frac{A \cos (3 d x) \sin (3 c)}{28 d}+\frac{(23 A+28 B) \cos (c) \sin (d x)}{84 d}+\frac{(A+B) \cos (2 c) \sin (2 d x)}{10 d}+\frac{A \cos (3 c) \sin (3 d x)}{28 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{5 A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (5 A+7 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 a (5 A+7 B) \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}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*((-3*(A + B)*Cot[c])/(5*d) + ((23*A + 28*B)*Cos[d*x]*Sin[c])/(84*d) + ((A + B)*Cos[2*d*x]*Sin[2*c])/(10*d) + (A*Cos[3*d*x]*Sin[3*c])/(28*d) + ((23*A + 28*B)*Cos[c]*Sin[d*x])/(84*d) + ((A + B)*Cos[2*c]*Sin[2*d*x])/(10*d) + (A*Cos[3*c]*Sin[3*d*x])/(28*d)) - (5*A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
484,1,830,101,6.2214975,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(-\frac{(3 A+5 B) \cot (c)}{5 d}+\frac{(A+B) \cos (d x) \sin (c)}{3 d}+\frac{A \cos (2 d x) \sin (2 c)}{10 d}+\frac{(A+B) \cos (c) \sin (d x)}{3 d}+\frac{A \cos (2 c) \sin (2 d x)}{10 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)-\frac{3 A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (3 A+5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(-1/5*((3*A + 5*B)*Cot[c])/d + ((A + B)*Cos[d*x]*Sin[c])/(3*d) + (A*Cos[2*d*x]*Sin[2*c])/(10*d) + ((A + B)*Cos[c]*Sin[d*x])/(3*d) + (A*Cos[2*c]*Sin[2*d*x])/(10*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (3*A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d) - (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
485,1,309,70,6.239831,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(\sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)} \left(-4 (A+3 B) \sin (c) \sqrt{\csc ^2(c)} \sqrt{\sec ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+9 (A+B) \csc (c) \sec (c) \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)-12 A \cot (c) \sqrt{\sec ^2(c)} \cos (c+d x)+4 A \sqrt{\sec ^2(c)} \sin (c+d x) \cos (c+d x)+3 A \csc (c) \sec (c) \cos \left(c+\tan ^{-1}(\tan (c))+d x\right)-12 B \cot (c) \sqrt{\sec ^2(c)} \cos (c+d x)+3 B \csc (c) \sec (c) \cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)-6 (A+B) \sec (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)\right)}{12 d \sqrt{\sec ^2(c)} \sqrt{\cos (c+d x)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}","\frac{2 a (A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a (A+B) 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}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*(-6*(A + B)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sec[c]*Sin[d*x + ArcTan[Tan[c]]] + (9*(A + B)*Cos[c - d*x - ArcTan[Tan[c]]]*Csc[c]*Sec[c] + 3*A*Cos[c + d*x + ArcTan[Tan[c]]]*Csc[c]*Sec[c] + 3*B*Cos[c + d*x + ArcTan[Tan[c]]]*Csc[c]*Sec[c] - 12*A*Cos[c + d*x]*Cot[c]*Sqrt[Sec[c]^2] - 12*B*Cos[c + d*x]*Cot[c]*Sqrt[Sec[c]^2] - 4*(A + 3*B)*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Sec[c]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] + 4*A*Cos[c + d*x]*Sqrt[Sec[c]^2]*Sin[c + d*x])*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2]))/(12*d*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2])","C",0
486,1,252,66,6.0554988,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{a (\cos (c+d x)+1) \sec ^2\left(\frac{1}{2} (c+d x)\right) \left(-\frac{2 (A-B) \sec (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\sec ^2(c)} \sqrt{\sin ^2\left(\tan ^{-1}(\tan (c))+d x\right)}}-4 (A+B) \sin (c) \sqrt{\csc ^2(c)} \cos (c+d x) \sqrt{\cos ^2\left(d x-\tan ^{-1}(\cot (c))\right)} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)+\frac{\csc (c) \left(-2 \sqrt{\sec ^2(c)} ((A-2 B) \cos (d x)+A \cos (2 c+d x))+3 (A-B) \sec (c) \cos \left(c-\tan ^{-1}(\tan (c))-d x\right)+(A-B) \sec (c) \cos \left(c+\tan ^{-1}(\tan (c))+d x\right)\right)}{\sqrt{\sec ^2(c)}}\right)}{4 d \sqrt{\cos (c+d x)}}","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(a*(1 + Cos[c + d*x])*Sec[(c + d*x)/2]^2*((Csc[c]*(3*(A - B)*Cos[c - d*x - ArcTan[Tan[c]]]*Sec[c] + (A - B)*Cos[c + d*x + ArcTan[Tan[c]]]*Sec[c] - 2*((A - 2*B)*Cos[d*x] + A*Cos[2*c + d*x])*Sqrt[Sec[c]^2]))/Sqrt[Sec[c]^2] - 4*(A + B)*Cos[c + d*x]*Sqrt[Cos[d*x - ArcTan[Cot[c]]]^2]*Sqrt[Csc[c]^2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[d*x - ArcTan[Cot[c]]]*Sin[c] - (2*(A - B)*HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sec[c]*Sin[d*x + ArcTan[Tan[c]]])/(Sqrt[Sec[c]^2]*Sqrt[Sin[d*x + ArcTan[Tan[c]]]^2])))/(4*d*Sqrt[Cos[c + d*x]])","C",0
487,1,813,95,6.3729654,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{B \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{\sec (c) (B \sin (c)+3 A \sin (d x)+3 B \sin (d x)) \sec (c+d x)}{3 d}+\frac{(A+B) \csc (c) \sec (c)}{d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a (A+B) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((A + B)*Csc[c]*Sec[c])/d + (B*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (Sec[c]*Sec[c + d*x]*(B*Sin[c] + 3*A*Sin[d*x] + 3*B*Sin[d*x]))/(3*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d))","C",0
488,1,865,132,6.4273396,"\int \frac{(a+a \sec (c+d x)) (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","a \left(\sqrt{\cos (c+d x)} (\cos (c+d x)+1) \left(\frac{B \sec (c) \sin (d x) \sec ^3(c+d x)}{5 d}+\frac{\sec (c) (3 B \sin (c)+5 A \sin (d x)+5 B \sin (d x)) \sec ^2(c+d x)}{15 d}+\frac{\sec (c) (5 A \sin (c)+5 B \sin (c)+15 A \sin (d x)+9 B \sin (d x)) \sec (c+d x)}{15 d}+\frac{(5 A+3 B) \csc (c) \sec (c)}{5 d}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)+\frac{A (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d}+\frac{3 B (\cos (c+d x)+1) \csc (c) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d}-\frac{A (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}-\frac{B (\cos (c+d x)+1) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d \sqrt{\cot ^2(c)+1}}\right)","\frac{2 a (A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 a (5 A+3 B) 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 \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (5 A+3 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 a B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"a*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sec[c/2 + (d*x)/2]^2*(((5*A + 3*B)*Csc[c]*Sec[c])/(5*d) + (B*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(5*d) + (Sec[c]*Sec[c + d*x]^2*(3*B*Sin[c] + 5*A*Sin[d*x] + 5*B*Sin[d*x]))/(15*d) + (Sec[c]*Sec[c + d*x]*(5*A*Sin[c] + 5*B*Sin[c] + 15*A*Sin[d*x] + 9*B*Sin[d*x]))/(15*d)) - (A*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) - (B*(1 + Cos[c + d*x])*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^2*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*Sqrt[1 + Cot[c]^2]) + (A*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d) + (3*B*(1 + Cos[c + d*x])*Csc[c]*Sec[c/2 + (d*x)/2]^2*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d))","C",0
489,1,1086,194,6.3065533,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{7}{2}}(c+d x) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(-\frac{(8 A+9 B) \cot (c)}{15 d}+\frac{(46 A+51 B) \cos (d x) \sin (c)}{168 d}+\frac{(37 A+36 B) \cos (2 d x) \sin (2 c)}{360 d}+\frac{(2 A+B) \cos (3 d x) \sin (3 c)}{56 d}+\frac{A \cos (4 d x) \sin (4 c)}{144 d}+\frac{(46 A+51 B) \cos (c) \sin (d x)}{168 d}+\frac{(37 A+36 B) \cos (2 c) \sin (2 d x)}{360 d}+\frac{(2 A+B) \cos (3 c) \sin (3 d x)}{56 d}+\frac{A \cos (4 c) \sin (4 d x)}{144 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{B+A \cos (c+d x)}-\frac{4 A \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d (B+A \cos (c+d x))}-\frac{3 B \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (B+A \cos (c+d x))}-\frac{5 A \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{21 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{2 B \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (5 A+6 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (8 A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{15 d}+\frac{2 a^2 (11 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d}+\frac{4 a^2 (8 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{45 d}+\frac{4 a^2 (5 A+6 B) \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) \left(a^2 \cos (c+d x)+a^2\right)}{9 d}",1,"(Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(-1/15*((8*A + 9*B)*Cot[c])/d + ((46*A + 51*B)*Cos[d*x]*Sin[c])/(168*d) + ((37*A + 36*B)*Cos[2*d*x]*Sin[2*c])/(360*d) + ((2*A + B)*Cos[3*d*x]*Sin[3*c])/(56*d) + (A*Cos[4*d*x]*Sin[4*c])/(144*d) + ((46*A + 51*B)*Cos[c]*Sin[d*x])/(168*d) + ((37*A + 36*B)*Cos[2*c]*Sin[2*d*x])/(360*d) + ((2*A + B)*Cos[3*c]*Sin[3*d*x])/(56*d) + (A*Cos[4*c]*Sin[4*d*x])/(144*d)))/(B + A*Cos[c + d*x]) - (5*A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(21*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (4*A*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(15*d*(B + A*Cos[c + d*x])) - (3*B*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(B + A*Cos[c + d*x]))","C",0
490,1,1040,161,6.2667628,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{7}{2}}(c+d x) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(-\frac{(3 A+4 B) \cot (c)}{5 d}+\frac{(51 A+56 B) \cos (d x) \sin (c)}{168 d}+\frac{(2 A+B) \cos (2 d x) \sin (2 c)}{20 d}+\frac{A \cos (3 d x) \sin (3 c)}{56 d}+\frac{(51 A+56 B) \cos (c) \sin (d x)}{168 d}+\frac{(2 A+B) \cos (2 c) \sin (2 d x)}{20 d}+\frac{A \cos (3 c) \sin (3 d x)}{56 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{B+A \cos (c+d x)}-\frac{3 A \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (B+A \cos (c+d x))}-\frac{2 B \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x))}-\frac{2 A \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{B \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (6 A+7 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{4 a^2 (3 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (9 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{4 a^2 (6 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{7 d}",1,"(Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(-1/5*((3*A + 4*B)*Cot[c])/d + ((51*A + 56*B)*Cos[d*x]*Sin[c])/(168*d) + ((2*A + B)*Cos[2*d*x]*Sin[2*c])/(20*d) + (A*Cos[3*d*x]*Sin[3*c])/(56*d) + ((51*A + 56*B)*Cos[c]*Sin[d*x])/(168*d) + ((2*A + B)*Cos[2*c]*Sin[2*d*x])/(20*d) + (A*Cos[3*c]*Sin[3*d*x])/(56*d)))/(B + A*Cos[c + d*x]) - (2*A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (3*A*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(B + A*Cos[c + d*x])) - (2*B*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(B + A*Cos[c + d*x]))","C",0
491,1,994,126,6.3098258,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{7}{2}}(c+d x) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(-\frac{(4 A+5 B) \cot (c)}{5 d}+\frac{(2 A+B) \cos (d x) \sin (c)}{6 d}+\frac{A \cos (2 d x) \sin (2 c)}{20 d}+\frac{(2 A+B) \cos (c) \sin (d x)}{6 d}+\frac{A \cos (2 c) \sin (2 d x)}{20 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{B+A \cos (c+d x)}-\frac{2 A \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x))}-\frac{B \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (B+A \cos (c+d x))}-\frac{A \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{2 B \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 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^2 (7 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \left(a^2 \cos (c+d x)+a^2\right)}{5 d}",1,"(Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(-1/5*((4*A + 5*B)*Cot[c])/d + ((2*A + B)*Cos[d*x]*Sin[c])/(6*d) + (A*Cos[2*d*x]*Sin[2*c])/(20*d) + ((2*A + B)*Cos[c]*Sin[d*x])/(6*d) + (A*Cos[2*c]*Sin[2*d*x])/(20*d)))/(B + A*Cos[c + d*x]) - (A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (2*A*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(B + A*Cos[c + d*x])) - (B*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(B + A*Cos[c + d*x]))","C",0
492,1,735,116,6.3921509,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","-\frac{A \csc (c) \cos ^3(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{2 d (A \cos (c+d x)+B)}-\frac{2 A \csc (c) \cos ^3(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1} (A \cos (c+d x)+B)}-\frac{B \csc (c) \cos ^3(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1} (A \cos (c+d x)+B)}+\frac{\cos ^{\frac{7}{2}}(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \left(-\frac{\csc (c) \sec (c) (2 A \cos (2 c)+2 A+B \cos (2 c)-B)}{4 d}+\frac{A \sin (c) \cos (d x)}{6 d}+\frac{A \cos (c) \sin (d x)}{6 d}+\frac{B \sec (c) \sin (d x) \sec (c+d x)}{2 d}\right)}{A \cos (c+d x)+B}","\frac{4 a^2 (2 A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{4 a^2 A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{d \sqrt{\cos (c+d x)}}",1,"(Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(-1/4*((2*A - B + 2*A*Cos[2*c] + B*Cos[2*c])*Csc[c]*Sec[c])/d + (A*Cos[d*x]*Sin[c])/(6*d) + (A*Cos[c]*Sin[d*x])/(6*d) + (B*Sec[c]*Sec[c + d*x]*Sin[d*x])/(2*d)))/(B + A*Cos[c + d*x]) - (2*A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (A*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(B + A*Cos[c + d*x]))","C",0
493,1,736,120,6.4800102,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{B \csc (c) \cos ^3(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \left(\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right) \, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right)}{\sqrt{\tan ^2(c)+1} \sqrt{1-\cos \left(\tan ^{-1}(\tan (c))+d x\right)} \sqrt{\cos \left(\tan ^{-1}(\tan (c))+d x\right)+1} \sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}-\frac{\frac{\tan (c) \sin \left(\tan ^{-1}(\tan (c))+d x\right)}{\sqrt{\tan ^2(c)+1}}+\frac{2 \cos ^2(c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}{\sin ^2(c)+\cos ^2(c)}}{\sqrt{\cos (c) \sqrt{\tan ^2(c)+1} \cos \left(\tan ^{-1}(\tan (c))+d x\right)}}\right)}{2 d (A \cos (c+d x)+B)}-\frac{A \csc (c) \cos ^3(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{d \sqrt{\cot ^2(c)+1} (A \cos (c+d x)+B)}-\frac{2 B \csc (c) \cos ^3(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin (c) \left(-\sqrt{\cot ^2(c)+1}\right) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec \left(d x-\tan ^{-1}(\cot (c))\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right)}{3 d \sqrt{\cot ^2(c)+1} (A \cos (c+d x)+B)}+\frac{\cos ^{\frac{7}{2}}(c+d x) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right) (a \sec (c+d x)+a)^2 (A+B \sec (c+d x)) \left(\frac{\sec (c) \sec (c+d x) (3 A \sin (d x)+B \sin (c)+6 B \sin (d x))}{6 d}-\frac{\csc (c) \sec (c) (A \cos (2 c)-A-4 B)}{4 d}+\frac{B \sec (c) \sin (d x) \sec ^2(c+d x)}{6 d}\right)}{A \cos (c+d x)+B}","\frac{4 a^2 (3 A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 a^2 (3 A+5 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)}}-\frac{4 a^2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 B \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(-1/4*((-A - 4*B + A*Cos[2*c])*Csc[c]*Sec[c])/d + (B*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(6*d) + (Sec[c]*Sec[c + d*x]*(B*Sin[c] + 3*A*Sin[d*x] + 6*B*Sin[d*x]))/(6*d)))/(B + A*Cos[c + d*x]) - (A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (B*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(B + A*Cos[c + d*x]))","C",0
494,1,1025,159,6.5847841,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{\cos ^{\frac{7}{2}}(c+d x) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{B \sec (c) \sin (d x) \sec ^3(c+d x)}{10 d}+\frac{\sec (c) (3 B \sin (c)+5 A \sin (d x)+10 B \sin (d x)) \sec ^2(c+d x)}{30 d}+\frac{\sec (c) (5 A \sin (c)+10 B \sin (c)+30 A \sin (d x)+24 B \sin (d x)) \sec (c+d x)}{30 d}+\frac{(5 A+4 B) \csc (c) \sec (c)}{5 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{B+A \cos (c+d x)}+\frac{A \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 d (B+A \cos (c+d x))}+\frac{2 B \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x))}-\frac{2 A \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{B \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{4 a^2 (5 A+4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a^2 (5 A+7 B) \sin (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{4 a^2 (5 A+4 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(((5*A + 4*B)*Csc[c]*Sec[c])/(5*d) + (B*Sec[c]*Sec[c + d*x]^3*Sin[d*x])/(10*d) + (Sec[c]*Sec[c + d*x]^2*(3*B*Sin[c] + 5*A*Sin[d*x] + 10*B*Sin[d*x]))/(30*d) + (Sec[c]*Sec[c + d*x]*(5*A*Sin[c] + 10*B*Sin[c] + 30*A*Sin[d*x] + 24*B*Sin[d*x]))/(30*d)))/(B + A*Cos[c + d*x]) - (2*A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (A*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(2*d*(B + A*Cos[c + d*x])) + (2*B*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(B + A*Cos[c + d*x]))","C",0
495,1,1067,194,6.6407517,"\int \frac{(a+a \sec (c+d x))^2 (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{\cos ^{\frac{7}{2}}(c+d x) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{B \sec (c) \sin (d x) \sec ^4(c+d x)}{14 d}+\frac{\sec (c) (5 B \sin (c)+7 A \sin (d x)+14 B \sin (d x)) \sec ^3(c+d x)}{70 d}+\frac{\sec (c) (21 A \sin (c)+42 B \sin (c)+70 A \sin (d x)+60 B \sin (d x)) \sec ^2(c+d x)}{210 d}+\frac{\sec (c) (35 A \sin (c)+30 B \sin (c)+84 A \sin (d x)+63 B \sin (d x)) \sec (c+d x)}{105 d}+\frac{(4 A+3 B) \csc (c) \sec (c)}{5 d}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{B+A \cos (c+d x)}+\frac{2 A \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d (B+A \cos (c+d x))}+\frac{3 B \cos ^3(c+d x) \csc (c) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \left(\frac{\, _2F_1\left(-\frac{1}{2},-\frac{1}{4};\frac{3}{4};\cos ^2\left(d x+\tan ^{-1}(\tan (c))\right)\right) \sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{1-\cos \left(d x+\tan ^{-1}(\tan (c))\right)} \sqrt{\cos \left(d x+\tan ^{-1}(\tan (c))\right)+1} \sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}} \sqrt{\tan ^2(c)+1}}-\frac{\frac{2 \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1} \cos ^2(c)}{\cos ^2(c)+\sin ^2(c)}+\frac{\sin \left(d x+\tan ^{-1}(\tan (c))\right) \tan (c)}{\sqrt{\tan ^2(c)+1}}}{\sqrt{\cos (c) \cos \left(d x+\tan ^{-1}(\tan (c))\right) \sqrt{\tan ^2(c)+1}}}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 d (B+A \cos (c+d x))}-\frac{A \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}-\frac{2 B \cos ^3(c+d x) \csc (c) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) (\sec (c+d x) a+a)^2 (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{7 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1}}","\frac{4 a^2 (7 A+6 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{4 a^2 (4 A+3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{4 a^2 (7 A+6 B) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a^2 (7 A+9 B) \sin (c+d x)}{35 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{4 a^2 (4 A+3 B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 B \sin (c+d x) \left(a^2 \cos (c+d x)+a^2\right)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(Cos[c + d*x]^(7/2)*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*(((4*A + 3*B)*Csc[c]*Sec[c])/(5*d) + (B*Sec[c]*Sec[c + d*x]^4*Sin[d*x])/(14*d) + (Sec[c]*Sec[c + d*x]^3*(5*B*Sin[c] + 7*A*Sin[d*x] + 14*B*Sin[d*x]))/(70*d) + (Sec[c]*Sec[c + d*x]^2*(21*A*Sin[c] + 42*B*Sin[c] + 70*A*Sin[d*x] + 60*B*Sin[d*x]))/(210*d) + (Sec[c]*Sec[c + d*x]*(35*A*Sin[c] + 30*B*Sin[c] + 84*A*Sin[d*x] + 63*B*Sin[d*x]))/(105*d)))/(B + A*Cos[c + d*x]) - (A*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) - (2*B*Cos[c + d*x]^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(7*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]) + (2*A*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(5*d*(B + A*Cos[c + d*x])) + (3*B*Cos[c + d*x]^3*Csc[c]*Sec[c/2 + (d*x)/2]^4*(a + a*Sec[c + d*x])^2*(A + B*Sec[c + d*x])*((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 + Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/Sqrt[1 + Tan[c]^2] + (2*Cos[c]^2*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2])/(Cos[c]^2 + Sin[c]^2))/Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 + Tan[c]^2]]))/(10*d*(B + A*Cos[c + d*x]))","C",0
496,1,1292,157,6.6928069,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{21 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{20 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(\frac{2 (-16 \cos (c) A-5 A+5 B+10 B \cos (c)) \csc (c)}{5 d}+\frac{4 (B-A) \cos (d x) \sin (c)}{3 d}+\frac{2 A \cos (2 d x) \sin (2 c)}{5 d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{4 (B-A) \cos (c) \sin (d x)}{3 d}+\frac{2 A \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{5 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","-\frac{5 (A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}+\frac{3 (7 A-5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a d}-\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(7 A-5 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}-\frac{5 (A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((21*I)/20)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x])*((2*(-5*A + 5*B - 16*A*Cos[c] + 10*B*Cos[c])*Csc[c])/(5*d) + (4*(-A + B)*Cos[d*x]*Sin[c])/(3*d) + (2*A*Cos[2*d*x]*Sin[2*c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/d + (4*(-A + B)*Cos[c]*Sin[d*x])/(3*d) + (2*A*Cos[2*c]*Sin[2*d*x])/(5*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (5*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
497,1,1239,124,6.5965164,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","-\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(-\frac{2 (B-A) (2 \cos (c)+1) \csc (c)}{d}+\frac{4 A \cos (d x) \sin (c)}{3 d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right)}{d}+\frac{4 A \cos (c) \sin (d x)}{3 d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{5 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}+\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","\frac{(5 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac{(5 A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((-3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x])*((-2*(-A + B)*(1 + 2*Cos[c])*Csc[c])/d + (4*A*Cos[d*x]*Sin[c])/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/d + (4*A*Cos[c]*Sin[d*x])/(3*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - (5*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",0
498,1,1208,88,6.472781,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{a+a \sec (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x]),x]","\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right)}{d}-\frac{2 (2 \cos (c) A+A-B) \csc (c)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","-\frac{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(3 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"(((3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - ((I/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x])*((-2*(A - B + 2*A*Cos[c])*Csc[c])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/d))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",1
499,1,1204,83,6.5094214,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(-\frac{2 (B-A) \csc (c)}{d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","\frac{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d (a \cos (c+d x)+a)}",1,"((-1/4*I)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + ((I/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x])*((-2*(-A + B)*Csc[c])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/d))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",1
500,1,1240,113,6.7050454,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(\frac{(\cos (c) B+2 B-A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{4 B \sec (c+d x) \sin (d x) \sec (c)}{d}+\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)}-\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}+\frac{B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","\frac{(A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-3 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{(A-3 B) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}+\frac{(A-B) \sin (c+d x)}{d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)}",1,"((I/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x])*(((2*B - A*Cos[c] + B*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/d + (4*B*Sec[c]*Sec[c + d*x]*Sin[d*x])/d))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) - (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) + (B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",1
501,1,1277,152,7.1731375,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])),x]","-\frac{3 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{3 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{4 (B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(\frac{4 B \sec (c) \sin (d x) \sec ^2(c+d x)}{3 d}+\frac{4 \sec (c) (B \sin (c)+3 A \sin (d x)-3 B \sin (d x)) \sec (c+d x)}{3 d}-\frac{(B-A) (\cos (c)+2) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}-\frac{2 \sec \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right)}{d}\right) \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)}+\frac{A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}-\frac{5 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)}","-\frac{(3 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a d}-\frac{3 (A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}+\frac{(A-B) \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)}-\frac{(3 A-5 B) \sin (c+d x)}{3 a d \cos ^{\frac{3}{2}}(c+d x)}+\frac{3 (A-B) \sin (c+d x)}{a d \sqrt{\cos (c+d x)}}",1,"(((-3*I)/4)*A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (((3*I)/4)*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*Sec[c/2]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (Cos[c/2 + (d*x)/2]^2*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x])*(-(((-A + B)*(2 + Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/d + (4*B*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (4*Sec[c]*Sec[c + d*x]*(B*Sin[c] + 3*A*Sin[d*x] - 3*B*Sin[d*x]))/(3*d)))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])) + (A*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])) - (5*B*Cos[c/2 + (d*x)/2]^2*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x]))","C",1
502,1,1396,204,7.0778152,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{28 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{7 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{(A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(3 B \sin \left(\frac{d x}{2}\right)-4 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{4 (-36 \cos (c) A-20 A+15 B+20 B \cos (c)) \csc (c)}{5 d}+\frac{8 (B-2 A) \cos (d x) \sin (c)}{3 d}+\frac{4 A \cos (2 d x) \sin (2 c)}{5 d}+\frac{8 (B-2 A) \cos (c) \sin (d x)}{3 d}+\frac{4 A \cos (2 c) \sin (2 d x)}{5 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{10 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{20 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","-\frac{5 (3 A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{7 (8 A-5 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^2 d}-\frac{(3 A-2 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{a^2 d (\cos (c+d x)+1)}+\frac{7 (8 A-5 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a^2 d}-\frac{5 (3 A-2 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(((28*I)/5)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (((7*I)/2)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (10*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (20*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*(A + B*Sec[c + d*x])*((4*(-20*A + 15*B - 36*A*Cos[c] + 20*B*Cos[c])*Csc[c])/(5*d) + (8*(-2*A + B)*Cos[d*x]*Sin[c])/(3*d) + (4*A*Cos[2*d*x]*Sin[2*c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-4*A*Sin[(d*x)/2] + 3*B*Sin[(d*x)/2]))/d + (8*(-2*A + B)*Cos[c]*Sin[d*x])/(3*d) + (4*A*Cos[2*c]*Sin[2*d*x])/(5*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",0
503,1,1352,171,6.8062261,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","-\frac{7 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{2 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{(A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(2 B \sin \left(\frac{d x}{2}\right)-3 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (-4 \cos (c) A-3 A+2 B+2 B \cos (c)) \csc (c)}{d}+\frac{8 A \cos (d x) \sin (c)}{3 d}+\frac{8 A \cos (c) \sin (d x)}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{20 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}+\frac{10 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{5 (2 A-B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(7 A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(7 A-4 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac{5 (2 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d}-\frac{(A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"(((-7*I)/2)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + ((2*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (20*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (10*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*(A + B*Sec[c + d*x])*((-4*(-3*A + 2*B - 4*A*Cos[c] + 2*B*Cos[c])*Csc[c])/d + (8*A*Cos[d*x]*Sin[c])/(3*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-3*A*Sin[(d*x)/2] + 2*B*Sin[(d*x)/2]))/d + (8*A*Cos[c]*Sin[d*x])/(3*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",0
504,1,1318,137,6.6996019,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^2,x]","\frac{2 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{(A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-2 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 (2 \cos (c) A+2 A-B) \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{10 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{4 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","-\frac{(5 A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(4 A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(5 A-2 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}",1,"((2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - ((I/2)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (10*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (4*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*(A + B*Sec[c + d*x])*((-4*(2*A - B + 2*A*Cos[c])*Csc[c])/d + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-2*A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",0
505,1,921,121,6.5703496,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^2),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{(A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 A \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{4 A \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{(2 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{A E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{A \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((-1/2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (2*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*(A + B*Sec[c + d*x])*((4*A*Csc[c])/d + (4*A*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",0
506,1,921,121,6.586259,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^2),x]","\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{(A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 B \sec \left(\frac{c}{2}\right) \sin \left(\frac{d x}{2}\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{4 B \csc (c)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{4 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{(A+2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{B \sin (c+d x) \sqrt{\cos (c+d x)}}{a^2 d (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d (a \cos (c+d x)+a)^2}",1,"((I/2)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (2*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (4*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*(A + B*Sec[c + d*x])*((-4*B*Csc[c])/d - (4*B*Sec[c/2]*Sec[c/2 + (d*x)/2]*Sin[(d*x)/2])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",0
507,1,1351,164,6.8997116,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^2),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{2 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{(A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(2 B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}+\frac{2 (2 \cos (c) B+2 B-A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{8 B \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}-\frac{4 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}+\frac{10 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","\frac{(2 A-5 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}+\frac{(A-4 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}-\frac{(A-4 B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{(2 A-5 B) \sin (c+d x)}{3 a^2 d \sqrt{\cos (c+d x)} (\cos (c+d x)+1)}+\frac{(A-B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}",1,"((I/2)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - ((2*I)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) - (4*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (10*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*(A + B*Sec[c + d*x])*((2*(2*B - A*Cos[c] + 2*B*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + 2*B*Sin[(d*x)/2]))/d + (8*B*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",0
508,1,1392,197,7.4999824,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^2),x]","-\frac{2 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{(B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{7 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{2 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{(A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(3 B \sin \left(\frac{d x}{2}\right)-2 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{d}-\frac{2 (-2 \cos (c) A-2 A+4 B+3 B \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{d}+\frac{8 B \sec (c) \sec ^2(c+d x) \sin (d x)}{3 d}+\frac{8 \sec (c) \sec (c+d x) (B \sin (c)+3 A \sin (d x)-6 B \sin (d x))}{3 d}\right) \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{\sqrt{\cos (c+d x)} (B+A \cos (c+d x)) (\sec (c+d x) a+a)^2}+\frac{10 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}-\frac{20 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec (c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^2}","-\frac{5 (A-2 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^2 d}-\frac{(4 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{(4 A-7 B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x) (\cos (c+d x)+1)}-\frac{5 (A-2 B) \sin (c+d x)}{3 a^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{(4 A-7 B) \sin (c+d x)}{a^2 d \sqrt{\cos (c+d x)}}+\frac{(A-B) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+a)^2}",1,"((-2*I)*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (((7*I)/2)*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2) + (10*A*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) - (20*B*Cos[c/2 + (d*x)/2]^4*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^2) + (Cos[c/2 + (d*x)/2]^4*(A + B*Sec[c + d*x])*((-2*(-2*A + 4*B - 2*A*Cos[c] + 3*B*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/d - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(3*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-2*A*Sin[(d*x)/2] + 3*B*Sin[(d*x)/2]))/d + (8*B*Sec[c]*Sec[c + d*x]^2*Sin[d*x])/(3*d) + (8*Sec[c]*Sec[c + d*x]*(B*Sin[c] + 3*A*Sin[d*x] - 6*B*Sin[d*x]))/(3*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(3*d)))/(Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^2)","C",0
509,1,1448,221,7.0788327,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","-\frac{119 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{49 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{(A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(14 B \sin \left(\frac{d x}{2}\right)-19 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (14 B-19 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(29 B \sin \left(\frac{d x}{2}\right)-59 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (-60 \cos (c) A-59 A+29 B+20 B \cos (c)) \csc (c)}{5 d}+\frac{16 A \cos (d x) \sin (c)}{3 d}+\frac{16 A \cos (c) \sin (d x)}{3 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{22 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}+\frac{26 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(33 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{7 (17 A-7 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{7 (17 A-7 B) \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) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a^3 d}-\frac{(A-B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(2 A-B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{3 a d (a \cos (c+d x)+a)^2}",1,"(((-119*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (((49*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (22*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (26*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x])*((-4*(-59*A + 29*B - 60*A*Cos[c] + 20*B*Cos[c])*Csc[c])/(5*d) + (16*A*Cos[d*x]*Sin[c])/(3*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-19*A*Sin[(d*x)/2] + 14*B*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-59*A*Sin[(d*x)/2] + 29*B*Sin[(d*x)/2]))/(5*d) + (16*A*Cos[c]*Sin[d*x])/(3*d) + (4*(-19*A + 14*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(3/2)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
510,1,1415,188,6.9317351,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^3,x]","\frac{49 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{9 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{(A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(9 B \sin \left(\frac{d x}{2}\right)-14 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (9 B-14 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(9 B \sin \left(\frac{d x}{2}\right)-29 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (20 \cos (c) A+29 A-9 B) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{26 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","-\frac{(13 A-3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(49 A-9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(13 A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B) \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) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{15 a d (a \cos (c+d x)+a)^2}",1,"(((49*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (((9*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (26*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x])*((-4*(29*A - 9*B + 20*A*Cos[c])*Csc[c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-29*A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-14*A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2]))/(15*d) - (4*(-14*A + 9*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(3/2)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
511,1,1407,182,6.9573059,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^3),x]","-\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{(A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(4 B \sin \left(\frac{d x}{2}\right)-9 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (4 B-9 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(9 A \sin \left(\frac{d x}{2}\right)+B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 (9 A+B) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(3 A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(9 A+B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(9 A+B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}-\frac{(A-B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}-\frac{(6 A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}",1,"(((-9*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - ((I/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x])*((4*(9*A + B)*Csc[c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(9*A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-9*A*Sin[(d*x)/2] + 4*B*Sin[(d*x)/2]))/(15*d) + (4*(-9*A + 4*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(3/2)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
512,1,1406,178,6.8622189,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^3),x]","-\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{(A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(4 A \sin \left(\frac{d x}{2}\right)+B \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (4 A+B) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (B-A) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(A+B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}-\frac{(A-B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}+\frac{(A-B) \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) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((-1/10*I)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + ((I/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x])*((-4*(-A + B)*Csc[c])/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(4*A*Sin[(d*x)/2] + B*Sin[(d*x)/2]))/(15*d) + (4*(4*A + B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(3/2)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
513,1,1407,180,6.8389109,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^3),x]","\frac{i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{9 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{(A+B \sec (c+d x)) \left(-\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(6 B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 (6 B-A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}-\frac{4 \sec \left(\frac{c}{2}\right) \left(A \sin \left(\frac{d x}{2}\right)+9 B \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}-\frac{4 (A+9 B) \csc (c)}{5 d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}-\frac{2 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(A+3 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(A+9 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(A+9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{10 d \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(A-6 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 a d (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{5 d (a \cos (c+d x)+a)^3}",1,"((I/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) + (((9*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) - (2*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x])*((-4*(A + 9*B)*Csc[c])/(5*d) - (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-(A*Sin[(d*x)/2]) + 6*B*Sin[(d*x)/2]))/(15*d) - (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(A*Sin[(d*x)/2] + 9*B*Sin[(d*x)/2]))/(5*d) - (4*(-A + 6*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) - (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(3/2)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
514,1,1447,221,7.1942661,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^3),x]","\frac{9 i A \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{49 i B \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \left(\frac{2 e^{2 i d x} \, _2F_1\left(\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{3 i d \left(1+e^{2 i d x}\right) \cos (c)-3 d \left(-1+e^{2 i d x}\right) \sin (c)}-\frac{2 \, _2F_1\left(-\frac{1}{4},\frac{1}{2};\frac{3}{4};-e^{2 i d x} (\cos (c)+i \sin (c))^2\right) \sqrt{e^{-i d x} \left(2 \left(1+e^{2 i d x}\right) \cos (c)+2 i \left(-1+e^{2 i d x}\right) \sin (c)\right)} \sqrt{e^{2 i d x} \cos (2 c)+i e^{2 i d x} \sin (2 c)+1}}{d \left(-1+e^{2 i d x}\right) \sin (c)-i d \left(1+e^{2 i d x}\right) \cos (c)}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{10 (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}+\frac{(A+B \sec (c+d x)) \left(\frac{2 \sec \left(\frac{c}{2}\right) \left(B \sin \left(\frac{d x}{2}\right)-A \sin \left(\frac{d x}{2}\right)\right) \sec ^5\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (B-A) \tan \left(\frac{c}{2}\right) \sec ^4\left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(11 B \sin \left(\frac{d x}{2}\right)-6 A \sin \left(\frac{d x}{2}\right)\right) \sec ^3\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 (11 B-6 A) \tan \left(\frac{c}{2}\right) \sec ^2\left(\frac{c}{2}+\frac{d x}{2}\right)}{15 d}+\frac{4 \sec \left(\frac{c}{2}\right) \left(29 B \sin \left(\frac{d x}{2}\right)-9 A \sin \left(\frac{d x}{2}\right)\right) \sec \left(\frac{c}{2}+\frac{d x}{2}\right)}{5 d}+\frac{2 (29 \cos (c) B+20 B-9 A \cos (c)) \csc \left(\frac{c}{2}\right) \sec \left(\frac{c}{2}\right) \sec (c)}{5 d}+\frac{16 B \sec (c) \sec (c+d x) \sin (d x)}{d}\right) \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{\cos ^{\frac{3}{2}}(c+d x) (B+A \cos (c+d x)) (\sec (c+d x) a+a)^3}-\frac{2 A \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}+\frac{26 B \csc \left(\frac{c}{2}\right) \, _2F_1\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};\sin ^2\left(d x-\tan ^{-1}(\cot (c))\right)\right) \sec \left(\frac{c}{2}\right) \sec ^2(c+d x) (A+B \sec (c+d x)) \sec \left(d x-\tan ^{-1}(\cot (c))\right) \sqrt{1-\sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{-\sqrt{\cot ^2(c)+1} \sin (c) \sin \left(d x-\tan ^{-1}(\cot (c))\right)} \sqrt{\sin \left(d x-\tan ^{-1}(\cot (c))\right)+1} \cos ^6\left(\frac{c}{2}+\frac{d x}{2}\right)}{3 d (B+A \cos (c+d x)) \sqrt{\cot ^2(c)+1} (\sec (c+d x) a+a)^3}","\frac{(3 A-13 B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{6 a^3 d}+\frac{(9 A-49 B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{10 a^3 d}-\frac{(9 A-49 B) \sin (c+d x)}{10 a^3 d \sqrt{\cos (c+d x)}}+\frac{(3 A-13 B) \sin (c+d x)}{6 d \sqrt{\cos (c+d x)} \left(a^3 \cos (c+d x)+a^3\right)}+\frac{(3 A-8 B) \sin (c+d x)}{15 a d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^2}+\frac{(A-B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)} (a \cos (c+d x)+a)^3}",1,"(((9*I)/10)*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (((49*I)/10)*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*((2*E^((2*I)*d*x)*Hypergeometric2F1[1/2, 3/4, 7/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((3*I)*d*(1 + E^((2*I)*d*x))*Cos[c] - 3*d*(-1 + E^((2*I)*d*x))*Sin[c]) - (2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(E^((2*I)*d*x)*(Cos[c] + I*Sin[c])^2)]*Sqrt[(2*(1 + E^((2*I)*d*x))*Cos[c] + (2*I)*(-1 + E^((2*I)*d*x))*Sin[c])/E^(I*d*x)]*Sqrt[1 + E^((2*I)*d*x)*Cos[2*c] + I*E^((2*I)*d*x)*Sin[2*c]])/((-I)*d*(1 + E^((2*I)*d*x))*Cos[c] + d*(-1 + E^((2*I)*d*x))*Sin[c])))/((B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3) - (2*A*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (26*B*Cos[c/2 + (d*x)/2]^6*Csc[c/2]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2]*Sec[c + d*x]^2*(A + B*Sec[c + d*x])*Sec[d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(3*d*(B + A*Cos[c + d*x])*Sqrt[1 + Cot[c]^2]*(a + a*Sec[c + d*x])^3) + (Cos[c/2 + (d*x)/2]^6*(A + B*Sec[c + d*x])*((2*(20*B - 9*A*Cos[c] + 29*B*Cos[c])*Csc[c/2]*Sec[c/2]*Sec[c])/(5*d) + (2*Sec[c/2]*Sec[c/2 + (d*x)/2]^5*(-(A*Sin[(d*x)/2]) + B*Sin[(d*x)/2]))/(5*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]^3*(-6*A*Sin[(d*x)/2] + 11*B*Sin[(d*x)/2]))/(15*d) + (4*Sec[c/2]*Sec[c/2 + (d*x)/2]*(-9*A*Sin[(d*x)/2] + 29*B*Sin[(d*x)/2]))/(5*d) + (16*B*Sec[c]*Sec[c + d*x]*Sin[d*x])/d + (4*(-6*A + 11*B)*Sec[c/2 + (d*x)/2]^2*Tan[c/2])/(15*d) + (2*(-A + B)*Sec[c/2 + (d*x)/2]^4*Tan[c/2])/(5*d)))/(Cos[c + d*x]^(3/2)*(B + A*Cos[c + d*x])*(a + a*Sec[c + d*x])^3)","C",0
515,1,119,220,0.5454712,"\int \cos ^{\frac{9}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(9/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} (94 (8 A+9 B) \cos (c+d x)+4 (83 A+54 B) \cos (2 (c+d x))+80 A \cos (3 (c+d x))+35 A \cos (4 (c+d x))+1321 A+90 B \cos (3 (c+d x))+1368 B)}{1260 d (\cos (c+d x)+1)}","\frac{2 a (8 A+9 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{63 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (8 A+9 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a (8 A+9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{32 a (8 A+9 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{9 d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*(1321*A + 1368*B + 94*(8*A + 9*B)*Cos[c + d*x] + 4*(83*A + 54*B)*Cos[2*(c + d*x)] + 80*A*Cos[3*(c + d*x)] + 90*B*Cos[3*(c + d*x)] + 35*A*Cos[4*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(1260*d*(1 + Cos[c + d*x]))","A",1
516,1,96,175,0.366811,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} ((141 A+112 B) \cos (c+d x)+6 (6 A+7 B) \cos (2 (c+d x))+15 A \cos (3 (c+d x))+228 A+266 B)}{210 d (\cos (c+d x)+1)}","\frac{2 a (6 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a (6 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a (6 A+7 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*(228*A + 266*B + (141*A + 112*B)*Cos[c + d*x] + 6*(6*A + 7*B)*Cos[2*(c + d*x)] + 15*A*Cos[3*(c + d*x)])*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(210*d*(1 + Cos[c + d*x]))","A",1
517,1,79,130,0.1603323,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left((4 A+5 B) \cos (c+d x)+3 A \cos ^2(c+d x)+8 A+10 B\right)}{15 d (\cos (c+d x)+1)}","\frac{2 a (4 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a (4 A+5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(2*Sqrt[Cos[c + d*x]]*(8*A + 10*B + (4*A + 5*B)*Cos[c + d*x] + 3*A*Cos[c + d*x]^2)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(15*d*(1 + Cos[c + d*x]))","A",1
518,1,56,82,0.1971948,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} (A \cos (c+d x)+2 A+3 B)}{3 d}","\frac{2 a (A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*Sqrt[Cos[c + d*x]]*(2*A + 3*B + A*Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(3*d)","A",1
519,1,94,96,0.3230254,"\int \sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(A \sqrt{1-\sec (c+d x)}-B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{2 a A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 \sqrt{a} 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)}{d}",1,"(2*Sqrt[Cos[c + d*x]]*(A*Sqrt[1 - Sec[c + d*x]] - B*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sqrt[a*(1 + Sec[c + d*x])]*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","A",1
520,1,89,98,0.3993651,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (2 A+B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 B \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)}{2 d}","\frac{\sqrt{a} (2 A+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)}{d}+\frac{a B \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(2*A + B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*B*Sec[c + d*x]*Sin[(c + d*x)/2]))/(2*d)","A",1
521,1,106,151,0.6554275,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{\sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (4 A+3 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 A+2 B \sec (c+d x)+3 B)\right)}{8 d}","\frac{a (4 A+3 B) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (4 A+3 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)}{4 d}+\frac{a B \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(4*A + 3*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*A + 3*B + 2*B*Sec[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
522,1,131,196,1.2241364,"\int \frac{\sqrt{a+a \sec (c+d x)} (A+B \sec (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + a*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (4 (6 A+5 B) \cos (c+d x)+3 (6 A+5 B) \cos (2 (c+d x))+18 A+31 B)+3 \sqrt{2} (6 A+5 B) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a (6 A+5 B) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a (6 A+5 B) \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{a} (6 A+5 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)}{8 d}+\frac{a B \sin (c+d x)}{3 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(6*A + 5*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (18*A + 31*B + 4*(6*A + 5*B)*Cos[c + d*x] + 3*(6*A + 5*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
523,1,131,275,0.5402531,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(35 (21 A+11 B) \cos ^4(c+d x)+(840 A+935 B) \cos ^3(c+d x)+6 (168 A+187 B) \cos ^2(c+d x)+8 (168 A+187 B) \cos (c+d x)+315 A \cos ^5(c+d x)+2688 A+2992 B\right)}{3465 d (\cos (c+d x)+1)}","\frac{2 a^2 (12 A+11 B) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x)}{99 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (168 A+187 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (168 A+187 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{32 a^2 (168 A+187 B) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{11 d}",1,"(2*a*Sqrt[Cos[c + d*x]]*(2688*A + 2992*B + 8*(168*A + 187*B)*Cos[c + d*x] + 6*(168*A + 187*B)*Cos[c + d*x]^2 + (840*A + 935*B)*Cos[c + d*x]^3 + 35*(21*A + 11*B)*Cos[c + d*x]^4 + 315*A*Cos[c + d*x]^5)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(3465*d*(1 + Cos[c + d*x]))","A",1
524,1,118,228,0.4232353,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(5 (17 A+9 B) \cos ^3(c+d x)+3 (34 A+39 B) \cos ^2(c+d x)+4 (34 A+39 B) \cos (c+d x)+8 (34 A+39 B)+35 A \cos ^4(c+d x)\right)}{315 d (\cos (c+d x)+1)}","\frac{2 a^2 (10 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^2 (34 A+39 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^2 (34 A+39 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (34 A+39 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{9 d}",1,"(2*a*Sqrt[Cos[c + d*x]]*(8*(34*A + 39*B) + 4*(34*A + 39*B)*Cos[c + d*x] + 3*(34*A + 39*B)*Cos[c + d*x]^2 + 5*(17*A + 9*B)*Cos[c + d*x]^3 + 35*A*Cos[c + d*x]^4)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(315*d*(1 + Cos[c + d*x]))","A",1
525,1,100,181,0.324164,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(3 (13 A+7 B) \cos ^2(c+d x)+(52 A+63 B) \cos (c+d x)+2 (52 A+63 B)+15 A \cos ^3(c+d x)\right)}{105 d (\cos (c+d x)+1)}","\frac{2 a^2 (8 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^2 (52 A+63 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^2 (52 A+63 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}{7 d}",1,"(2*a*Sqrt[Cos[c + d*x]]*(2*(52*A + 63*B) + (52*A + 63*B)*Cos[c + d*x] + 3*(13*A + 7*B)*Cos[c + d*x]^2 + 15*A*Cos[c + d*x]^3)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(105*d*(1 + Cos[c + d*x]))","A",1
526,1,80,131,0.2877599,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left((9 A+5 B) \cos (c+d x)+3 A \cos ^2(c+d x)+18 A+25 B\right)}{15 d (\cos (c+d x)+1)}","\frac{8 a^2 (3 A+5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a (3 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d}",1,"(2*a*Sqrt[Cos[c + d*x]]*(18*A + 25*B + (9*A + 5*B)*Cos[c + d*x] + 3*A*Cos[c + d*x]^2)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(15*d*(1 + Cos[c + d*x]))","A",1
527,1,101,145,0.4403777,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 \sin (c+d x) \left(\sqrt{1-\sec (c+d x)} (A \cos (c+d x)+5 A+3 B)+3 B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{3 d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{3/2} 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)}{d}+\frac{2 a^2 (4 A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{3 d}",1,"(2*a^2*((5*A + 3*B + A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] + 3*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(3*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
528,1,133,144,0.8551769,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{a \sqrt{\cos (c+d x)} \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{1-\sec (c+d x)} (2 A+B \sec (c+d x))+2 A \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-3 B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)}}","\frac{a^{3/2} (2 A+3 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)}{d}+\frac{a^2 (2 A-B) \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{d \sqrt{\cos (c+d x)}}",1,"(a*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])]*(2*A*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]] - 3*B*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Sec[c + d*x]] + Sqrt[1 - Sec[c + d*x]]*(2*A + B*Sec[c + d*x]))*Tan[(c + d*x)/2])/(d*Sqrt[1 - Sec[c + d*x]])","A",1
529,1,107,153,0.8448355,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{a \sqrt{\cos (c+d x)} \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sqrt{2} (12 A+7 B) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)+2 \sin \left(\frac{1}{2} (c+d x)\right) \sec (c+d x) (4 A+2 B \sec (c+d x)+7 B)\right)}{8 d}","\frac{a^{3/2} (12 A+7 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)}{4 d}+\frac{a^2 (4 A+5 B) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{2 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a*Sqrt[Cos[c + d*x]]*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(Sqrt[2]*(12*A + 7*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]] + 2*Sec[c + d*x]*(4*A + 7*B + 2*B*Sec[c + d*x])*Sin[(c + d*x)/2]))/(8*d)","A",1
530,1,134,200,1.2780194,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (4 (6 A+11 B) \cos (c+d x)+(42 A+33 B) \cos (2 (c+d x))+7 (6 A+7 B))+3 \sqrt{2} (14 A+11 B) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a^{3/2} (14 A+11 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)}{8 d}+\frac{a^2 (14 A+11 B) \sin (c+d x)}{8 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (6 A+7 B) \sin (c+d x)}{12 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(14*A + 11*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (7*(6*A + 7*B) + 4*(6*A + 11*B)*Cos[c + d*x] + (42*A + 33*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
531,1,153,247,1.963598,"\int \frac{(a+a \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{a \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((1048 A+1155 B) \cos (c+d x)+4 (88 A+75 B) \cos (2 (c+d x))+264 A \cos (3 (c+d x))+352 A+225 B \cos (3 (c+d x))+492 B)+6 \sqrt{2} (88 A+75 B) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a^{3/2} (88 A+75 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)}{64 d}+\frac{a^2 (88 A+75 B) \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (88 A+75 B) \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (8 A+9 B) \sin (c+d x)}{24 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a B \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(88*A + 75*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (352*A + 492*B + (1048*A + 1155*B)*Cos[c + d*x] + 4*(88*A + 75*B)*Cos[2*(c + d*x)] + 264*A*Cos[3*(c + d*x)] + 225*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","A",1
532,1,137,275,0.5976715,"\int \cos ^{\frac{11}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(35 (32 A+11 B) \cos ^4(c+d x)+5 (355 A+286 B) \cos ^3(c+d x)+3 (710 A+803 B) \cos ^2(c+d x)+4 (710 A+803 B) \cos (c+d x)+8 (710 A+803 B)+315 A \cos ^5(c+d x)\right)}{3465 d (\cos (c+d x)+1)}","\frac{2 a^3 (194 A+209 B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{693 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (710 A+803 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{1155 d \sqrt{a \sec (c+d x)+a}}+\frac{8 a^3 (710 A+803 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3465 d \sqrt{a \sec (c+d x)+a}}+\frac{16 a^3 (710 A+803 B) \sin (c+d x)}{3465 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (14 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 A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{11 d}",1,"(2*a^2*Sqrt[Cos[c + d*x]]*(8*(710*A + 803*B) + 4*(710*A + 803*B)*Cos[c + d*x] + 3*(710*A + 803*B)*Cos[c + d*x]^2 + 5*(355*A + 286*B)*Cos[c + d*x]^3 + 35*(32*A + 11*B)*Cos[c + d*x]^4 + 315*A*Cos[c + d*x]^5)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(3465*d*(1 + Cos[c + d*x]))","A",1
533,1,116,228,0.5496455,"\int \cos ^{\frac{9}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(5 (26 A+9 B) \cos ^3(c+d x)+3 (73 A+60 B) \cos ^2(c+d x)+(292 A+345 B) \cos (c+d x)+35 A \cos ^4(c+d x)+584 A+690 B\right)}{315 d (\cos (c+d x)+1)}","\frac{2 a^3 (124 A+135 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{2 a^3 (292 A+345 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{315 d \sqrt{a \sec (c+d x)+a}}+\frac{4 a^3 (292 A+345 B) \sin (c+d x)}{315 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (4 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 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{9 d}",1,"(2*a^2*Sqrt[Cos[c + d*x]]*(584*A + 690*B + (292*A + 345*B)*Cos[c + d*x] + 3*(73*A + 60*B)*Cos[c + d*x]^2 + 5*(26*A + 9*B)*Cos[c + d*x]^3 + 35*A*Cos[c + d*x]^4)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(315*d*(1 + Cos[c + d*x]))","A",1
534,1,99,178,0.4140327,"\int \cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)} \left(3 (20 A+7 B) \cos ^2(c+d x)+(115 A+98 B) \cos (c+d x)+15 A \cos ^3(c+d x)+230 A+301 B\right)}{105 d (\cos (c+d x)+1)}","\frac{64 a^3 (5 A+7 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{16 a^2 (5 A+7 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{105 d}+\frac{2 a (5 A+7 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{35 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{7 d}",1,"(2*a^2*Sqrt[Cos[c + d*x]]*(230*A + 301*B + (115*A + 98*B)*Cos[c + d*x] + 3*(20*A + 7*B)*Cos[c + d*x]^2 + 15*A*Cos[c + d*x]^3)*Sqrt[a*(1 + Sec[c + d*x])]*Sin[c + d*x])/(105*d*(1 + Cos[c + d*x]))","A",1
535,1,118,192,0.6782737,"\int \cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{2 a^3 \sin (c+d x) \left(\sqrt{1-\sec (c+d x)} \left((14 A+5 B) \cos (c+d x)+3 A \cos ^2(c+d x)+43 A+40 B\right)+15 B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{15 d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","\frac{2 a^{5/2} 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)}{d}+\frac{2 a^3 (32 A+35 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{2 a^2 (8 A+5 B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}{15 d}+\frac{2 a 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*((43*A + 40*B + (14*A + 5*B)*Cos[c + d*x] + 3*A*Cos[c + d*x]^2)*Sqrt[1 - Sec[c + d*x]] + 15*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(15*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
536,1,117,197,0.681599,"\int \cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^3 \sin (c+d x) \left(\sqrt{1-\sec (c+d x)} (2 A \cos (c+d x)+16 A+3 B \sec (c+d x)+6 B)+3 (2 A+5 B) \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)\right)}{3 d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","\frac{a^{5/2} (2 A+5 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)}{d}+\frac{a^3 (14 A+3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{a^2 (2 A-3 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}{3 d}",1,"(a^3*(3*(2*A + 5*B)*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]] + Sqrt[1 - Sec[c + d*x]]*(16*A + 6*B + 2*A*Cos[c + d*x] + 3*B*Sec[c + d*x]))*Sin[c + d*x])/(3*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",0
537,1,173,200,0.9510904,"\int \sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{a^3 \sin (c+d x) \sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(\sqrt{1-\sec (c+d x)} \left((4 A+11 B) \sec (c+d x)+8 A+2 B \sec ^2(c+d x)\right)+20 A \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-19 B \sqrt{\sec (c+d x)} \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)} (A \cos (c+d x)+B)}","\frac{a^{5/2} (20 A+19 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)}{4 d}+\frac{a^3 (4 A-9 B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (4 A+7 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \sqrt{\cos (c+d x)}}+\frac{a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{2 d \sqrt{\cos (c+d x)}}",1,"(a^3*Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x])*(20*A*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]] - 19*B*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Sec[c + d*x]] + Sqrt[1 - Sec[c + d*x]]*(8*A + (4*A + 11*B)*Sec[c + d*x] + 2*B*Sec[c + d*x]^2))*Sin[c + d*x])/(4*d*(B + A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
538,1,133,200,1.3439754,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (4 (6 A+17 B) \cos (c+d x)+(66 A+75 B) \cos (2 (c+d x))+66 A+91 B)+3 \sqrt{2} (38 A+25 B) \cos ^3(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{48 d \cos ^{\frac{5}{2}}(c+d x)}","\frac{a^{5/2} (38 A+25 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)}{8 d}+\frac{a^3 (54 A+49 B) \sin (c+d x)}{24 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (2 A+3 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(3*Sqrt[2]*(38*A + 25*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^3 + (66*A + 91*B + 4*(6*A + 17*B)*Cos[c + d*x] + (66*A + 75*B)*Cos[2*(c + d*x)])*Sin[(c + d*x)/2]))/(48*d*Cos[c + d*x]^(5/2))","A",1
539,1,154,247,1.9834183,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) ((2056 A+2203 B) \cos (c+d x)+(544 A+652 B) \cos (2 (c+d x))+600 A \cos (3 (c+d x))+544 A+489 B \cos (3 (c+d x))+844 B)+6 \sqrt{2} (200 A+163 B) \cos ^4(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{768 d \cos ^{\frac{7}{2}}(c+d x)}","\frac{a^{5/2} (200 A+163 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)}{64 d}+\frac{a^3 (200 A+163 B) \sin (c+d x)}{64 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (104 A+95 B) \sin (c+d x)}{96 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (8 A+11 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{24 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(6*Sqrt[2]*(200*A + 163*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^4 + (544*A + 844*B + (2056*A + 2203*B)*Cos[c + d*x] + (544*A + 652*B)*Cos[2*(c + d*x)] + 600*A*Cos[3*(c + d*x)] + 489*B*Cos[3*(c + d*x)])*Sin[(c + d*x)/2]))/(768*d*Cos[c + d*x]^(7/2))","A",1
540,1,178,294,2.883483,"\int \frac{(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\cos ^{\frac{5}{2}}(c+d x)} \, dx","Integrate[((a + a*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(5/2),x]","\frac{a^2 \sec \left(\frac{1}{2} (c+d x)\right) \sqrt{a (\sec (c+d x)+1)} \left(\sin \left(\frac{1}{2} (c+d x)\right) (36 (650 A+781 B) \cos (c+d x)+4 (6730 A+6509 B) \cos (2 (c+d x))+6520 A \cos (3 (c+d x))+4890 A \cos (4 (c+d x))+22030 A+5660 B \cos (3 (c+d x))+4245 B \cos (4 (c+d x))+24863 B)+60 \sqrt{2} (326 A+283 B) \cos ^5(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{15360 d \cos ^{\frac{9}{2}}(c+d x)}","\frac{a^{5/2} (326 A+283 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)}{128 d}+\frac{a^3 (326 A+283 B) \sin (c+d x)}{128 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (326 A+283 B) \sin (c+d x)}{192 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^3 (170 A+157 B) \sin (c+d x)}{240 d \cos ^{\frac{7}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{a^2 (10 A+13 B) \sin (c+d x) \sqrt{a \sec (c+d x)+a}}{40 d \cos ^{\frac{7}{2}}(c+d x)}+\frac{a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{5 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(a^2*Sec[(c + d*x)/2]*Sqrt[a*(1 + Sec[c + d*x])]*(60*Sqrt[2]*(326*A + 283*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^5 + (22030*A + 24863*B + 36*(650*A + 781*B)*Cos[c + d*x] + 4*(6730*A + 6509*B)*Cos[2*(c + d*x)] + 6520*A*Cos[3*(c + d*x)] + 5660*B*Cos[3*(c + d*x)] + 4890*A*Cos[4*(c + d*x)] + 4245*B*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]))/(15360*d*Cos[c + d*x]^(9/2))","A",1
541,1,170,250,1.3781816,"\int \frac{\cos ^{\frac{7}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(7/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left((91 B-43 A) \sec ^3(c+d x)+(31 A-7 B) \sec ^2(c+d x)-3 (A-7 B) \sec (c+d x)+15 A\right)-105 \sqrt{2} (A-B) \sec ^{\frac{7}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{105 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 (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 (31 A-7 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{105 d \sqrt{a \sec (c+d x)+a}}-\frac{2 (43 A-91 B) \sin (c+d x)}{105 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{7 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^(5/2)*(-105*Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(7/2) + 2*Sqrt[1 - Sec[c + d*x]]*(15*A - 3*(A - 7*B)*Sec[c + d*x] + (31*A - 7*B)*Sec[c + d*x]^2 + (-43*A + 91*B)*Sec[c + d*x]^3))*Sin[c + d*x])/(105*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
542,1,154,207,0.7955392,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \left(2 \sqrt{1-\sec (c+d x)} \left((13 A-5 B) \sec ^2(c+d x)-(A-5 B) \sec (c+d x)+3 A\right)+15 \sqrt{2} (A-B) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{15 d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 (A-5 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d \sqrt{a \sec (c+d x)+a}}+\frac{2 (13 A-5 B) \sin (c+d x)}{15 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (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 \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d \sqrt{a \sec (c+d x)+a}}",1,"(Cos[c + d*x]^(3/2)*(15*Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sec[c + d*x]^(5/2) + 2*Sqrt[1 - Sec[c + d*x]]*(3*A - (A - 5*B)*Sec[c + d*x] + (13*A - 5*B)*Sec[c + d*x]^2))*Sin[c + d*x])/(15*d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
543,1,124,162,0.3511759,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \left(2 \sqrt{1-\sec (c+d x)} (A \cos (c+d x)-A+3 B)-3 \sqrt{2} (A-B) \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{3 d \sqrt{\cos (c+d x)-1} \sqrt{a (\sec (c+d x)+1)}}","-\frac{2 (A-3 B) \sin (c+d x)}{3 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (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 \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d \sqrt{a \sec (c+d x)+a}}",1,"((2*(-A + 3*B + A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]] - 3*Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*Sin[c + d*x])/(3*d*Sqrt[-1 + Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
544,1,140,119,0.3084623,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + a*Sec[c + d*x]],x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} (A+B \sec (c+d x)) \left(\sqrt{2} (A-B) \sqrt{\sec (c+d x)} \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 A \sqrt{1-\sec (c+d x)}\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)} (A \cos (c+d x)+B)}","\frac{2 A \sin (c+d x)}{d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{\sqrt{2} (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}",1,"(Sqrt[Cos[c + d*x]]*(2*A*Sqrt[1 - Sec[c + d*x]] + Sqrt[2]*(A - B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Sec[c + d*x]])*(A + B*Sec[c + d*x])*Sin[c + d*x])/(d*(B + A*Cos[c + d*x])*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
545,1,115,140,0.2267452,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \left(\sqrt{2} (B-A) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)-2 B \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{d \sqrt{1-\sec (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{\sqrt{2} (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 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*ArcSin[Sqrt[Sec[c + d*x]]] + Sqrt[2]*(-A + B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]])*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
546,1,114,181,0.6681693,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[a + a*Sec[c + d*x]]),x]","-\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 (A-B) \cos (c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} (2 A-B) \cos (c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)-2 B \sin \left(\frac{1}{2} (c+d x)\right)\right)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","-\frac{\sqrt{2} (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-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{B \sin (c+d x)}{d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"-((Cos[(c + d*x)/2]*(2*(A - B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x] - Sqrt[2]*(2*A - B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x] - 2*B*Sin[(c + d*x)/2]))/(d*Cos[c + d*x]^(3/2)*Sqrt[a*(1 + Sec[c + d*x])]))","A",1
547,1,137,230,1.0461837,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+a \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[a + a*Sec[c + d*x]]),x]","\frac{\cos \left(\frac{1}{2} (c+d x)\right) \left(2 \sin \left(\frac{1}{2} (c+d x)\right) ((4 A-B) \cos (c+d x)+2 B)+8 (A-B) \cos ^2(c+d x) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)-\sqrt{2} (4 A-7 B) \cos ^2(c+d x) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a (\sec (c+d x)+1)}}","\frac{(4 A-B) \sin (c+d x)}{4 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{\sqrt{2} (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{(4 A-7 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)}{4 \sqrt{a} d}+\frac{B \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}",1,"(Cos[(c + d*x)/2]*(8*(A - B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[c + d*x]^2 - Sqrt[2]*(4*A - 7*B)*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[c + d*x]^2 + 2*(2*B + (4*A - B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(4*d*Cos[c + d*x]^(5/2)*Sqrt[a*(1 + Sec[c + d*x])])","A",1
548,1,178,270,1.3459075,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \tan (c+d x) \sqrt{1-\sec (c+d x)} (3 (39 A-20 B) \cos (c+d x)+(10 B-6 A) \cos (2 (c+d x))+3 A \cos (3 (c+d x))+141 A-85 B)+30 \sqrt{2} (15 A-11 B) \sin (c+d x) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{60 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{3/2}}","-\frac{(15 A-11 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(9 A-5 B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{10 a d \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \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) \sin (c+d x) \sqrt{\cos (c+d x)}}{30 a d \sqrt{a \sec (c+d x)+a}}+\frac{(147 A-95 B) \sin (c+d x)}{30 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}",1,"(30*Sqrt[2]*(15*A - 11*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sec[c + d*x]^(3/2)*Sin[c + d*x] + 2*(141*A - 85*B + 3*(39*A - 20*B)*Cos[c + d*x] + (-6*A + 10*B)*Cos[2*(c + d*x)] + 3*A*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Tan[c + d*x])/(60*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
549,1,155,223,1.237069,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{\sin (c+d x) \left(\sqrt{1-\sec (c+d x)} (\sec (c+d x) (2 A \cos (2 (c+d x))-17 A+15 B)+12 (B-A))-3 \sqrt{2} (11 A-7 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)\right)}{6 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{3/2}}","\frac{(11 A-7 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(7 A-3 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{6 a d \sqrt{a \sec (c+d x)+a}}-\frac{(19 A-15 B) \sin (c+d x)}{6 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d (a \sec (c+d x)+a)^{3/2}}",1,"((-3*Sqrt[2]*(11*A - 7*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2*Sec[c + d*x]^(3/2) + Sqrt[1 - Sec[c + d*x]]*(12*(-A + B) + (-17*A + 15*B + 2*A*Cos[2*(c + d*x)])*Sec[c + d*x]))*Sin[c + d*x])/(6*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
550,1,198,176,1.9428888,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(3/2),x]","\frac{2 \tan (c+d x) \sqrt{1-\sec (c+d x)} \left(2 A^2 \cos (2 (c+d x))+2 A^2+A (5 A+3 B) \cos (c+d x)+5 A B-B^2\right)+4 \sqrt{2} (7 A-3 B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^3\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) (A \cos (c+d x)+B)}{4 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{3/2} (A \cos (c+d x)+B)}","-\frac{(7 A-3 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(5 A-B) \sin (c+d x)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(A-B) \sin (c+d x)}{2 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}",1,"(4*Sqrt[2]*(7*A - 3*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^3*(B + A*Cos[c + d*x])*Sec[c + d*x]^(3/2)*Sin[(c + d*x)/2] + 2*(2*A^2 + 5*A*B - B^2 + A*(5*A + 3*B)*Cos[c + d*x] + 2*A^2*Cos[2*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Tan[c + d*x])/(4*d*Sqrt[-1 + Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
551,1,86,127,0.5234656,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\frac{1}{2} (B-A) \sin (c+d x)+(3 A+B) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)}{a d \sqrt{\cos (c+d x)} (\cos (c+d x)+1) \sqrt{a (\sec (c+d x)+1)}}","\frac{(3 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(A-B) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"((3*A + B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + ((-A + B)*Sin[c + d*x])/2)/(a*d*Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])*Sqrt[a*(1 + Sec[c + d*x])])","A",1
552,1,113,185,1.1368551,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{(A-B) \tan \left(\frac{1}{2} (c+d x)\right)+(A-5 B) \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)+4 \sqrt{2} B \cos \left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sqrt{2} \sin \left(\frac{1}{2} (c+d x)\right)\right)}{2 a d \sqrt{\cos (c+d x)} \sqrt{a (\sec (c+d x)+1)}}","\frac{(A-5 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)}{2 \sqrt{2} a^{3/2} d}+\frac{2 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)}{a^{3/2} d}+\frac{(A-B) \sin (c+d x)}{2 d \cos ^{\frac{3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"((A - 5*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + 4*Sqrt[2]*B*ArcTanh[Sqrt[2]*Sin[(c + d*x)/2]]*Cos[(c + d*x)/2] + (A - B)*Tan[(c + d*x)/2])/(2*a*d*Sqrt[Cos[c + d*x]]*Sqrt[a*(1 + Sec[c + d*x])])","A",1
553,1,288,237,2.2815622,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(3/2)),x]","-\frac{\sin (c+d x) \sqrt{\cos (c+d x)} \sec ^{\frac{5}{2}}(c+d x) \left(4 (A-3 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-2 \sqrt{2} (5 A-9 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+2 A \cos (c+d x) \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}+20 A \cos ^2\left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)-4 B \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-6 B \cos (c+d x) \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}-36 B \cos ^2\left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{4 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{3/2}}","-\frac{(5 A-9 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)}{2 \sqrt{2} a^{3/2} d}+\frac{(2 A-3 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)}{a^{3/2} d}-\frac{(A-3 B) \sin (c+d x)}{2 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A-B) \sin (c+d x)}{2 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"-1/4*(Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(5/2)*(4*(A - 3*B)*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 + 20*A*ArcSin[Sqrt[Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 - 36*B*ArcSin[Sqrt[Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 - 2*Sqrt[2]*(5*A - 9*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 - 4*B*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])] + 2*A*Cos[c + d*x]*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2] - 6*B*Cos[c + d*x]*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2])*Sin[c + d*x])/(d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
554,1,328,287,3.5186868,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(3/2)),x]","\frac{\sin (c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(2 (6 A-7 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right)-2 \sqrt{2} (9 A-13 B) \cos ^2\left(\frac{1}{2} (c+d x)\right) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)+4 A \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}+6 A \cos (c+d x) \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}+36 A \cos ^2\left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)+2 B \sqrt{1-\sec (c+d x)} \sec ^{\frac{3}{2}}(c+d x)-3 B \sqrt{-((\sec (c+d x)-1) \sec (c+d x))}-7 B \cos (c+d x) \sqrt{(\cos (c+d x)-1) \sec ^2(c+d x)}-52 B \cos ^2\left(\frac{1}{2} (c+d x)\right) \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right)\right)}{4 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{3/2}}","\frac{(9 A-13 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)}{2 \sqrt{2} a^{3/2} d}-\frac{(12 A-19 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)}{4 a^{3/2} d}+\frac{(6 A-7 B) \sin (c+d x)}{4 a d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}-\frac{(A-2 B) \sin (c+d x)}{2 a d \cos ^{\frac{5}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(A-B) \sin (c+d x)}{2 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}",1,"(Sec[c + d*x]^(3/2)*(2*(6*A - 7*B)*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 + 36*A*ArcSin[Sqrt[Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 - 52*B*ArcSin[Sqrt[Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 - 2*Sqrt[2]*(9*A - 13*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^2 + 2*B*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]^(3/2) + 4*A*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])] - 3*B*Sqrt[-((-1 + Sec[c + d*x])*Sec[c + d*x])] + 6*A*Cos[c + d*x]*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2] - 7*B*Cos[c + d*x]*Sqrt[(-1 + Cos[c + d*x])*Sec[c + d*x]^2])*Sin[c + d*x])/(4*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(3/2))","A",0
555,1,207,317,2.3435024,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \tan (c+d x) \sqrt{1-\sec (c+d x)} \sec (c+d x) (5 (887 A-479 B) \cos (c+d x)+16 (52 A-25 B) \cos (2 (c+d x))-40 A \cos (3 (c+d x))+12 A \cos (4 (c+d x))+3491 A+40 B \cos (3 (c+d x))-1895 B)+60 \sqrt{2} (283 A-163 B) \sin (c+d x) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{480 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{5/2}}","-\frac{(283 A-163 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(157 A-85 B) \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) \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) \sin (c+d x)}{240 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(21 A-13 B) \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) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(60*Sqrt[2]*(283*A - 163*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^(5/2)*Sin[c + d*x] + 2*(3491*A - 1895*B + 5*(887*A - 479*B)*Cos[c + d*x] + 16*(52*A - 25*B)*Cos[2*(c + d*x)] - 40*A*Cos[3*(c + d*x)] + 40*B*Cos[3*(c + d*x)] + 12*A*Cos[4*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(480*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
556,1,183,270,1.6982546,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{2 \tan (c+d x) \sqrt{1-\sec (c+d x)} \sec (c+d x) ((255 B-479 A) \cos (c+d x)+(48 B-80 A) \cos (2 (c+d x))+8 A \cos (3 (c+d x))-379 A+195 B)-12 \sqrt{2} (163 A-75 B) \sin (c+d x) \cos ^4\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right)}{96 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{5/2}}","\frac{(163 A-75 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(95 A-39 B) \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) \sin (c+d x)}{48 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(17 A-9 B) \sin (c+d x) \sqrt{\cos (c+d x)}}{16 a d (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 d (a \sec (c+d x)+a)^{5/2}}",1,"(-12*Sqrt[2]*(163*A - 75*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^4*Sec[c + d*x]^(5/2)*Sin[c + d*x] + 2*(-379*A + 195*B + (-479*A + 255*B)*Cos[c + d*x] + (-80*A + 48*B)*Cos[2*(c + d*x)] + 8*A*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(96*d*Sqrt[-1 + Cos[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
557,1,228,223,2.6937816,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + a*Sec[c + d*x])^(5/2),x]","\frac{\tan (c+d x) \sqrt{1-\sec (c+d x)} \sec (c+d x) \left(2 \left(73 A^2+76 A B-13 B^2\right) \cos (c+d x)+16 A^2 \cos (3 (c+d x))+85 A^2+A (85 A+19 B) \cos (2 (c+d x))+117 A B-18 B^2\right)+8 \sqrt{2} (75 A-19 B) \sin \left(\frac{1}{2} (c+d x)\right) \cos ^5\left(\frac{1}{2} (c+d x)\right) \sec ^{\frac{5}{2}}(c+d x) \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) (A \cos (c+d x)+B)}{32 d \sqrt{\cos (c+d x)-1} (a (\sec (c+d x)+1))^{5/2} (A \cos (c+d x)+B)}","-\frac{(75 A-19 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(49 A-9 B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}-\frac{(13 A-5 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(8*Sqrt[2]*(75*A - 19*B)*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Cos[(c + d*x)/2]^5*(B + A*Cos[c + d*x])*Sec[c + d*x]^(5/2)*Sin[(c + d*x)/2] + (85*A^2 + 117*A*B - 18*B^2 + 2*(73*A^2 + 76*A*B - 13*B^2)*Cos[c + d*x] + A*(85*A + 19*B)*Cos[2*(c + d*x)] + 16*A^2*Cos[3*(c + d*x)])*Sqrt[1 - Sec[c + d*x]]*Sec[c + d*x]*Tan[c + d*x])/(32*d*Sqrt[-1 + Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a*(1 + Sec[c + d*x]))^(5/2))","A",0
558,1,108,223,1.0155321,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\sec \left(\frac{1}{2} (c+d x)\right) \left(4 \sin \left(\frac{1}{2} (c+d x)\right) ((5 B-13 A) \cos (c+d x)-9 A+B)+8 (19 A+5 B) \cos ^4\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{64 a d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{3/2}}","\frac{(19 A+5 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)}{16 \sqrt{2} a^{5/2} d}-\frac{(9 A-B) \sin (c+d x)}{16 a^2 d \sqrt{\cos (c+d x)} \sqrt{a \sec (c+d x)+a}}+\frac{(5 A+3 B) \sin (c+d x)}{16 a d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{3/2}}+\frac{(A-B) \sin (c+d x)}{4 d \sqrt{\cos (c+d x)} (a \sec (c+d x)+a)^{5/2}}",1,"(Sec[(c + d*x)/2]*(8*(19*A + 5*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^4 + 4*(-9*A + B + (-13*A + 5*B)*Cos[c + d*x])*Sin[(c + d*x)/2]))/(64*a*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(3/2))","A",1
559,1,108,176,0.8335158,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{\cos ^2\left(\frac{1}{2} (c+d x)\right) \left(\frac{1}{2} \tan \left(\frac{1}{2} (c+d x)\right) ((5 A+3 B) \cos (c+d x)+A+7 B)+(5 A+3 B) \cos ^3\left(\frac{1}{2} (c+d x)\right) \tanh ^{-1}\left(\sin \left(\frac{1}{2} (c+d x)\right)\right)\right)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}","\frac{(5 A+3 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(5 A+3 B) \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) \sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"(Cos[(c + d*x)/2]^2*((5*A + 3*B)*ArcTanh[Sin[(c + d*x)/2]]*Cos[(c + d*x)/2]^3 + ((A + 7*B + (5*A + 3*B)*Cos[c + d*x])*Tan[(c + d*x)/2])/2))/(4*d*Cos[c + d*x]^(5/2)*(a*(1 + Sec[c + d*x]))^(5/2))","A",1
560,1,965,234,6.1701443,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + a*Sec[c + d*x])^(5/2)),x]","\frac{3 A \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{11 B \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{3 A \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{43 B \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{3 A \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{43 B \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{3 A \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{11 B \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{A \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{5}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{7 B \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{5}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{3 B \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{7}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{A \sin (c+d x) (\sec (c+d x)+1)}{16 d \cos ^{\frac{7}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{3 B \sin (c+d x) (\sec (c+d x)+1)}{16 d \cos ^{\frac{9}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{A \sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{B \sin (c+d x)}{4 d \cos ^{\frac{9}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}","\frac{(3 A-43 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)}{16 \sqrt{2} a^{5/2} d}+\frac{2 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)}{a^{5/2} d}+\frac{(3 A-11 B) \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) \sin (c+d x)}{4 d \cos ^{\frac{5}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"-1/4*(B*Sin[c + d*x])/(d*Cos[c + d*x]^(9/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (A*Sin[c + d*x])/(4*d*Cos[c + d*x]^(7/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*B*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*Cos[c + d*x]^(9/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (A*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*Cos[c + d*x]^(7/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (3*B*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(7/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (A*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(5/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (7*B*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(5/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*B*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (43*B*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (3*A*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (43*B*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","B",1
561,1,1061,286,6.2031762,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + a*Sec[c + d*x])^(5/2)),x]","-\frac{11 A \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{35 B \sin ^{-1}\left(\sqrt{1-\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{43 A \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{115 B \sin ^{-1}\left(\sqrt{\sec (c+d x)}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}+\frac{43 A \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{115 B \tan ^{-1}\left(\frac{\sqrt{2} \sqrt{\sec (c+d x)}}{\sqrt{1-\sec (c+d x)}}\right) \sqrt{\cos (c+d x)} \sec ^{\frac{3}{2}}(c+d x) \sin (c+d x) (\sec (c+d x)+1)^2}{16 \sqrt{2} d \sqrt{1-\sec (c+d x)} (a (\sec (c+d x)+1))^{5/2}}-\frac{11 A \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{35 B \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{3}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{7 A \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{5}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{15 B \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{5}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{3 A \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{7}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{11 B \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{7}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{7 B \sin (c+d x) (\sec (c+d x)+1)^2}{16 d \cos ^{\frac{9}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{3 A \sin (c+d x) (\sec (c+d x)+1)}{16 d \cos ^{\frac{9}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}+\frac{7 B \sin (c+d x) (\sec (c+d x)+1)}{16 d \cos ^{\frac{11}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{A \sin (c+d x)}{4 d \cos ^{\frac{9}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}-\frac{B \sin (c+d x)}{4 d \cos ^{\frac{11}{2}}(c+d x) (a (\sec (c+d x)+1))^{5/2}}","-\frac{(43 A-115 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)}{16 \sqrt{2} a^{5/2} d}+\frac{(2 A-5 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)}{a^{5/2} d}-\frac{(11 A-35 B) \sin (c+d x)}{16 a^2 d \cos ^{\frac{3}{2}}(c+d x) \sqrt{a \sec (c+d x)+a}}+\frac{(7 A-15 B) \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) \sin (c+d x)}{4 d \cos ^{\frac{7}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}",1,"-1/4*(B*Sin[c + d*x])/(d*Cos[c + d*x]^(11/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (A*Sin[c + d*x])/(4*d*Cos[c + d*x]^(9/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (7*B*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*Cos[c + d*x]^(11/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (3*A*(1 + Sec[c + d*x])*Sin[c + d*x])/(16*d*Cos[c + d*x]^(9/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (7*B*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(9/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (3*A*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(7/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (11*B*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(7/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (7*A*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(5/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (15*B*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(5/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*A*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(5/2)) + (35*B*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Cos[c + d*x]^(3/2)*(a*(1 + Sec[c + d*x]))^(5/2)) - (11*A*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (35*B*ArcSin[Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (43*A*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (115*B*ArcSin[Sqrt[Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) + (43*A*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2)) - (115*B*ArcTan[(Sqrt[2]*Sqrt[Sec[c + d*x]])/Sqrt[1 - Sec[c + d*x]]]*Sqrt[Cos[c + d*x]]*Sec[c + d*x]^(3/2)*(1 + Sec[c + d*x])^2*Sin[c + d*x])/(16*Sqrt[2]*d*Sqrt[1 - Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(5/2))","B",1
562,1,103,140,0.9185942,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{10 (5 a A+7 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+126 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (42 (a B+A b) \cos (c+d x)+15 a A \cos (2 (c+d x))+65 a A+70 b B)}{105 d}","\frac{2 (5 a A+7 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{6 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 d}+\frac{2 (5 a A+7 b B) \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}",1,"(126*(A*b + a*B)*EllipticE[(c + d*x)/2, 2] + 10*(5*a*A + 7*b*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(65*a*A + 70*b*B + 42*(A*b + a*B)*Cos[c + d*x] + 15*a*A*Cos[2*(c + d*x)])*Sin[c + d*x])/(105*d)","A",1
563,1,86,108,0.4405702,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{2 \left(5 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 (3 a A+5 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} (3 a A \cos (c+d x)+5 a B+5 A b)\right)}{15 d}","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (3 a A+5 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{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*(3*a*A + 5*b*B)*EllipticE[(c + d*x)/2, 2] + 5*(A*b + a*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(5*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
564,1,67,75,0.2726447,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{2 \left((a A+3 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a A \sin (c+d x) \sqrt{\cos (c+d x)}\right)}{3 d}","\frac{2 (a A+3 b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 (a B+A b) 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}",1,"(2*(3*(A*b + a*B)*EllipticE[(c + d*x)/2, 2] + (a*A + 3*b*B)*EllipticF[(c + d*x)/2, 2] + a*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x]))/(3*d)","A",1
565,1,64,71,0.3716646,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x)) (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]),x]","\frac{2 \left((a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+(a A-b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b B \sin (c+d x)}{\sqrt{\cos (c+d x)}}\right)}{d}","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (a A-b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*((a*A - b*B)*EllipticE[(c + d*x)/2, 2] + (A*b + a*B)*EllipticF[(c + d*x)/2, 2] + (b*B*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/d","A",1
566,1,107,103,0.5986042,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{2 \left((3 a A+b B) \sqrt{\cos (c+d x)} F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-3 (a B+A b) \sqrt{\cos (c+d x)} E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 a B \sin (c+d x)+3 A b \sin (c+d x)+b B \tan (c+d x)\right)}{3 d \sqrt{\cos (c+d x)}}","\frac{2 (3 a A+b B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (a B+A b) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 (a B+A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(-3*(A*b + a*B)*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + (3*a*A + b*B)*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + 3*A*b*Sin[c + d*x] + 3*a*B*Sin[c + d*x] + b*B*Tan[c + d*x]))/(3*d*Sqrt[Cos[c + d*x]])","A",1
567,1,134,140,1.0177669,"\int \frac{(a+b \sec (c+d x)) (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{10 (a B+A b) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 (5 a A+3 b B) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 a A \sin (2 (c+d x))+10 a B \sin (c+d x)+10 A b \sin (c+d x)+9 b B \sin (2 (c+d x))+6 b B \tan (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 (a B+A b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 (5 a A+3 b B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 (a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 (5 a A+3 b B) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(5*a*A + 3*b*B)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(A*b + a*B)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*A*b*Sin[c + d*x] + 10*a*B*Sin[c + d*x] + 15*a*A*Sin[2*(c + d*x)] + 9*b*B*Sin[2*(c + d*x)] + 6*b*B*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
568,1,139,182,1.274565,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{10 \left(5 a^2 A+14 a b B+7 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+42 \left(3 a^2 B+6 a A b+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\sin (c+d x) \sqrt{\cos (c+d x)} \left(5 \left(3 a^2 A \cos (2 (c+d x))+13 a^2 A+28 a b B+14 A b^2\right)+42 a (a B+2 A b) \cos (c+d x)\right)}{105 d}","\frac{2 \left(3 a^2 B+6 a A b+5 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 A+7 b (2 a B+A b)\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}+\frac{2 \left(5 a^2 A+7 b (2 a B+A b)\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{21 d}+\frac{2 a (7 a B+9 A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{35 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}{7 d}",1,"(42*(6*a*A*b + 3*a^2*B + 5*b^2*B)*EllipticE[(c + d*x)/2, 2] + 10*(5*a^2*A + 7*A*b^2 + 14*a*b*B)*EllipticF[(c + d*x)/2, 2] + Sqrt[Cos[c + d*x]]*(42*a*(2*A*b + a*B)*Cos[c + d*x] + 5*(13*a^2*A + 14*A*b^2 + 28*a*b*B + 3*a^2*A*Cos[2*(c + d*x)]))*Sin[c + d*x])/(105*d)","A",1
569,1,106,140,0.672772,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{2 \left(5 \left(a^2 B+2 a A b+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \left(3 a^2 A+10 a b B+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+a \sin (c+d x) \sqrt{\cos (c+d x)} (3 a A \cos (c+d x)+5 a B+10 A b)\right)}{15 d}","\frac{2 \left(a^2 B+2 a A b+3 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(3 a^2 A+5 b (2 a B+A b)\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 a (5 a B+7 A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{15 d}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}{5 d}",1,"(2*(3*(3*a^2*A + 5*A*b^2 + 10*a*b*B)*EllipticE[(c + d*x)/2, 2] + 5*(2*a*A*b + a^2*B + 3*b^2*B)*EllipticF[(c + d*x)/2, 2] + a*Sqrt[Cos[c + d*x]]*(10*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x]))/(15*d)","A",1
570,1,102,121,0.6641532,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{2 \left(\left(a^2 A+6 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \left(a^2 B+2 a A b-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{\sin (c+d x) \left(a^2 A \cos (c+d x)+3 b^2 B\right)}{\sqrt{\cos (c+d x)}}\right)}{3 d}","\frac{2 \left(a^2 A+6 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(a^2 B+2 a A b-b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 a^2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 d}+\frac{2 b^2 B \sin (c+d x)}{d \sqrt{\cos (c+d x)}}",1,"(2*(3*(2*a*A*b + a^2*B - b^2*B)*EllipticE[(c + d*x)/2, 2] + (a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticF[(c + d*x)/2, 2] + ((3*b^2*B + a^2*A*Cos[c + d*x])*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(3*d)","A",1
571,1,105,126,1.2247532,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2 (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]),x]","\frac{2 \left(\left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)+3 \left(a^2 A-2 a b B-A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{b \sin (c+d x) (3 (2 a B+A b) \cos (c+d x)+b B)}{\cos ^{\frac{3}{2}}(c+d x)}\right)}{3 d}","\frac{2 \left(3 a^2 B+6 a A b+b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}+\frac{2 \left(a^2 A-2 a b B-A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{d}+\frac{2 b (2 a B+A b) \sin (c+d x)}{d \sqrt{\cos (c+d x)}}+\frac{2 b^2 B \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"(2*(3*(a^2*A - A*b^2 - 2*a*b*B)*EllipticE[(c + d*x)/2, 2] + (6*a*A*b + 3*a^2*B + b^2*B)*EllipticF[(c + d*x)/2, 2] + (b*(b*B + 3*(A*b + 2*a*B)*Cos[c + d*x])*Sin[c + d*x])/Cos[c + d*x]^(3/2)))/(3*d)","A",1
572,1,175,172,1.2802011,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\frac{10 \left(3 a^2 A+2 a b B+A b^2\right) \cos ^{\frac{3}{2}}(c+d x) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-6 \left(5 a^2 B+10 a A b+3 b^2 B\right) \cos ^{\frac{3}{2}}(c+d x) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+15 a^2 B \sin (2 (c+d x))+30 a A b \sin (2 (c+d x))+20 a b B \sin (c+d x)+10 A b^2 \sin (c+d x)+9 b^2 B \sin (2 (c+d x))+6 b^2 B \tan (c+d x)}{15 d \cos ^{\frac{3}{2}}(c+d x)}","\frac{2 \left(3 a^2 A+2 a b B+A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 d}-\frac{2 \left(5 a^2 B+10 a A b+3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(5 a^2 B+10 a A b+3 b^2 B\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b (2 a B+A b) \sin (c+d x)}{3 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 b^2 B \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}",1,"(-6*(10*a*A*b + 5*a^2*B + 3*b^2*B)*Cos[c + d*x]^(3/2)*EllipticE[(c + d*x)/2, 2] + 10*(3*a^2*A + A*b^2 + 2*a*b*B)*Cos[c + d*x]^(3/2)*EllipticF[(c + d*x)/2, 2] + 10*A*b^2*Sin[c + d*x] + 20*a*b*B*Sin[c + d*x] + 30*a*A*b*Sin[2*(c + d*x)] + 15*a^2*B*Sin[2*(c + d*x)] + 9*b^2*B*Sin[2*(c + d*x)] + 6*b^2*B*Tan[c + d*x])/(15*d*Cos[c + d*x]^(3/2))","A",1
573,1,191,214,5.259695,"\int \frac{(a+b \sec (c+d x))^2 (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^2*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\frac{2 \left(5 \left(7 a^2 B+14 a A b+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-21 \left(5 a^2 A+6 a b B+3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)+\frac{5 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{21 \left(5 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{21 b (2 a B+A b) \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}+\frac{15 b^2 B \sin (c+d x)}{\cos ^{\frac{7}{2}}(c+d x)}\right)}{105 d}","\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{21 d}-\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 d}+\frac{2 \left(7 a^2 B+14 a A b+5 b^2 B\right) \sin (c+d x)}{21 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 \left(5 a^2 A+6 a b B+3 A b^2\right) \sin (c+d x)}{5 d \sqrt{\cos (c+d x)}}+\frac{2 b (2 a B+A b) \sin (c+d x)}{5 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{2 b^2 B \sin (c+d x)}{7 d \cos ^{\frac{7}{2}}(c+d x)}",1,"(2*(-21*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*EllipticE[(c + d*x)/2, 2] + 5*(14*a*A*b + 7*a^2*B + 5*b^2*B)*EllipticF[(c + d*x)/2, 2] + (15*b^2*B*Sin[c + d*x])/Cos[c + d*x]^(7/2) + (21*b*(A*b + 2*a*B)*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (5*(14*a*A*b + 7*a^2*B + 5*b^2*B)*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (21*(5*a^2*A + 3*A*b^2 + 6*a*b*B)*Sin[c + d*x])/Sqrt[Cos[c + d*x]]))/(105*d)","A",1
574,1,260,182,2.8036913,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{\frac{2 a^2 \left(9 a^2 A-5 a b B+5 A b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(3 a^2 A-5 a b B+5 A b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{b \sqrt{\sin ^2(c+d x)}}+4 a^2 \sin (c+d x) \sqrt{\cos (c+d x)} (3 a A \cos (c+d x)+5 a B-5 A b)+2 a^2 (5 a B+4 A b) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{30 a^4 d}","\frac{2 b^3 (A b-a B) \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 \left(a^2+3 b^2\right) (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d}+\frac{2 \left(3 a^2 A-5 a b B+5 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 a^3 d}+\frac{2 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{5 a d}",1,"((2*a^2*(9*a^2*A + 5*A*b^2 - 5*a*b*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + 2*a^2*(4*A*b + 5*a*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + 4*a^2*Sqrt[Cos[c + d*x]]*(-5*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x] + (6*(3*a^2*A + 5*A*b^2 - 5*a*b*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(b*Sqrt[Sin[c + d*x]^2]))/(30*a^4*d)","A",1
575,1,207,136,1.4722802,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{\frac{3 (a B-A b) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+\frac{(3 a B-A b) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+A \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)+2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}","-\frac{2 b^2 (A b-a B) \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 \left(a^2 A-3 a b B+3 A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^3 d}+\frac{2 A \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d}",1,"(((-(A*b) + 3*a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + A*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) + 2*A*Sqrt[Cos[c + d*x]]*Sin[c + d*x] + (3*(-(A*b) + a*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/(3*a*d)","A",1
576,1,128,89,0.9858914,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{a+b \sec (c+d x)} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x]),x]","\frac{a B \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)-\frac{2 A \sin (c+d x) \left(-(a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+b \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+a E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{\sqrt{\sin ^2(c+d x)}}}{a^2 d}","-\frac{2 (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d}+\frac{2 b (A b-a B) \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 E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}",1,"(a*B*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)) - (2*A*(a*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] - (a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + b*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/Sqrt[Sin[c + d*x]^2])/(a^2*d)","A",1
577,1,58,61,0.2327512,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])),x]","\frac{2 \left((a B-A b) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)+A (a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a d (a+b)}","\frac{2 A F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d}-\frac{2 (A b-a B) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a d (a+b)}",1,"(2*(A*(a + b)*EllipticF[(c + d*x)/2, 2] + (-(A*b) + a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)*d)","A",1
578,1,206,86,2.8003785,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])),x]","\frac{-\frac{2 B \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 (2 A b-3 a B) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 b B \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{4 B \sin (c+d x)}{\sqrt{\cos (c+d x)}}}{2 b d}","\frac{2 (A b-a B) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b d (a+b)}-\frac{2 B E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d}+\frac{2 B \sin (c+d x)}{b d \sqrt{\cos (c+d x)}}",1,"((2*(2*A*b - 3*a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) - (2*b*B*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (4*B*Sin[c + d*x])/Sqrt[Cos[c + d*x]] - (2*B*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/(2*b*d)","B",1
579,1,260,150,2.3054533,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])),x]","\frac{\frac{2 b \left(9 a^2 B-9 a A b+2 b^2 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 (a B-A b) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{b \left(8 a b B-6 A b^2\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{12 b (A b-a B) \sin (c+d x)}{\sqrt{\cos (c+d x)}}+\frac{4 b^2 B \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}}{6 b^3 d}","-\frac{2 (A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d}-\frac{2 a (A b-a B) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a+b)}+\frac{2 (A b-a B) \sin (c+d x)}{b^2 d \sqrt{\cos (c+d x)}}+\frac{2 B F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b d}+\frac{2 B \sin (c+d x)}{3 b d \cos ^{\frac{3}{2}}(c+d x)}",1,"((2*b*(-9*a*A*b + 9*a^2*B + 2*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (b*(-6*A*b^2 + 8*a*b*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (4*b^2*B*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (12*b*(A*b - a*B)*Sin[c + d*x])/Sqrt[Cos[c + d*x]] + (6*(-(A*b) + a*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/(6*b^3*d)","A",1
580,1,326,217,4.8612106,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])),x]","\frac{\frac{6 b \left(5 a^2 B-5 a A b+3 b^2 B\right) \sin (c+d x)}{\sqrt{\cos (c+d x)}}-\frac{b^2 \left(20 a^2 B-20 a A b+9 b^2 B\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}-\frac{3 \left(5 a^2 B-5 a A b+3 b^2 B\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a \sqrt{\sin ^2(c+d x)}}+\frac{b \left(-45 a^3 B+45 a^2 A b-19 a b^2 B+10 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{10 b^2 (A b-a B) \sin (c+d x)}{\cos ^{\frac{3}{2}}(c+d x)}+\frac{6 b^3 B \sin (c+d x)}{\cos ^{\frac{5}{2}}(c+d x)}}{15 b^4 d}","\frac{2 a^2 (A b-a B) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a+b)}+\frac{2 \left(-5 a^2 B+5 a A b-3 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{5 b^3 d}-\frac{2 \left(-5 a^2 B+5 a A b-3 b^2 B\right) \sin (c+d x)}{5 b^3 d \sqrt{\cos (c+d x)}}+\frac{2 (A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d}+\frac{2 (A b-a B) \sin (c+d x)}{3 b^2 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 B \sin (c+d x)}{5 b d \cos ^{\frac{5}{2}}(c+d x)}",1,"((b*(45*a^2*A*b + 10*A*b^3 - 45*a^3*B - 19*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) - (b^2*(-20*a*A*b + 20*a^2*B + 9*b^2*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (6*b^3*B*Sin[c + d*x])/Cos[c + d*x]^(5/2) + (10*b^2*(A*b - a*B)*Sin[c + d*x])/Cos[c + d*x]^(3/2) + (6*b*(-5*a*A*b + 5*a^2*B + 3*b^2*B)*Sin[c + d*x])/Sqrt[Cos[c + d*x]] - (3*(-5*a*A*b + 5*a^2*B + 3*b^2*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*Sqrt[Sin[c + d*x]^2]))/(15*b^4*d)","A",1
581,1,318,305,3.7820709,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(\frac{3 b^2 (A b-a B)}{\left(b^2-a^2\right) (a \cos (c+d x)+b)}+2 A\right)-\frac{\frac{8 \left(a^2 A-3 a b B+2 A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(6 a^3 B-8 a^2 A b-3 a b^2 B+5 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(2 a^3 B-4 a^2 A b-3 a b^2 B+5 A b^3\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(b-a) (a+b)}}{12 a^2 d}","\frac{b (A b-a B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\left(2 a^2 A+3 a b B-5 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}-\frac{\left(-2 a^3 B+4 a^2 A b+3 a b^2 B-5 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}-\frac{b^2 \left(-5 a^3 B+7 a^2 A b+3 a b^2 B-5 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^4 d (a-b) (a+b)^2}+\frac{\left(2 a^4 A-12 a^3 b B+16 a^2 A b^2+9 a b^3 B-15 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 a^4 d \left(a^2-b^2\right)}",1,"(4*Sqrt[Cos[c + d*x]]*(2*A + (3*b^2*(A*b - a*B))/((-a^2 + b^2)*(b + a*Cos[c + d*x])))*Sin[c + d*x] - ((2*(-8*a^2*A*b + 5*A*b^3 + 6*a^3*B - 3*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(a^2*A + 2*A*b^2 - 3*a*b*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-4*a^2*A*b + 5*A*b^3 + 2*a^3*B - 3*a*b^2*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(12*a^2*d)","A",1
582,1,281,223,2.8513136,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^2} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^2,x]","\frac{\frac{4 b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\frac{2 \left(2 a^2 A-a b B-A b^2\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 \left(2 a^2 A+a b B-3 A b^2\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+\frac{8 (a B-A b) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}}{(a-b) (a+b)}}{4 a d}","\frac{\left(2 a^2 A+a b B-3 A b^2\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{\left(-2 a^3 B+4 a^2 A b+a b^2 B-3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d \left(a^2-b^2\right)}+\frac{b \left(-3 a^3 B+5 a^2 A b+a b^2 B-3 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^3 d (a-b) (a+b)^2}",1,"((4*b*(A*b - a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + ((2*(2*a^2*A - A*b^2 - a*b*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-(A*b) + a*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (2*(2*a^2*A - 3*A*b^2 + a*b*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*a*d)","A",1
583,1,260,203,2.7108632,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{4 (a B-A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{\frac{2 (A b-a B) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}+\frac{2 (a B-A b) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{(4 a A-4 b B) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}}{(b-a) (a+b)}}{4 d}","\frac{\left(2 a^2 A-a b B-A b^2\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d \left(a^2-b^2\right)}+\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{(A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{d \left(a^2-b^2\right) (a \cos (c+d x)+b)}-\frac{\left(a^3 (-B)+3 a^2 A b-a b^2 B-A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a^2 d (a-b) (a+b)^2}",1,"((4*(-(A*b) + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) - ((2*(-(A*b) + a*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + ((4*a*A - 4*b*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(A*b - a*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((-a + b)*(a + b)))/(4*d)","A",1
584,1,273,197,2.796905,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\frac{\frac{2 \left(3 a^2 B+a A b-4 b^2 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{2 (a B-A b) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{4 b (a B-A b) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}}{(a-b) (a+b)}-\frac{4 a (a B-A b) \sin (c+d x) \sqrt{\cos (c+d x)}}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}}{4 b d}","-\frac{(A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{a d \left(a^2-b^2\right)}-\frac{(A b-a B) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{a (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{b d \left(a^2-b^2\right) (a \cos (c+d x)+b)}+\frac{\left(a^3 B+a^2 A b-3 a b^2 B+A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a b d (a-b) (a+b)^2}",1,"((-4*a*(-(A*b) + a*B)*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + ((2*(a*A*b + 3*a^2*B - 4*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (4*b*(-(A*b) + a*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(-(A*b) + a*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b)))/(4*b*d)","A",1
585,1,317,255,4.7309016,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^2),x]","\frac{4 \sqrt{\cos (c+d x)} \left(\frac{a^2 (a B-A b) \sin (c+d x)}{\left(a^2-b^2\right) (a \cos (c+d x)+b)}+2 B \tan (c+d x)\right)-\frac{-\frac{8 b \left(-2 a^2 B+a A b+b^2 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}+\frac{2 \left(3 a^2 B-a A b-2 b^2 B\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{2 \left(9 a^3 B-3 a^2 A b-10 a b^2 B+4 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b) (a+b)}}{4 b^2 d}","\frac{(A b-a B) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b d \left(a^2-b^2\right)}+\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d \left(a^2-b^2\right)}-\frac{\left(-3 a^2 B+a A b+2 b^2 B\right) \sin (c+d x)}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}+\frac{\left(-3 a^3 B+a^2 A b+5 a b^2 B-3 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^2 d (a-b) (a+b)^2}",1,"(-(((2*(-3*a^2*A*b + 4*A*b^3 + 9*a^3*B - 10*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) - (8*b*(a*A*b - 2*a^2*B + b^2*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + (2*(-(a*A*b) + 3*a^2*B - 2*b^2*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)*(a + b))) + 4*Sqrt[Cos[c + d*x]]*((a^2*(-(A*b) + a*B)*Sin[c + d*x])/((a^2 - b^2)*(b + a*Cos[c + d*x])) + 2*B*Tan[c + d*x]))/(4*b^2*d)","A",1
586,1,427,346,6.940927,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^2} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^2),x]","\frac{\sqrt{\cos (c+d x)} \left(\frac{a^4 B \sin (c+d x)-a^3 A b \sin (c+d x)}{b^3 \left(b^2-a^2\right) (a \cos (c+d x)+b)}+\frac{2 \sec (c+d x) (A b \sin (c+d x)-2 a B \sin (c+d x))}{b^3}+\frac{2 B \tan (c+d x) \sec (c+d x)}{3 b^2}\right)}{d}+\frac{\frac{\left(40 a^3 b B-24 a^2 A b^2-28 a b^3 B+12 A b^4\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{2 \left(15 a^4 B-9 a^3 A b-12 a^2 b^2 B+6 a A b^3\right) \sin (c+d x) \cos (2 (c+d x)) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right)}+\frac{2 \left(45 a^4 B-27 a^3 A b-44 a^2 b^2 B+30 a A b^3-4 b^4 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{12 b^3 d (a-b) (a+b)}","-\frac{\left(-5 a^2 B+3 a A b+2 b^2 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{3 b^2 d \left(a^2-b^2\right)}+\frac{a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac{\left(-5 a^2 B+3 a A b+2 b^2 B\right) \sin (c+d x)}{3 b^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}-\frac{\left(-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d \left(a^2-b^2\right)}-\frac{a \left(-5 a^3 B+3 a^2 A b+7 a b^2 B-5 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{b^3 d (a-b) (a+b)^2}+\frac{\left(-5 a^3 B+3 a^2 A b+4 a b^2 B-2 A b^3\right) \sin (c+d x)}{b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}",1,"((2*(-27*a^3*A*b + 30*a*A*b^3 + 45*a^4*B - 44*a^2*b^2*B - 4*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + ((-24*a^2*A*b^2 + 12*A*b^4 + 40*a^3*b*B - 28*a*b^3*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(-9*a^3*A*b + 6*a*A*b^3 + 15*a^4*B - 12*a^2*b^2*B)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(12*(a - b)*b^3*(a + b)*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]*(A*b*Sin[c + d*x] - 2*a*B*Sin[c + d*x]))/b^3 + (-(a^3*A*b*Sin[c + d*x]) + a^4*B*Sin[c + d*x])/(b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(3*b^2)))/d","A",1
587,1,461,461,6.1398785,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(4 a^6 A+4 A \left(a^3-a b^2\right)^2 \cos (2 (c+d x))+33 a^3 b^3 B-57 a^2 A b^4+a b \left(16 a^4 A+39 a^3 b B-83 a^2 A b^2-21 a b^3 B+49 A b^4\right) \cos (c+d x)-15 a b^5 B+35 A b^6\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{\frac{16 \left(2 a^4 A-12 a^3 b B+14 a^2 A b^2+3 a b^3 B-7 A b^4\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(24 a^5 B-56 a^4 A b-21 a^3 b^2 B+73 a^2 A b^3+15 a b^4 B-35 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{6 \left(8 a^5 B-24 a^4 A b-29 a^3 b^2 B+65 a^2 A b^3+15 a b^4 B-35 A b^5\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{48 a^3 d}","\frac{b (A b-a B) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{b \left(-9 a^3 B+13 a^2 A b+3 a b^2 B-7 A b^3\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{4 a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{\left(8 a^4 A+33 a^3 b B-61 a^2 A b^2-15 a b^3 B+35 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{12 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-8 a^5 B+24 a^4 A b+29 a^3 b^2 B-65 a^2 A b^3-15 a b^4 B+35 A b^5\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}-\frac{b^2 \left(-35 a^5 B+63 a^4 A b+38 a^3 b^2 B-86 a^2 A b^3-15 a b^4 B+35 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^5 d (a-b)^2 (a+b)^3}+\frac{\left(8 a^6 A-72 a^5 b B+128 a^4 A b^2+99 a^3 b^3 B-223 a^2 A b^4-45 a b^5 B+105 A b^6\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 a^5 d \left(a^2-b^2\right)^2}",1,"((4*Sqrt[Cos[c + d*x]]*(4*a^6*A - 57*a^2*A*b^4 + 35*A*b^6 + 33*a^3*b^3*B - 15*a*b^5*B + a*b*(16*a^4*A - 83*a^2*A*b^2 + 49*A*b^4 + 39*a^3*b*B - 21*a*b^3*B)*Cos[c + d*x] + 4*A*(a^3 - a*b^2)^2*Cos[2*(c + d*x)])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + ((2*(-56*a^4*A*b + 73*a^2*A*b^3 - 35*A*b^5 + 24*a^5*B - 21*a^3*b^2*B + 15*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(2*a^4*A + 14*a^2*A*b^2 - 7*A*b^4 - 12*a^3*b*B + 3*a*b^3*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + (6*(-24*a^4*A*b + 65*a^2*A*b^3 - 35*A*b^5 + 8*a^5*B - 29*a^3*b^2*B + 15*a*b^4*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(48*a^3*d)","A",1
588,1,390,367,5.1265313,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^3} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^3,x]","\frac{\frac{\frac{8 \left(2 a^3 B-4 a^2 A b+a b^2 B+A b^3\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{\left(8 a^4 A-5 a^3 b B-7 a^2 A b^2-a b^3 B+5 A b^4\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+15 A b^4\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}-\frac{2 b \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \left(9 a^3 B-13 a^2 A b-3 a b^2 B+7 A b^3\right) \cos (c+d x)+b \left(7 a^3 B-11 a^2 A b-a b^2 B+5 A b^3\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{8 a^2 d}","\frac{b (A b-a B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{2 a d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{b \left(-7 a^3 B+11 a^2 A b+a b^2 B-5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{\left(8 a^4 A+9 a^3 b B-29 a^2 A b^2-3 a b^3 B+15 A b^4\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-8 a^5 B+24 a^4 A b+5 a^3 b^2 B-33 a^2 A b^3-3 a b^4 B+15 A b^5\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d \left(a^2-b^2\right)^2}+\frac{b \left(-15 a^5 B+35 a^4 A b+6 a^3 b^2 B-38 a^2 A b^3-3 a b^4 B+15 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^4 d (a-b)^2 (a+b)^3}",1,"((-2*b*Sqrt[Cos[c + d*x]]*(b*(-11*a^2*A*b + 5*A*b^3 + 7*a^3*B - a*b^2*B) + a*(-13*a^2*A*b + 7*A*b^3 + 9*a^3*B - 3*a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (((8*a^4*A - 7*a^2*A*b^2 + 5*A*b^4 - 5*a^3*b*B - a*b^3*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*(-4*a^2*A*b + A*b^3 + 2*a^3*B + a*b^2*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) + ((8*a^4*A - 29*a^2*A*b^2 + 15*A*b^4 + 9*a^3*b*B - 3*a*b^3*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*a^2*d)","A",1
589,1,361,346,4.8515254,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{4 \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \left(5 a^3 B-9 a^2 A b+a b^2 B+3 A b^3\right) \cos (c+d x)+b \left(3 a^3 B-7 a^2 A b+3 a b^2 B+A b^3\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{\frac{16 \left(2 a^2 A-3 a b B+A b^2\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a+b}+\frac{2 \left(a^3 B-5 a^2 A b+5 a b^2 B-A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}-\frac{2 \left(5 a^3 B-9 a^2 A b+a b^2 B+3 A b^3\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{16 a d}","\frac{b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 a d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{\left(-5 a^3 B+9 a^2 A b-a b^2 B-3 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(-5 a^3 B+9 a^2 A b-a b^2 B-3 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 a d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{\left(8 a^4 A-7 a^3 b B-5 a^2 A b^2+a b^3 B+3 A b^4\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d \left(a^2-b^2\right)^2}-\frac{\left(-3 a^5 B+15 a^4 A b-10 a^3 b^2 B-6 a^2 A b^3+a b^4 B+3 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^3 d (a-b)^2 (a+b)^3}",1,"((4*Sqrt[Cos[c + d*x]]*(b*(-7*a^2*A*b + A*b^3 + 3*a^3*B + 3*a*b^2*B) + a*(-9*a^2*A*b + 3*A*b^3 + 5*a^3*B + a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + ((2*(-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (16*(2*a^2*A + A*b^2 - 3*a*b*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a + b) - (2*(-9*a^2*A*b + 3*A*b^3 + 5*a^3*B + a*b^2*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(16*a*d)","A",1
590,1,364,338,4.9316867,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{2 \sin (c+d x) \sqrt{\cos (c+d x)} \left(b \left(a^3 B+3 a^2 A b-7 a b^2 B+3 A b^3\right)-a \left(a^3 B-5 a^2 A b+5 a b^2 B-A b^3\right) \cos (c+d x)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}+\frac{\frac{8 b \left(a^2 B-3 a A b+2 b^2 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}+\frac{\left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}+\frac{\left(a^3 B-5 a^2 A b+5 a b^2 B-A b^3\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{\sin ^2(c+d x)}}}{(a-b)^2 (a+b)^2}}{8 b d}","-\frac{(A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}-\frac{\left(-3 a^3 B+7 a^2 A b-3 a b^2 B-A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 d \left(a^2-b^2\right)^2}-\frac{\left(a^3 (-B)+5 a^2 A b-5 a b^2 B+A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}+\frac{\left(a^3 (-B)+5 a^2 A b-5 a b^2 B+A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{\left(a^5 B+3 a^4 A b-10 a^3 b^2 B+10 a^2 A b^3-3 a b^4 B-A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a^2 b d (a-b)^2 (a+b)^3}",1,"((2*Sqrt[Cos[c + d*x]]*(b*(3*a^2*A*b + 3*A*b^3 + a^3*B - 7*a*b^2*B) - a*(-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-3*a*A*b + a^2*B + 2*b^2*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + ((-5*a^2*A*b - A*b^3 + a^3*B + 5*a*b^2*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b*d)","A",1
591,1,383,342,5.1363991,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{\frac{8 b \left(a^3 B+a^2 A b-4 a b^2 B+2 A b^3\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}+\frac{\left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{\left(9 a^4 B+3 a^3 A b-19 a^2 b^2 B-9 a A b^3+16 b^4 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b)^2 (a+b)^2}-\frac{2 a \sin (c+d x) \sqrt{\cos (c+d x)} \left(a \left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) \cos (c+d x)+b \left(5 a^3 B-a^2 A b-11 a b^2 B+7 A b^3\right)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}}{8 b^2 d}","\frac{a (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{2 b d \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}+\frac{\left(a^3 B+3 a^2 A b-7 a b^2 B+3 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b d \left(a^2-b^2\right)^2}+\frac{\left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}-\frac{a \left(3 a^3 B+a^2 A b-9 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{4 b^2 d \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{\left(3 a^5 B+a^4 A b-6 a^3 b^2 B-10 a^2 A b^3+15 a b^4 B-3 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 a b^2 d (a-b)^2 (a+b)^3}",1,"((-2*a*Sqrt[Cos[c + d*x]]*(b*(-(a^2*A*b) + 7*A*b^3 + 5*a^3*B - 11*a*b^2*B) + a*(a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2) + (((3*a^3*A*b - 9*a*A*b^3 + 9*a^4*B - 19*a^2*b^2*B + 16*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(a^2*A*b + 2*A*b^3 + a^3*B - 4*a*b^2*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + ((a^2*A*b + 5*A*b^3 + 3*a^3*B - 9*a*b^2*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2))/(8*b^2*d)","A",1
592,1,458,420,6.1849911,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\frac{\sqrt{\cos (c+d x)} \left(16 B \left(b^3-a^2 b\right)^2 \tan (c+d x)+a^2 \left(15 a^4 B-3 a^3 A b-29 a^2 b^2 B+9 a A b^3+8 b^4 B\right) \sin (2 (c+d x))+2 a b \left(25 a^4 B-5 a^3 A b-47 a^2 b^2 B+11 a A b^3+16 b^4 B\right) \sin (c+d x)\right)}{\left(a^2-b^2\right)^2 (a \cos (c+d x)+b)^2}-\frac{\frac{8 b \left(5 a^4 B-a^3 A b-10 a^2 b^2 B+4 a A b^3+2 b^4 B\right) \left((a+b) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)\right)}{a (a+b)}+\frac{\left(15 a^4 B-3 a^3 A b-29 a^2 b^2 B+9 a A b^3+8 b^4 B\right) \sin (c+d x) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a b \sqrt{\sin ^2(c+d x)}}+\frac{\left(45 a^5 B-9 a^4 A b-95 a^3 b^2 B+19 a^2 A b^3+56 a b^4 B-16 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{(a-b)^2 (a+b)^2}}{8 b^3 d}","\frac{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)^2}+\frac{\left(-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^2 d \left(a^2-b^2\right)^2}+\frac{a \left(-5 a^3 B+a^2 A b+11 a b^2 B-7 A b^3\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} (a \cos (c+d x)+b)}+\frac{\left(-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(-15 a^4 B+3 a^3 A b+29 a^2 b^2 B-9 a A b^3-8 b^4 B\right) \sin (c+d x)}{4 b^3 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}+\frac{\left(-15 a^5 B+3 a^4 A b+38 a^3 b^2 B-6 a^2 A b^3-35 a b^4 B+15 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^3 d (a-b)^2 (a+b)^3}",1,"(-((((-9*a^4*A*b + 19*a^2*A*b^3 - 16*A*b^5 + 45*a^5*B - 95*a^3*b^2*B + 56*a*b^4*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + (8*b*(-(a^3*A*b) + 4*a*A*b^3 + 5*a^4*B - 10*a^2*b^2*B + 2*b^4*B)*((a + b)*EllipticF[(c + d*x)/2, 2] - b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2]))/(a*(a + b)) + ((-3*a^3*A*b + 9*a*A*b^3 + 15*a^4*B - 29*a^2*b^2*B + 8*b^4*B)*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a*b*Sqrt[Sin[c + d*x]^2]))/((a - b)^2*(a + b)^2)) + (Sqrt[Cos[c + d*x]]*(2*a*b*(-5*a^3*A*b + 11*a*A*b^3 + 25*a^4*B - 47*a^2*b^2*B + 16*b^4*B)*Sin[c + d*x] + a^2*(-3*a^3*A*b + 9*a*A*b^3 + 15*a^4*B - 29*a^2*b^2*B + 8*b^4*B)*Sin[2*(c + d*x)] + 16*(-(a^2*b) + b^3)^2*B*Tan[c + d*x]))/((a^2 - b^2)^2*(b + a*Cos[c + d*x])^2))/(8*b^3*d)","A",1
593,1,570,523,7.3553425,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^3} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^3),x]","\frac{\sqrt{\cos (c+d x)} \left(\frac{a^4 B \sin (c+d x)-a^3 A b \sin (c+d x)}{2 b^3 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{-11 a^6 B \sin (c+d x)+7 a^5 A b \sin (c+d x)+17 a^4 b^2 B \sin (c+d x)-13 a^3 A b^3 \sin (c+d x)}{4 b^4 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 \sec (c+d x) (A b \sin (c+d x)-3 a B \sin (c+d x))}{b^4}+\frac{2 B \tan (c+d x) \sec (c+d x)}{3 b^3}\right)}{d}+\frac{\frac{\left(280 a^5 b B-120 a^4 A b^2-512 a^3 b^3 B+240 a^2 A b^4+160 a b^5 B-48 A b^6\right) \left(2 F\left(\left.\frac{1}{2} (c+d x)\right|2\right)-\frac{2 b \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}\right)}{a}+\frac{2 \left(105 a^6 B-45 a^5 A b-195 a^4 b^2 B+87 a^3 A b^3+72 a^2 b^4 B-24 a A b^5\right) \sin (c+d x) \cos (2 (c+d x)) \left(\left(a^2-2 b^2\right) \Pi \left(-\frac{a}{b};\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)+2 b (a+b) F\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)-2 a b E\left(\left.\sin ^{-1}\left(\sqrt{\cos (c+d x)}\right)\right|-1\right)\right)}{a^2 b \sqrt{1-\cos ^2(c+d x)} \left(2 \cos ^2(c+d x)-1\right)}+\frac{2 \left(315 a^6 B-135 a^5 A b-641 a^4 b^2 B+285 a^3 A b^3+328 a^2 b^4 B-168 a A b^5+16 b^6 B\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{a+b}}{48 b^4 d (a-b)^2 (a+b)^2}","\frac{a (A b-a B) \sin (c+d x)}{2 b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}+\frac{a \left(-7 a^3 B+3 a^2 A b+13 a b^2 B-9 A b^3\right) \sin (c+d x)}{4 b^2 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}-\frac{\left(-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right) F\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{12 b^3 d \left(a^2-b^2\right)^2}-\frac{\left(-35 a^4 B+15 a^3 A b+61 a^2 b^2 B-33 a A b^3-8 b^4 B\right) \sin (c+d x)}{12 b^3 d \left(a^2-b^2\right)^2 \cos ^{\frac{3}{2}}(c+d x)}-\frac{\left(-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right) E\left(\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d \left(a^2-b^2\right)^2}-\frac{a \left(-35 a^5 B+15 a^4 A b+86 a^3 b^2 B-38 a^2 A b^3-63 a b^4 B+35 A b^5\right) \Pi \left(\frac{2 a}{a+b};\left.\frac{1}{2} (c+d x)\right|2\right)}{4 b^4 d (a-b)^2 (a+b)^3}+\frac{\left(-35 a^5 B+15 a^4 A b+65 a^3 b^2 B-29 a^2 A b^3-24 a b^4 B+8 A b^5\right) \sin (c+d x)}{4 b^4 d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)}}",1,"((2*(-135*a^5*A*b + 285*a^3*A*b^3 - 168*a*A*b^5 + 315*a^6*B - 641*a^4*b^2*B + 328*a^2*b^4*B + 16*b^6*B)*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b) + ((-120*a^4*A*b^2 + 240*a^2*A*b^4 - 48*A*b^6 + 280*a^5*b*B - 512*a^3*b^3*B + 160*a*b^5*B)*(2*EllipticF[(c + d*x)/2, 2] - (2*b*EllipticPi[(2*a)/(a + b), (c + d*x)/2, 2])/(a + b)))/a + (2*(-45*a^5*A*b + 87*a^3*A*b^3 - 24*a*A*b^5 + 105*a^6*B - 195*a^4*b^2*B + 72*a^2*b^4*B)*Cos[2*(c + d*x)]*(-2*a*b*EllipticE[ArcSin[Sqrt[Cos[c + d*x]]], -1] + 2*b*(a + b)*EllipticF[ArcSin[Sqrt[Cos[c + d*x]]], -1] + (a^2 - 2*b^2)*EllipticPi[-(a/b), ArcSin[Sqrt[Cos[c + d*x]]], -1])*Sin[c + d*x])/(a^2*b*Sqrt[1 - Cos[c + d*x]^2]*(-1 + 2*Cos[c + d*x]^2)))/(48*(a - b)^2*b^4*(a + b)^2*d) + (Sqrt[Cos[c + d*x]]*((2*Sec[c + d*x]*(A*b*Sin[c + d*x] - 3*a*B*Sin[c + d*x]))/b^4 + (-(a^3*A*b*Sin[c + d*x]) + a^4*B*Sin[c + d*x])/(2*b^3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (7*a^5*A*b*Sin[c + d*x] - 13*a^3*A*b^3*Sin[c + d*x] - 11*a^6*B*Sin[c + d*x] + 17*a^4*b^2*B*Sin[c + d*x])/(4*b^4*(-a^2 + b^2)^2*(b + a*Cos[c + d*x])) + (2*B*Sec[c + d*x]*Tan[c + d*x])/(3*b^3)))/d","A",1
594,1,455,343,18.0809979,"\int \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(\frac{\left(115 a^2 A+28 a b B-16 A b^2\right) \sin (c+d x)}{210 a^2}+\frac{(7 a B+A b) \sin (2 (c+d x))}{35 a}+\frac{1}{14} A \sin (3 (c+d x))\right)}{d}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \sqrt{a+b \sec (c+d x)} \left(i a (a+b) \left(a^2 (25 A+63 B)-2 a b (3 A+7 B)+8 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(63 a^3 B+19 a^2 A b-14 a b^2 B+8 A b^3\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(63 a^3 B+19 a^2 A b-14 a b^2 B+8 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{105 a^3 d \sqrt{\sec (c+d x)} (a \cos (c+d x)+b)}","\frac{2 \left(25 a^2 A+7 a b B-4 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{105 a^2 d}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A-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 \left(63 a^3 B+19 a^2 A b-14 a b^2 B+8 A b^3\right) \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)}{105 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+A b) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{35 a d}+\frac{2 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}",1,"(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(((115*a^2*A - 16*A*b^2 + 28*a*b*B)*Sin[c + d*x])/(210*a^2) + ((A*b + 7*a*B)*Sin[2*(c + d*x)])/(35*a) + (A*Sin[3*(c + d*x)])/14))/d - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sqrt[a + b*Sec[c + d*x]]*((-I)*(a + b)*(19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^2 - 2*a*b*(3*A + 7*B) + a^2*(25*A + 63*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (19*a^2*A*b + 8*A*b^3 + 63*a^3*B - 14*a*b^2*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^3*d*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]])","C",0
595,1,353,267,14.9231834,"\int \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(a \sin (c+d x) (3 a A \cos (c+d x)+5 a B+A b)-\frac{\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(-\left(9 a^2 A+5 a b B-2 A b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(9 a^2 A+5 a b B-2 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i a (a+b) (9 a A+5 a B-2 A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}\right)}{15 a^2 d}","-\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 \left(9 a^2 A+5 a b B-2 A b^2\right) \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)}{15 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\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*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(a*(A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x] - ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(9*a^2*A - 2*A*b^2 + 5*a*b*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(9*a*A - 2*A*b + 5*a*B)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^2*A - 2*A*b^2 + 5*a*b*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/((b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2))))/(15*a^2*d)","C",0
596,1,305,201,8.9494731,"\int \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} \left(A \sin (c+d x)+\frac{\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left((3 a B+A b) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a (a+b) (A+3 B) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i (a+b) (3 a B+A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)}\right)}{3 d}","\frac{2 A \left(a^2-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 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}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A*Sin[c + d*x] + ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(I*(a + b)*(A*b + 3*a*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(A + 3*B)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (A*b + 3*a*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2))))/(3*d)","C",0
597,1,25347,208,29.933582,"\int \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 A \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)} E\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 b B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
598,1,52603,253,32.4523269,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{(2 a A+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)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a B+2 A b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}-\frac{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)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
599,1,77879,336,33.0191564,"\int \frac{\sqrt{a+b \sec (c+d x)} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[(Sqrt[a + b*Sec[c + d*x]]*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\left(a^2 (-B)+4 a A b+4 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(a B+4 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b d \sqrt{\cos (c+d x)}}+\frac{(3 a B+4 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)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{(a B+4 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)}{4 b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
600,1,540,427,18.6517416,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(\frac{\left(133 a^2 A+144 a b B+6 A b^2\right) \sin (2 (c+d x))}{630 a}+\frac{\left(345 a^3 B+402 a^2 A b+36 a b^2 B-16 A b^3\right) \sin (c+d x)}{630 a^2}+\frac{1}{126} (9 a B+10 A b) \sin (3 (c+d x))+\frac{1}{36} a A \sin (4 (c+d x))\right)}{d (a \cos (c+d x)+b)}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{3/2} \left(i a (a+b) \left(3 a^3 (49 A+25 B)+3 a^2 b (13 A+57 B)-6 a b^2 (A+3 B)+8 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{315 a^3 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(49 a^2 A+72 a b B+3 A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{315 a d}+\frac{2 \left(75 a^3 B+88 a^2 A b+9 a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{315 a^2 d}+\frac{2 \left(a^2-b^2\right) \left(75 a^3 B+39 a^2 A 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 \left(147 a^4 A+246 a^3 b B+33 a^2 A b^2-18 a b^3 B+8 A b^4\right) \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)}{315 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (9 a B+10 A b) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{63 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{9 d}",1,"(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(((402*a^2*A*b - 16*A*b^3 + 345*a^3*B + 36*a*b^2*B)*Sin[c + d*x])/(630*a^2) + ((133*a^2*A + 6*A*b^2 + 144*a*b*B)*Sin[2*(c + d*x)])/(630*a) + ((10*A*b + 9*a*B)*Sin[3*(c + d*x)])/126 + (a*A*Sin[4*(c + d*x)])/36))/(d*(b + a*Cos[c + d*x])) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 - 6*a*b^2*(A + 3*B) + 3*a^3*(49*A + 25*B) + 3*a^2*b*(13*A + 57*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (147*a^4*A + 33*a^2*A*b^2 + 8*A*b^4 + 246*a^3*b*B - 18*a*b^3*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^3*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","C",0
601,1,466,342,17.2537762,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(\frac{\left(115 a^2 A+168 a b B+12 A b^2\right) \sin (c+d x)}{210 a}+\frac{1}{35} (7 a B+8 A b) \sin (2 (c+d x))+\frac{1}{14} a A \sin (3 (c+d x))\right)}{d (a \cos (c+d x)+b)}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{3/2} \left(i a (a+b) \left(a^2 (25 A+63 B)+3 a b (19 A+7 B)-6 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(63 a^3 B+82 a^2 A b+21 a b^2 B-6 A b^3\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(63 a^3 B+82 a^2 A b+21 a b^2 B-6 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{105 a^2 d \sec ^{\frac{3}{2}}(c+d x) (a \cos (c+d x)+b)^2}","\frac{2 \left(25 a^2 A+42 a b B+3 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{105 a d}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A+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 \left(63 a^3 B+82 a^2 A b+21 a b^2 B-6 A b^3\right) \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)}{105 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (7 a B+8 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 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{7 d}",1,"(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(((115*a^2*A + 12*A*b^2 + 168*a*b*B)*Sin[c + d*x])/(210*a) + ((8*A*b + 7*a*B)*Sin[2*(c + d*x)])/35 + (a*A*Sin[3*(c + d*x)])/14))/(d*(b + a*Cos[c + d*x])) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-6*A*b^2 + 3*a*b*(19*A + 7*B) + a^2*(25*A + 63*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (82*a^2*A*b - 6*A*b^3 + 63*a^3*B + 21*a*b^2*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a^2*d*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2))","C",0
602,1,369,266,14.538556,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \left(2 \sin (c+d x) (a \cos (c+d x)+b) (3 a A \cos (c+d x)+5 a B+6 A b)-\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(-\left(9 a^2 A+20 a b B+3 A b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(9 a^2 A+20 a b B+3 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i a (a+b) (a (9 A+5 B)+3 b (A+5 B)) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a \sec ^{\frac{3}{2}}(c+d x)}\right)}{15 d (a \cos (c+d x)+b)^2}","\frac{2 \left(a^2-b^2\right) (5 a B+3 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 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(9 a^2 A+20 a b B+3 A b^2\right) \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)}{15 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 (5 a B+6 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 d}",1,"(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(2*(b + a*Cos[c + d*x])*(6*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x] - (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(9*a^2*A + 3*A*b^2 + 20*a*b*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(3*b*(A + 5*B) + a*(9*A + 5*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^2*A + 3*A*b^2 + 20*a*b*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a*Sec[c + d*x]^(3/2))))/(15*d*(b + a*Cos[c + d*x])^2)","C",0
603,1,45958,276,34.2204634,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 \left(a^2 A+3 a b B-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 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 (3 a B+4 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 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 d}+\frac{2 b^2 B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
604,1,66581,272,33.1803372,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{\left(2 a^2 B+2 a A b+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)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(2 a A-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)}{d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b (3 a B+2 A b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{d \sqrt{\cos (c+d x)}}",1,"Result too large to show","C",0
605,1,79375,339,33.6171061,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{\left(8 a^2 A+7 a b B+4 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)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(3 a^2 B+12 a A b+4 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(5 a B+4 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}-\frac{(5 a B+4 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)}{4 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
606,1,104716,421,33.9032799,"\int \frac{(a+b \sec (c+d x))^{3/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\left(3 a^2 B+30 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 b d \sqrt{\cos (c+d x)}}+\frac{\left(17 a^2 B+42 a A b+16 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)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left(3 a^2 B+30 a A b+16 b^2 B\right) \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)}{24 b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(a^3 (-B)+6 a^2 A b+12 a b^2 B+8 A b^3\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{(7 a B+6 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{12 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Result too large to show","C",0
607,1,626,519,20.6445142,"\int \cos ^{\frac{11}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(11/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{\left(513 a^2 A+836 a b B+452 A b^2\right) \sin (3 (c+d x))}{5544}+\frac{1}{88} a^2 A \sin (5 (c+d x))+\frac{\left(1463 a^3 B+3095 a^2 A b+1650 a b^2 B+30 A b^3\right) \sin (2 (c+d x))}{6930 a}+\frac{\left(6525 a^4 A+16434 a^3 b B+9330 a^2 A b^2+440 a b^3 B-160 A b^4\right) \sin (c+d x)}{13860 a^2}+\frac{1}{396} a (11 a B+23 A b) \sin (4 (c+d x))\right)}{d (a \cos (c+d x)+b)^2}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} \left(i a (a+b) \left(3 a^4 (225 A+539 B)+6 a^3 b (505 A+209 B)+15 a^2 b^2 (19 A+121 B)-10 a b^3 (3 A+11 B)+40 A b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3465 a^3 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(81 a^2 A+209 a b B+113 A b^2\right) \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{693 d}+\frac{2 \left(539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{3465 a d}+\frac{2 \left(675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3465 a^2 d}+\frac{2 \left(a^2-b^2\right) \left(675 a^4 A+1254 a^3 b B+285 a^2 A b^2-110 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 \left(1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right) \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)}{3465 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a (11 a B+14 A b) \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{99 d}+\frac{2 a A \sin (c+d x) \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{11 d}",1,"(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(((6525*a^4*A + 9330*a^2*A*b^2 - 160*A*b^4 + 16434*a^3*b*B + 440*a*b^3*B)*Sin[c + d*x])/(13860*a^2) + ((3095*a^2*A*b + 30*A*b^3 + 1463*a^3*B + 1650*a*b^2*B)*Sin[2*(c + d*x)])/(6930*a) + ((513*a^2*A + 452*A*b^2 + 836*a*b*B)*Sin[3*(c + d*x)])/5544 + (a*(23*A*b + 11*a*B)*Sin[4*(c + d*x)])/396 + (a^2*A*Sin[5*(c + d*x)])/88))/(d*(b + a*Cos[c + d*x])^2) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*((-I)*(a + b)*(3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(40*A*b^4 - 10*a*b^3*(3*A + 11*B) + 15*a^2*b^2*(19*A + 121*B) + 6*a^3*b*(505*A + 209*B) + 3*a^4*(225*A + 539*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (3705*a^4*A*b + 255*a^2*A*b^3 + 40*A*b^5 + 1617*a^5*B + 3069*a^3*b^2*B - 110*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3465*a^3*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","C",0
608,1,542,425,19.4101713,"\int \cos ^{\frac{9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(9/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{1}{630} \left(133 a^2 A+270 a b B+150 A b^2\right) \sin (2 (c+d x))+\frac{1}{36} a^2 A \sin (4 (c+d x))+\frac{\left(345 a^3 B+747 a^2 A b+540 a b^2 B+20 A b^3\right) \sin (c+d x)}{630 a}+\frac{1}{126} a (9 a B+19 A b) \sin (3 (c+d x))\right)}{d (a \cos (c+d x)+b)^2}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} \left(i a (a+b) \left(3 a^3 (49 A+25 B)+6 a^2 b (19 A+60 B)+15 a b^2 (11 A+3 B)-10 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{315 a^2 d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(49 a^2 A+135 a b B+75 A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{315 d}+\frac{2 \left(75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{315 a d}+\frac{2 \left(a^2-b^2\right) \left(75 a^3 B+114 a^2 A 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 \left(147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right) \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)}{315 a^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a (3 a B+4 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 A \sin (c+d x) \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{9 d}",1,"(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(((747*a^2*A*b + 20*A*b^3 + 345*a^3*B + 540*a*b^2*B)*Sin[c + d*x])/(630*a) + ((133*a^2*A + 150*A*b^2 + 270*a*b*B)*Sin[2*(c + d*x)])/630 + (a*(19*A*b + 9*a*B)*Sin[3*(c + d*x)])/126 + (a^2*A*Sin[4*(c + d*x)])/36))/(d*(b + a*Cos[c + d*x])^2) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*((-I)*(a + b)*(147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-10*A*b^3 + 15*a*b^2*(11*A + 3*B) + 3*a^3*(49*A + 25*B) + 6*a^2*b*(19*A + 60*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (147*a^4*A + 279*a^2*A*b^2 - 10*A*b^4 + 435*a^3*b*B + 45*a*b^3*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(315*a^2*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","C",0
609,1,470,340,19.4120422,"\int \cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\frac{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left(\frac{1}{210} \left(115 a^2 A+308 a b B+180 A b^2\right) \sin (c+d x)+\frac{1}{14} a^2 A \sin (3 (c+d x))+\frac{1}{35} a (7 a B+15 A b) \sin (2 (c+d x))\right)}{d (a \cos (c+d x)+b)^2}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a+b \sec (c+d x))^{5/2} \left(i a (a+b) \left(a^2 (25 A+63 B)+8 a b (15 A+7 B)+15 b^2 (A+7 B)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(63 a^3 B+145 a^2 A b+161 a b^2 B+15 A b^3\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(63 a^3 B+145 a^2 A b+161 a b^2 B+15 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{105 a d \sec ^{\frac{5}{2}}(c+d x) (a \cos (c+d x)+b)^3}","\frac{2 \left(25 a^2 A+77 a b B+45 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{105 d}+\frac{2 \left(a^2-b^2\right) \left(25 a^2 A+56 a b B+15 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 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(63 a^3 B+145 a^2 A b+161 a b^2 B+15 A b^3\right) \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)}{105 a d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 a (7 a B+10 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 A \sin (c+d x) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{7 d}",1,"(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(((115*a^2*A + 180*A*b^2 + 308*a*b*B)*Sin[c + d*x])/210 + (a*(15*A*b + 7*a*B)*Sin[2*(c + d*x)])/35 + (a^2*A*Sin[3*(c + d*x)])/14))/(d*(b + a*Cos[c + d*x])^2) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(a + b*Sec[c + d*x])^(5/2)*((-I)*(a + b)*(145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(15*b^2*(A + 7*B) + 8*a*b*(15*A + 7*B) + a^2*(25*A + 63*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (145*a^2*A*b + 15*A*b^3 + 63*a^3*B + 161*a*b^2*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(105*a*d*(b + a*Cos[c + d*x])^3*Sec[c + d*x]^(5/2))","C",0
610,1,49609,342,35.4885829,"\int \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{2 \left(9 a^2 A+35 a b B+23 A b^2\right) \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)}{15 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(5 a^3 B+8 a^2 A 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 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a (5 a B+8 A b) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 d}+\frac{2 a 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^3 B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
611,1,73332,349,34.1522902,"\int \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{\left(6 a^2 B+14 a A b-3 b^2 B\right) \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 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(2 a^3 A+12 a^2 b B+4 a A b^2+3 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)}{3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b^2 (5 a B+2 A b) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{b (2 a A-3 b B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{3 d \sqrt{\cos (c+d x)}}+\frac{2 a A \sin (c+d x) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}{3 d}",1,"Result too large to show","C",0
612,1,97208,359,34.735125,"\int \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx","Integrate[Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]),x]","\text{Result too large to show}","\frac{\left(8 a^2 A-9 a b B-4 A b^2\right) \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 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{b \left(15 a^2 B+20 a A b+4 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(8 a^3 B+16 a^2 A b+11 a b^2 B+4 A b^3\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} F\left(\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b (7 a B+4 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \sqrt{\cos (c+d x)}}+\frac{b B \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{2 d \sqrt{\cos (c+d x)}}",1,"Result too large to show","C",0
613,1,106199,422,34.6194716,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sqrt{\cos (c+d x)}} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[Cos[c + d*x]],x]","\text{Result too large to show}","\frac{\left(33 a^2 B+54 a A b+16 b^2 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d \sqrt{\cos (c+d x)}}-\frac{\left(33 a^2 B+54 a A b+16 b^2 B\right) \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)}{24 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(48 a^3 A+59 a^2 b B+66 a A b^2+16 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)}{24 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(5 a^3 B+30 a^2 A b+20 a b^2 B+8 A b^3\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{b (3 a B+2 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{b B \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{3 d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
614,1,131553,513,35.4025233,"\int \frac{(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\cos ^{\frac{3}{2}}(c+d x)} \, dx","Integrate[((a + b*Sec[c + d*x])^(5/2)*(A + B*Sec[c + d*x]))/Cos[c + d*x]^(3/2),x]","\text{Result too large to show}","\frac{\left(59 a^2 B+104 a A b+36 b^2 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{96 d \cos ^{\frac{3}{2}}(c+d x)}+\frac{\left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{192 b d \sqrt{\cos (c+d x)}}+\frac{\left(133 a^3 B+472 a^2 A b+356 a b^2 B+128 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)}{192 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{\left(15 a^3 B+264 a^2 A b+284 a b^2 B+128 A b^3\right) \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)}{192 b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{\left(-5 a^4 B+40 a^3 A b+120 a^2 b^2 B+160 a A b^3+48 b^4 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)}{64 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{b (11 a B+8 A b) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{24 d \cos ^{\frac{5}{2}}(c+d x)}+\frac{b B \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{4 d \cos ^{\frac{5}{2}}(c+d x)}",1,"Result too large to show","C",0
615,1,363,280,16.4709619,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 a \sin (c+d x) (a \cos (c+d x)+b) (3 a A \cos (c+d x)+5 a B-4 A b)+\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(\left(9 a^2 A-10 a b B+8 A b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a \left(a^2 (9 A+5 B)+2 a b (A-5 B)+8 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i (a+b) \left(9 a^2 A-10 a b B+8 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}}{15 a^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","-\frac{2 (4 A b-5 a B) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^2 d}+\frac{2 \left(9 a^2 A-10 a b B+8 A b^2\right) \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)}{15 a^3 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}-\frac{2 \left(-5 a^3 B+7 a^2 A 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 A \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a d}",1,"(2*a*(b + a*Cos[c + d*x])*(-4*A*b + 5*a*B + 3*a*A*Cos[c + d*x])*Sin[c + d*x] + (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(I*(a + b)*(9*a^2*A + 8*A*b^2 - 10*a*b*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(8*A*b^2 + 2*a*b*(A - 5*B) + a^2*(9*A + 5*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (9*a^2*A + 8*A*b^2 - 10*a*b*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2))/(15*a^3*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","C",0
616,1,311,212,9.2334409,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \left(a A \sin (c+d x) (a \cos (c+d x)+b)-\frac{\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left((2 A b-3 a B) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)+i a (a (A+3 B)-2 A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i (a+b) (3 a B-2 A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\sec ^{\frac{3}{2}}(c+d x)}\right)}{3 a^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}","\frac{2 \left(a^2 A-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*(a*A*(b + a*Cos[c + d*x])*Sin[c + d*x] - ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(-2*A*b + 3*a*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(-2*A*b + a*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (2*A*b - 3*a*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/Sec[c + d*x]^(3/2)))/(3*a^2*d*Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","C",0
617,1,260,150,7.0508965,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{\sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/Sqrt[a + b*Sec[c + d*x]],x]","\frac{2 \sqrt{\cos (c+d x)} \sqrt{\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)} (A+B \sec (c+d x)) \left(-i a (A+B) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+A \tan \left(\frac{1}{2} (c+d x)\right) \sqrt{\sec ^2\left(\frac{1}{2} (c+d x)\right)} (a \cos (c+d x)+b)+i A (a+b) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a d \sqrt{\sec (c+d x)} \sqrt{a+b \sec (c+d x)} (A \cos (c+d x)+B)}","\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 (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)}}",1,"(2*Sqrt[Cos[c + d*x]]*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*(A + B*Sec[c + d*x])*(I*A*(a + b)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(A + B)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + A*(b + a*Cos[c + d*x])*Sqrt[Sec[(c + d*x)/2]^2]*Tan[(c + d*x)/2]))/(a*d*(B + A*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*Sqrt[a + b*Sec[c + d*x]])","C",1
618,1,16611,138,28.8085695,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","\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)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 B \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
619,1,51168,256,32.9538516,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","\frac{(2 A b-a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \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{B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b d \sqrt{\cos (c+d x)}}+\frac{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{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)}{b d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"Result too large to show","C",0
620,1,77909,344,32.915725,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*Sqrt[a + b*Sec[c + d*x]]),x]","\text{Result too large to show}","-\frac{\left(-3 a^2 B+4 a A b-4 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{(4 A b-3 a B) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b^2 d \sqrt{\cos (c+d x)}}-\frac{(4 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)}{4 b^2 d \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(4 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)}{4 b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{B \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 b d \cos ^{\frac{3}{2}}(c+d x)}",1,"Result too large to show","C",0
621,1,533,423,20.9849764,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a \cos (c+d x)+b)^2 \left(\frac{2 (5 a B-9 A b) \sin (c+d x)}{15 a^3}+\frac{A \sin (2 (c+d x))}{5 a^2}+\frac{2 \left(A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)\right)}{a^3 \left(a^2-b^2\right) (a \cos (c+d x)+b)}\right)}{d \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b) \left(i a (a+b) \left(a^3 (9 A+5 B)-6 a^2 b (2 A+5 B)+4 a b^2 (9 A+10 B)-48 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(9 a^4 A-25 a^3 b B+24 a^2 A b^2+40 a b^3 B-48 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(9 a^4 A-25 a^3 b B+24 a^2 A b^2+40 a b^3 B-48 A b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{15 a^4 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}","\frac{2 \left(a^2 A+5 a b B-6 A b^2\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{5 a^2 d \left(a^2-b^2\right)}+\frac{2 b (A b-a B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(-5 a^3 B+9 a^2 A b+20 a b^2 B-24 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)}-\frac{2 \left(-5 a^3 B+12 a^2 A 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 \left(9 a^4 A-25 a^3 b B+24 a^2 A b^2+40 a b^3 B-48 A b^4\right) \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)}{15 a^4 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^2*((2*(-9*A*b + 5*a*B)*Sin[c + d*x])/(15*a^3) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x]))/(a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(5*a^2)))/(d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-48*A*b^3 - 6*a^2*b*(2*A + 5*B) + a^3*(9*A + 5*B) + 4*a*b^2*(9*A + 10*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^4*A + 24*a^2*A*b^2 - 48*A*b^4 - 25*a^3*b*B + 40*a*b^3*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^4*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2))","C",0
622,1,469,326,18.0298884,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{(a \cos (c+d x)+b)^2 \left(\frac{2 A \sin (c+d x)}{3 a^2}-\frac{2 \left(A b^3 \sin (c+d x)-a b^2 B \sin (c+d x)\right)}{a^2 \left(a^2-b^2\right) (a \cos (c+d x)+b)}\right)}{d \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b) \left(i a \left(a^2-a b-2 b^2\right) (a (A+3 B)-4 A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(3 a^3 B-5 a^2 A b-6 a b^2 B+8 A b^3\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(3 a^3 B-5 a^2 A b-6 a b^2 B+8 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a^3 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}","\frac{2 \left(a^2 A+3 a b B-4 A b^2\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^2 d \left(a^2-b^2\right)}+\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{a d \left(a^2-b^2\right) \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2 A-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 \left(-3 a^3 B+5 a^2 A b+6 a b^2 B-8 A b^3\right) \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^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^2*((2*A*Sin[c + d*x])/(3*a^2) - (2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x]))/(a^2*(a^2 - b^2)*(b + a*Cos[c + d*x]))))/(d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)) - (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*Sec[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a^2 - a*b - 2*b^2)*(-4*A*b + a*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-5*a^2*A*b + 8*A*b^3 + 3*a^3*B - 6*a*b^2*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a^3*(a^2 - b^2)*d*(a + b*Sec[c + d*x])^(3/2))","C",0
623,1,445,235,16.5158328,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]","\frac{2 (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(A b^2 \sin (c+d x)-a b B \sin (c+d x)\right)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2} (A \cos (c+d x)+B)}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sqrt{\sec (c+d x)} \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b) (A+B \sec (c+d x)) \left(-\left(a^2 A+a b B-2 A b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(a^2 A+a b B-2 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i a (a+b) (a (A+B)-2 A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{a^2 d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2} (A \cos (c+d x)+B)}","\frac{2 b (A b-a B) \sin (c+d x)}{a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^2 A+a b B-2 A b^2\right) \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^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*(b + a*Cos[c + d*x])*(A + B*Sec[c + d*x])*(A*b^2*Sin[c + d*x] - a*b*B*Sin[c + d*x]))/(a*(a^2 - b^2)*d*Sqrt[Cos[c + d*x]]*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2)) - (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])*Sqrt[Sec[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])*((-I)*(a + b)*(a^2*A - 2*A*b^2 + a*b*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-2*A*b + a*(A + B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a^2*A - 2*A*b^2 + a*b*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^2*(a^2 - b^2)*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(3/2))","C",0
624,1,328,215,10.9248007,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(3/2)),x]","\frac{2 (a \cos (c+d x)+b) \left(\frac{(a B-A b) \sin (c+d x)}{a^2-b^2}+\frac{\left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left((A b-a B) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i a (a+b) (A-B) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-i (a+b) (a B-A b) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\left(a^3-a b^2\right) \sec ^{\frac{3}{2}}(c+d x)}\right)}{d \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}","-\frac{2 (A b-a B) \sin (c+d x)}{d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 (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)}{a 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)}}",1,"(2*(b + a*Cos[c + d*x])*(((-(A*b) + a*B)*Sin[c + d*x])/(a^2 - b^2) + ((Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(-(A*b) + a*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(A - B)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + (A*b - a*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/((a^3 - a*b^2)*Sec[c + d*x]^(3/2))))/(d*Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2))","C",0
625,1,50122,220,32.3686323,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 (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)}{b d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
626,1,95694,371,33.8406911,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}-\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}+\frac{\left(-3 a^2 B+2 a A b+b^2 B\right) \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^2 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 A b-3 a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{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)}{b d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
627,1,140027,487,35.2761488,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(3/2)),x]","\text{Result too large to show}","-\frac{\left(-5 a^2 B+4 a A b+b^2 B\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{2 b^2 d \left(a^2-b^2\right) \cos ^{\frac{3}{2}}(c+d x)}+\frac{2 a (A b-a B) \sin (c+d x)}{b d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}-\frac{\left(-15 a^2 B+12 a A b-4 b^2 B\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{4 b^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{\left(-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right) \sin (c+d x) \sqrt{a+b \sec (c+d x)}}{4 b^3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)}}-\frac{\left(-15 a^3 B+12 a^2 A b+7 a b^2 B-4 A b^3\right) \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^3 d \left(a^2-b^2\right) \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(4 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)}{4 b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
628,1,4179,588,25.4119175,"\int \frac{\cos ^{\frac{5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(5/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 b (A b-a B) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left(-9 a^3 B+12 a^2 A b+5 a b^2 B-8 A b^3\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x)}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(3 a^4 A+50 a^3 b B-71 a^2 A b^2-30 a b^3 B+48 A b^4\right) \sin (c+d x) \cos ^{\frac{3}{2}}(c+d x) \sqrt{a+b \sec (c+d x)}}{15 a^3 d \left(a^2-b^2\right)^2}-\frac{2 \left(-5 a^5 B+14 a^4 A b+65 a^3 b^2 B-98 a^2 A b^3-40 a b^4 B+64 A b^5\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{15 a^4 d \left(a^2-b^2\right)^2}-\frac{2 \left(-5 a^5 B+17 a^4 A b-80 a^3 b^2 B+116 a^2 A b^3+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 \left(9 a^6 A-40 a^5 b B+55 a^4 A b^2+140 a^3 b^3 B-212 a^2 A b^4-80 a b^5 B+128 A b^6\right) \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)}{15 a^5 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*((2*(-14*A*b + 5*a*B)*Sin[c + d*x])/(15*a^4) - (2*(A*b^5*Sin[c + d*x] - a*b^4*B*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-15*a^2*A*b^4*Sin[c + d*x] + 11*A*b^6*Sin[c + d*x] + 12*a^3*b^3*B*Sin[c + d*x] - 8*a*b^5*B*Sin[c + d*x]))/(3*a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + (A*Sin[2*(c + d*x)])/(5*a^3)))/(d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)) - (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*((3*a^2*A*Sqrt[Cos[c + d*x]])/(5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (11*A*b^2*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (212*A*b^4*Sqrt[Cos[c + d*x]])/(15*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (128*A*b^6*Sqrt[Cos[c + d*x]])/(15*a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a*b*B*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (28*b^3*B*Sqrt[Cos[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*b^5*B*Sqrt[Cos[c + d*x]])/(3*a^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (8*a*A*b*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (44*A*b^3*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (32*A*b^5*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(15*a^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (a^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (7*b^2*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*b^4*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + a^5*(9*A + 5*B) + 8*a^3*b^2*(9*A + 10*B) + 4*a^2*b^3*(-29*A + 15*B) - a^4*b*(17*A + 45*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(15*a^5*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*(-1/15*(Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + a^5*(9*A + 5*B) + 8*a^3*b^2*(9*A + 10*B) + 4*a^2*b^3*(-29*A + 15*B) - a^4*b*(17*A + 45*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^4*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)) + (Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + a^5*(9*A + 5*B) + 8*a^3*b^2*(9*A + 10*B) + 4*a^2*b^3*(-29*A + 15*B) - a^4*b*(17*A + 45*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(5*a^5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + a^5*(9*A + 5*B) + 8*a^3*b^2*(9*A + 10*B) + 4*a^2*b^3*(-29*A + 15*B) - a^4*b*(17*A + 45*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + a^5*(9*A + 5*B) + 8*a^3*b^2*(9*A + 10*B) + 4*a^2*b^3*(-29*A + 15*B) - a^4*b*(17*A + 45*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + a^5*(9*A + 5*B) + 8*a^3*b^2*(9*A + 10*B) + 4*a^2*b^3*(-29*A + 15*B) - a^4*b*(17*A + 45*B))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(15*a^5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(128*A*b^5 - 16*a*b^4*(6*A + 5*B) + a^5*(9*A + 5*B) + 8*a^3*b^2*(9*A + 10*B) + 4*a^2*b^3*(-29*A + 15*B) - a^4*b*(17*A + 45*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (9*a^6*A + 55*a^4*A*b^2 - 212*a^2*A*b^4 + 128*A*b^6 - 40*a^5*b*B + 140*a^3*b^3*B - 80*a*b^5*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(5*a^5*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]])))","C",0
629,1,3758,472,24.2614219,"\int \frac{\cos ^{\frac{3}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Cos[c + d*x]^(3/2)*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\text{Result too large to show}","\frac{2 b (A b-a B) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a d \left(a^2-b^2\right) (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left(-7 a^3 B+10 a^2 A b+3 a b^2 B-6 A b^3\right) \sin (c+d x) \sqrt{\cos (c+d x)}}{3 a^2 d \left(a^2-b^2\right)^2 \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(a^4 A+8 a^3 b B-13 a^2 A b^2-4 a b^3 B+8 A b^4\right) \sin (c+d x) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}{3 a^3 d \left(a^2-b^2\right)^2}+\frac{2 \left(a^4 A-9 a^3 b B+16 a^2 A b^2+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 \left(-3 a^5 B+8 a^4 A b+15 a^3 b^2 B-28 a^2 A b^3-8 a b^4 B+16 A b^5\right) \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^4 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*((2*A*Sin[c + d*x])/(3*a^3) + (2*(A*b^4*Sin[c + d*x] - a*b^3*B*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) + (2*(-12*a^2*A*b^3*Sin[c + d*x] + 8*A*b^5*Sin[c + d*x] + 9*a^3*b^2*B*Sin[c + d*x] - 5*a*b^4*B*Sin[c + d*x]))/(3*a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)) - (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*((-8*a*A*b*Sqrt[Cos[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (28*A*b^3*Sqrt[Cos[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (16*A*b^5*Sqrt[Cos[c + d*x]])/(3*a^3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*B*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) - (5*b^2*B*Sqrt[Cos[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (8*b^4*B*Sqrt[Cos[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]*Sqrt[Sec[c + d*x]]) + (a^2*A*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (7*A*b^2*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (4*A*b^4*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a^2*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*a*b*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/((a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) + (2*b^3*B*Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]])/(3*a*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]))*Sec[c + d*x]^(5/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a^4*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2)*(-1/3*(Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^3*(a^2 - b^2)^2*(b + a*Cos[c + d*x])^(3/2)) + (Sqrt[Cos[c + d*x]]*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*Sin[c + d*x]*((-I)*(a + b)*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (2*Cos[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(-1/2*((-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(5/2)) - I*(a + b)*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + I*a*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Tan[(c + d*x)/2] + a*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*(Sec[(c + d*x)/2]^2)^(3/2)*Sin[c + d*x]*Tan[(c + d*x)/2] - (3*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]^2)/2 - ((I/2)*(a + b)*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + ((I/2)*a*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*(-((a*Sec[(c + d*x)/2]^2*Sin[c + d*x])/(a + b)) + ((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2*Tan[(c + d*x)/2])/(a + b)))/Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (a*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)]) + ((a + b)*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*Sec[(c + d*x)/2]^4*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)]*Sqrt[1 + ((-a + b)*Tan[(c + d*x)/2]^2)/(a + b)])/(2*Sqrt[1 + Tan[(c + d*x)/2]^2])))/(3*a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]]) - (Cos[c + d*x]^(3/2)*Sqrt[Cos[(c + d*x)/2]^2*Sec[c + d*x]]*((-I)*(a + b)*(-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(-16*A*b^4 + 2*a^2*b^2*(8*A - 3*B) - 9*a^3*b*(A + B) + 4*a*b^3*(3*A + 2*B) + a^4*(A + 3*B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-8*a^4*A*b + 28*a^2*A*b^3 - 16*A*b^5 + 3*a^5*B - 15*a^3*b^2*B + 8*a*b^4*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2])*(-(Cos[(c + d*x)/2]*Sec[c + d*x]*Sin[(c + d*x)/2]) + Cos[(c + d*x)/2]^2*Sec[c + d*x]*Tan[c + d*x]))/(a^4*(a^2 - b^2)^2*Sqrt[b + a*Cos[c + d*x]])))","C",0
630,1,621,368,19.7834622,"\int \frac{\sqrt{\cos (c+d x)} (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(Sqrt[Cos[c + d*x]]*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(5/2),x]","\frac{(a \cos (c+d x)+b)^3 (A+B \sec (c+d x)) \left(-\frac{2 \left(A b^3 \sin (c+d x)-a b^2 B \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right) (a \cos (c+d x)+b)^2}-\frac{2 \left(6 a^3 b B \sin (c+d x)-9 a^2 A b^2 \sin (c+d x)-2 a b^3 B \sin (c+d x)+5 A b^4 \sin (c+d x)\right)}{3 a^2 \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}\right)}{d \cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A \cos (c+d x)+B)}-\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{3}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b)^2 (A+B \sec (c+d x)) \left(i a (a+b) \left(3 a^3 (A+B)+3 a^2 b (B-3 A)-2 a b^2 (3 A+B)+8 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 a b^3 B+8 A b^4\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 a b^3 B+8 A b^4\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a d \left(a^3-a b^2\right)^2 (a+b \sec (c+d x))^{5/2} (A \cos (c+d x)+B)}","\frac{2 b (A b-a B) \sin (c+d x)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 b \left(-5 a^3 B+8 a^2 A b+a b^2 B-4 A b^3\right) \sin (c+d x)}{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 \left(-3 a^3 B+9 a^2 A 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 \left(3 a^4 A+6 a^3 b B-15 a^2 A b^2-2 a b^3 B+8 A b^4\right) \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^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^3*(A + B*Sec[c + d*x])*((-2*(A*b^3*Sin[c + d*x] - a*b^2*B*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)*(b + a*Cos[c + d*x])^2) - (2*(-9*a^2*A*b^2*Sin[c + d*x] + 5*A*b^4*Sin[c + d*x] + 6*a^3*b*B*Sin[c + d*x] - 2*a*b^3*B*Sin[c + d*x]))/(3*a^2*(a^2 - b^2)^2*(b + a*Cos[c + d*x]))))/(d*Cos[c + d*x]^(3/2)*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(5/2)) - (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(3/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*(A + B*Sec[c + d*x])*((-I)*(a + b)*(3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(8*A*b^3 + 3*a^2*b*(-3*A + B) + 3*a^3*(A + B) - 2*a*b^2*(3*A + B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (3*a^4*A - 15*a^2*A*b^2 + 8*A*b^4 + 6*a^3*b*B - 2*a*b^3*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a*(a^3 - a*b^2)^2*d*(B + A*Cos[c + d*x])*(a + b*Sec[c + d*x])^(5/2))","C",0
631,1,463,346,18.0280628,"\int \frac{A+B \sec (c+d x)}{\sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Sqrt[Cos[c + d*x]]*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{(a \cos (c+d x)+b)^2 \left(\frac{2 \sin (c+d x) \left(a \left(3 a^3 B-6 a^2 A b+a b^2 B+2 A b^3\right) \cos (c+d x)+b \left(2 a^3 B-5 a^2 A b+2 a b^2 B+A b^3\right)\right)}{a \left(a^2-b^2\right)^2 (a \cos (c+d x)+b)}+\frac{2 \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} \left(-i a (a+b) \left(3 a^2 (A-B)+a b (3 A-B)-2 A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)-\left(3 a^3 B-6 a^2 A b+a b^2 B+2 A b^3\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(3 a^3 B-6 a^2 A b+a b^2 B+2 A b^3\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{\left(a^3-a b^2\right)^2 \sec ^{\frac{3}{2}}(c+d x)}\right)}{3 d \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}","-\frac{2 (A b-a B) \sin (c+d x)}{3 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}+\frac{2 \left(3 a^2 A-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 \left(-2 a^3 B+5 a^2 A b-2 a b^2 B-A b^3\right) \sin (c+d x)}{3 a d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 \left(-3 a^3 B+6 a^2 A b-a b^2 B-2 A b^3\right) \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 \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}",1,"((b + a*Cos[c + d*x])^2*((2*(b*(-5*a^2*A*b + A*b^3 + 2*a^3*B + 2*a*b^2*B) + a*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*Cos[c + d*x])*Sin[c + d*x])/(a*(a^2 - b^2)^2*(b + a*Cos[c + d*x])) + (2*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - I*a*(a + b)*(-2*A*b^2 + 3*a^2*(A - B) + a*b*(3*A - B))*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (-6*a^2*A*b + 2*A*b^3 + 3*a^3*B + a*b^2*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/((a^3 - a*b^2)^2*Sec[c + d*x]^(3/2))))/(3*d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2))","C",0
632,1,487,329,16.7959847,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{3}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(3/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\frac{(a \cos (c+d x)+b)^3 \left(\frac{2 (A b \sin (c+d x)-a B \sin (c+d x))}{3 \left(b^2-a^2\right) (a \cos (c+d x)+b)^2}+\frac{2 \left(3 a^2 A \sin (c+d x)-4 a b B \sin (c+d x)+A b^2 \sin (c+d x)\right)}{3 \left(b^2-a^2\right)^2 (a \cos (c+d x)+b)}\right)}{d \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}}+\frac{2 \cos ^{\frac{3}{2}}(c+d x) \sec ^{\frac{5}{2}}(c+d x) \left(\cos ^2\left(\frac{1}{2} (c+d x)\right) \sec (c+d x)\right)^{3/2} (a \cos (c+d x)+b)^2 \left(-\left(3 a^2 A-4 a b B+A b^2\right) \tan \left(\frac{1}{2} (c+d x)\right) \sec ^2\left(\frac{1}{2} (c+d x)\right)^{3/2} (a \cos (c+d x)+b)-i (a+b) \left(3 a^2 A-4 a b B+A b^2\right) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} E\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)+i a (a+b) (3 a A-a B+A b-3 b B) \sec ^2\left(\frac{1}{2} (c+d x)\right) \sqrt{\frac{\sec ^2\left(\frac{1}{2} (c+d x)\right) (a \cos (c+d x)+b)}{a+b}} F\left(i \sinh ^{-1}\left(\tan \left(\frac{1}{2} (c+d x)\right)\right)|\frac{b-a}{a+b}\right)\right)}{3 a d \left(a^2-b^2\right)^2 (a+b \sec (c+d x))^{5/2}}","\frac{2 a (A b-a B) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} (a+b \sec (c+d x))^{3/2}}-\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)}{3 a d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 \left(3 a^2 A-4 a b B+A b^2\right) \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 \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 \left(a^3 B+2 a^2 A b-5 a b^2 B+2 A b^3\right) \sin (c+d x)}{3 b d \left(a^2-b^2\right)^2 \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"((b + a*Cos[c + d*x])^3*((2*(A*b*Sin[c + d*x] - a*B*Sin[c + d*x]))/(3*(-a^2 + b^2)*(b + a*Cos[c + d*x])^2) + (2*(3*a^2*A*Sin[c + d*x] + A*b^2*Sin[c + d*x] - 4*a*b*B*Sin[c + d*x]))/(3*(-a^2 + b^2)^2*(b + a*Cos[c + d*x]))))/(d*Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)) + (2*Cos[c + d*x]^(3/2)*(b + a*Cos[c + d*x])^2*Sec[c + d*x]^(5/2)*(Cos[(c + d*x)/2]^2*Sec[c + d*x])^(3/2)*((-I)*(a + b)*(3*a^2*A + A*b^2 - 4*a*b*B)*EllipticE[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] + I*a*(a + b)*(3*a*A + A*b - a*B - 3*b*B)*EllipticF[I*ArcSinh[Tan[(c + d*x)/2]], (-a + b)/(a + b)]*Sec[(c + d*x)/2]^2*Sqrt[((b + a*Cos[c + d*x])*Sec[(c + d*x)/2]^2)/(a + b)] - (3*a^2*A + A*b^2 - 4*a*b*B)*(b + a*Cos[c + d*x])*(Sec[(c + d*x)/2]^2)^(3/2)*Tan[(c + d*x)/2]))/(3*a*(a^2 - b^2)^2*d*(a + b*Sec[c + d*x])^(5/2))","C",0
633,1,97528,399,34.3080002,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(5/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \sin (c+d x)}{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 (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)}{3 b d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}-\frac{2 a \left(3 a^3 B-7 a b^2 B+4 A b^3\right) \sin (c+d x)}{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(3 a^3 B-7 a b^2 B+4 A b^3\right) \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 b^2 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{2 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)}{b^2 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0
634,1,184379,526,37.095071,"\int \frac{A+B \sec (c+d x)}{\cos ^{\frac{7}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx","Integrate[(A + B*Sec[c + d*x])/(Cos[c + d*x]^(7/2)*(a + b*Sec[c + d*x])^(5/2)),x]","\text{Result too large to show}","\frac{2 a (A b-a B) \sin (c+d x)}{3 b d \left(a^2-b^2\right) \cos ^{\frac{5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac{\left(-5 a^2 B+2 a A b+3 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)}{3 b^2 d \left(a^2-b^2\right) \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}+\frac{2 a \left(-5 a^3 B+2 a^2 A b+9 a b^2 B-6 A b^3\right) \sin (c+d x)}{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{\left(-15 a^4 B+6 a^3 A b+26 a^2 b^2 B-14 a A b^3-3 b^4 B\right) \sin (c+d x) \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(-15 a^4 B+6 a^3 A b+26 a^2 b^2 B-14 a A b^3-3 b^4 B\right) \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 b^3 d \left(a^2-b^2\right)^2 \sqrt{\frac{a \cos (c+d x)+b}{a+b}}}+\frac{(2 A b-5 a B) \sqrt{\frac{a \cos (c+d x)+b}{a+b}} \Pi \left(2;\frac{1}{2} (c+d x)|\frac{2 a}{a+b}\right)}{b^3 d \sqrt{\cos (c+d x)} \sqrt{a+b \sec (c+d x)}}",1,"Result too large to show","C",0